Rellich inequalities in bounded domains

TL;DR

This paper establishes precise conditions for weighted Rellich inequalities in bounded domains, considering various operators and boundary behaviors, including critical cases and remainders.

Contribution

It provides necessary and sufficient conditions for Rellich inequalities in bounded domains for a class of differential operators, extending previous results.

Findings

Conditions for validity of Rellich inequalities derived

Analysis of critical cases and boundary behaviors conducted

Inclusion of remainder terms in inequalities

Abstract

We find necessary and sufficient conditions for the validity of weighted Rellich inequalities in Lp for functions in bounded domains vanishing at the boundary. General operators like L = Delta+ c\|x|^2x nabla-b\|x|^2 are considered. Critical cases and remainder terms are also investigated.

Equations

$$Lu=\Delta u+c\frac{x}{|x{|}^2}\cdot \nabla u-\frac{b}{|x{|}^2}u,c,b\in \mathbb{R}$$

$$\Vert |x{|}^{\alpha }Lu{\Vert }_p\ge C\Vert |x{|}^{\alpha -2}u{\Vert }_p.$$

$${\left(\frac{N(N-4)}{4}\right)}^2{\int }_{{\mathbb{R}}^N}|x{|}^{-4}|u{|}^{2}dx\le {\int }_{{\mathbb{R}}^N}|\Delta u{|}^{2}dx$$

$${\left(\frac{N}{p}-2\right)}^p{\left(\frac{N}{p^{\prime }}\right)}^p{\int }_{{\mathbb{R}}^N}|x{|}^{-2p}|u{|}^{p}dx\le {\int }_{{\mathbb{R}}^N}|\Delta u{|}^{p}dx$$

$$C^p(N,p,\alpha ){\int }_{{\mathbb{R}}^N}|x{|}^{(\alpha -2)p}|u{|}^{p}dx\le {\int }_{{\mathbb{R}}^N}|x{|}^{\alpha p}|\Delta u{|}^{p}dx$$

$$\alpha$$

$$\alpha\neq N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}\pm\textrm{\emph{Re}\,}\sqrt{D+\lambda_{n}},\quad\forall\,n\in\mathbb{N}_{0}.$$

$$N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}-\sqrt{D}<\alpha<N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}+\sqrt{D}.$$

$${\int }_{\Omega }|\Delta u{|}^{2}dx$$

$$+\left(1+\frac{N(N-4)}{8}\right)\sum _{i=1}^{\infty }{\int }_{\Omega }\frac{|u{|}^2}{|x{|}^4}{X}_1^2{X}_2^2\cdots X_i^{2}dx,$$

$$\Big{\|}|x|^{\alpha}Lu\Big{\|}_{p}^{p}-C^{p}\Big{\|}|x|^{\alpha-2}u\Big{\|}_{p}^{p}\geq c\Big{\|}|x|^{\alpha-2}\left|\log|R^{-1}x|\right|^{-\frac{2}{p}}u\Big{\|}_{p}^{p}$$

$$\alpha=N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}\pm\textrm{\emph{Re}\,}\sqrt{D+\lambda_{n}}$$

$$\||x|^{\alpha}Lu\|_{p}\geq C\Big{\|}|x|^{\alpha-2}\left|\log|R^{-1}x|\right|^{-2}|u|\Big{\|}_{p}\quad{\rm when}\ D+\lambda_{n}\leq 0$$

$$\||x|^{\alpha}Lu\|_{p}\geq C\Big{\|}|x|^{\alpha-2}\left|\log|R^{-1}x|\right|^{-1}|u|\Big{\|}_{p}\quad{\rm when}\ D+\lambda_{n}>0$$

$$D_{p,\alpha }(\Omega ):$$

$$D_{p,\alpha ,0}(\Omega ):=\{u\in D_{p,\alpha }(\Omega ),u=0\mathrm{in}\mathrm{a}\mathrm{neighborhood}\mathrm{of}\partial \Omega \}$$

$$v(x)=|x{|}^{\alpha -2}u(x),$$

$$A=|x{|}^2\Delta +(c+4-2\alpha )x\cdot \nabla \mathrm{and}\mu =b-(2-\alpha )(N-\alpha +c).$$

$$\Vert \mu v-Av{\Vert }_p\ge C\Vert v{\Vert }_p,v\in D_{p,max}(\Omega )$$

$$D_{p,max}(\Omega ):$$

$$N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}-\sqrt{D}<\alpha<N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}+\sqrt{D}.$$

$$L=D_{rr}+\frac{N-1+c}{r}{D}_r-\frac{b-{\Delta }_0}{r^2},$$

$$Lu=\left[v_{rr}+\frac{N-1+c}{r}{v}_r-\frac{b+{\lambda }_n}{r^2}v\right]P.$$

$$v_{rr}+\frac{N-1+c}{r}{v}_r-\frac{b+{\lambda }_n}{r^2}v=0.$$

$$s_1^n:=\frac{N-2+c}{2}-\sqrt{D+{\lambda }_n},s_2^n:=\frac{N-2+c}{2}+\sqrt{D+{\lambda }_n}$$

$$D:=b+{\left(\frac{N-2+c}{2}\right)}^2.$$

$$\displaystyle\alpha_{n}^{\pm}:=N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}\pm\textrm{\emph{Re}\,}\sqrt{D+\lambda_{n}},\quad\,n\in\mathbb{N}_{0},$$

$$u(r\omega ):=r^{\gamma }P(\omega ),x=r\omega \in B_R$$

$${\alpha }_n^{-}-2+\gamma =-\frac{N}{p},1\le p\le \infty .$$

$$L{u}_{ϵ}(r\omega )=P(\omega )\left[r^{\gamma }{\phi }_{ϵ}^{\prime \prime }+(2\gamma +N-1+c)r^{\gamma -1}{\phi }_{ϵ}^{\prime }\right].$$

Full text

Mathematics subject classification (2010): 26D10, 35PXX, 47F05.

Keywords: Rellich inequalities, Spectral theory.

1. 1 Introduction

2. 2 Basic results and methods

3. 3 The operator $A=|x|^{2}\Delta+cx\cdot\nabla$

4. 3.1 The Laplace-Beltrami operator $\Delta_{0}$ on $L^{p}_{J}(S^{N-1})$

5. 3.2 The operator $\Gamma$ on $L^{p}(I,r^{N-1}\,dr)$

6. 3.3 The operator $A=|x|^{2}\Delta+cx\cdot\nabla$ on $L^{p}_{J}(\mathbb{R}^{N})$ and $L^{p}_{J}(B)$

7. 3.4 The operator $A=|x|^{2}\Delta+cx\cdot\nabla$ on $L^{p}(\Omega)$

8. 4 Rellich inequalities in $\mathbb{R}^{N}$ and in $B$

9. 5 Rellich inequalities in general domains

10. 5.1 The operator $L$ in $L^{2}(\Omega,d\mu)$

11. 5.2 Main result

12. 5.3 Rellich inequalities in exterior domains

13. 6 Critical cases in $L^{p}(\mathbb{R}^{N})$

14. 7 Best constants and remainder terms

15. 8 Appendix

16. 8.1 Approximation on Sobolev spaces on domains

17. 8.2 Some results on spectral theory

18. 8.3 Spectrum of a second order ordinary differential operator

1 Introduction

In this paper we consider the operator

[TABLE]

acting in the space $L^{p}(\Omega)$, for $1\leq p\leq\infty$, endowed with Dirichlet boundary conditions and we determine all $\alpha^{\prime}s$ (depending on $N,p,c,b$) for which the following weighted Rellich inequalities hold

[TABLE]

Note that, when $c=0$, $L$ becomes a Schrödinger operator with inverse square potential. When best constants can be computed, we prove that they are not attained by adding remainder terms. Finally, when Rellich inequalities above fail, we prove modified inequalities which include logarithmic terms.

The first results in this direction have been obtained for the Laplacian in unweighted $L^{p}$-spaces and when $\Omega=\mathbb{R}^{N}$. In 1956, Rellich proved the inequalities

[TABLE]

for $N\not=2$ and for every $u\in C_{c}^{\infty}(\mathbb{R}^{N}\setminus\{0\})$, see [33]. These inequalities have been then extended to $L^{p}$-norms: in 1996, Okazawa proved in [31] the validity of

[TABLE]

for $1<p<\frac{N}{2}$, showing also the optimality of the constants.

Weighted Rellich inequalities have also been studied in [11] and later by Mitidieri who proved for $N\geq 3$ and for $2-\frac{N}{p}<\alpha<\frac{N}{p^{\prime}}$

[TABLE]

with the optimal constants $C^{p}(N,p,\alpha)=\left(\frac{N}{p}-2+\alpha\right)^{p}\left(\frac{N}{p^{\prime}}-\alpha\right)^{p}$, see [26, Theorem 3.1].

In the recent paper [7], Caldiroli and Musina improved weighted Rellich inequalities for $p=2$ by giving necessary and sufficient conditions on $\alpha$ for the validity of (3) and finding also the optimal constants $C^{2}(N,2,\alpha)$. In particular they proved that (3) is verified for $p=2$ if and only if $\alpha\neq N/2+n$, $\alpha\neq-N/2+2-n$ for every $n\in\mathbb{N}_{0}$.

In [24] the results in [7] are extended to $1\leq p\leq\infty$, computing also best constants in some cases. It is shown that (3) holds if and only if $\alpha\neq N/p^{\prime}+n$, $\alpha\neq-N/p+2-n$ for every $n\in\mathbb{N}_{0}$. Moreover, Rellich inequalities are employed to find necessary and sufficient conditions for the validity of weighted Calderón-Zygmund estimates when $1<p<\infty$. These methods can be applied to general operators as in (1), thus providing a complete solution to problem (2) with $\Omega=\mathbb{R}^{N}$.

Let us now consider bounded open sets $\Omega$ containing the origin and spaces of functions vanishing at the boundary. In contrast with Hardy inequality, where many results in bounded domains improving those in the whole space are known, Rellich inequalities do not seem to have been studied intensively. We quote however [28] for $L=\Delta$, where the author discovers a range of parameters $\alpha$ where Rellich inequalities hold in the whole space but not in a bounded $\Omega$, due to the boundary conditions.

In this paper we find all parameters $\alpha$ for which (2) hold for a general $L$ as in (1), assuming that $\Omega$ has a smooth boundary and the condition $D:=b+(N-2+c)^{2}/4\geq 0$ on the coefficients of $L$, which guarantees the solvability of related elliptic problems. When $\Omega$ is a ball, however, this restriction on $D$ is not necessary. Our method is based on the spectral analysis of the auxiliary operator $A=|x|^{2}\Delta+cx\cdot\nabla$, as explained in Section 2. In particular, we show that, setting $\lambda_{n}=n(N-2+n)$, (1) holds if and only if

[TABLE]

When $\Omega$ is a ball centered at the origin, the above characterization holds also when $D<0$ (changing the square roots with their real parts) and in the extreme cases $p=1,\infty$. However, when $\Omega=\mathbb{R}^{N}$ the results in [24] say that Rellich inequalities hold if and only if

[TABLE]

The reason for the difference between (6) and (5) is explained in Section 2 in an elementary way in the case of the ball, by showing explicit counterexamples due to the boundary.

Rellich inequalities can be proved by using integration by parts and applying Hardy-type inequalities only when

[TABLE]

This proof allows also to compute the best constant C:=b+\Bigl{(}\frac{N}{p}-2+\alpha\Bigr{)}\Bigl{(}\frac{N}{p^{\prime}}-\alpha+c\Bigr{)}. For the other values of $\alpha$ appearing in (1), the best constant is unknown unless $p=2$, see [7], [24], or when $p$ is generic but special subspaces of $L^{p}$ are considered, see [24].

In the range (6), Rellich inequalities have essentially a one dimensional structure, since the (approximate) extremants are radial functions and best constants can therefore be computed. Outside of this range, however, the problem loses its rotational symmetry and the extremants, in special subspaces, involve spherical harmonics, see [24], again. This explains also why symmetrization arguments based on spherical rearrangements do not work and a spectral analysis appears. Similarly, best constants can be computed on subspaces of $L^{p}$ which allow a one-dimensional reduction and then on the whole $L^{2}$, by orthogonal expansions.

Remainder terms are known for the Laplacian in the unweighted case. We quote [36] where the authors obtained in particular

[TABLE]

for bounded domains $\Omega$ in $\mathbb{R}^{N}$, $N\geq 5$, $u\in C_{c}^{\infty}(\Omega\setminus\{0\})$, where $X_{k}=X_{k}\left(\frac{|x|}{R(\Omega)}\right)$, $R(\Omega)=\sup_{x\in\Omega}|x|$, are iterated radial logarithmic functions. The result has been extended to $L^{p}$ norms in [6] under the restriction $p<\frac{N}{2}$, according to (6) when $\alpha=b=c=0$. A different proof which uses symmetrization and covers also the case $p=\frac{N}{2}$ is given in [3]. Rellich inequalites with remainder terms in the whole space have been investigated in [34], where the remainder is given in terms of weighted $L^{q}$ norms of the Schwartz symmetrization of the functions.

We prove a similar result for our operator $L$ in weighted $L^{p}$ norms, considering only one remainder term. When $\alpha$ satisfies (6) we obtain with $C$ above

[TABLE]

for $u\in C^{2}_{c}(B_{R/2}\setminus\{0\})$. Some explanation on the class of functions here considered is necessary. Since (6) is satisfied, Rellich inequalities hold for both $\Omega$ bounded or $\Omega=\mathbb{R}^{N}$ but we choose to formulate the above result with reference to the whole space, that is for functions having compact support. A similar formulation for functions only vanishing at $\partial\Omega$, when $\Omega$ is a ball, is also possible but we prefer to point out only the role of the singularity at [math], since the weight $|x|^{\alpha}$ has no effect on the boundary.

In the critical cases, when Rellich inequalities do not hold, we prove that modified inequalities with logarithmic correction terms are still valid. Again we focus on the singularity at [math] and consider functions with compact support in $\mathbb{R}^{N}$. If

[TABLE]

for some $n\in\mathbb{N}_{0}$, $1<p\leq\infty$, then

[TABLE]

[TABLE]

for $u\in C^{2}_{c}(B_{R/2}\setminus\{0\})$. When $p=1$, the previous inequalities hold with $|\log|R^{-1}x||^{-2}$ and $|\log|R^{-1}x||^{-1}$ replaced by $|\log|R^{-1}x||^{-2-\varepsilon}$ and $|\log|R^{-1}x||^{-1-\varepsilon}$, respectively.

In this way we extend the results already proved in [2] for the Laplace operator under the more restrictive conditions $\alpha=0$, $p=\frac{N}{2}$, $N\geq 3$. We also refer to [15] where Rellich inequalities for the Laplacian have been proved with different remainder terms for $\alpha=0$, $p\leq\frac{N}{2}$.

The treatment of the critical case does not rely on rearrangements, as already explained, but a reduction to the one-dimensional case is still possible via a spectral analysis. In fact we show that Rellich inequalities are true, even in the critical cases, if we consider subspaces of $L^{p}(\mathbb{R}^{N})$ spanned by functions like $f(r)P(\omega)$, where $P$ is a spherical harmonic of degree different from $n$ and the problem is then reduced to find the right inequalities for (linear combinations of) functions $g(r)Q(\omega)$ where $Q$ is a spherical harmonic of degree $n$, hence to a finite number of one-dimensional problems.

Let us explain why semigroups of linear operators appear often in the paper. When $p=2$, Rellich inequalities can be reduced to a countable set of one-dimensional inequalities, by an orthogonal expansion in spherical harmonics, see for example [24]. Moreover, it turns out that is more convenient to work with the operator $A=|x|^{2}L$ instead of $L$, so that the radial and the angular parts decouple. When $p\neq 2$ the one-dimensional analysis can be still performed but one needs a substitute for orthogonal expansions. This role is played by the semigroup $e^{tA}$ which allows to compute the spectrum of $A$, by tensor product arguments, since the radial and the angular parts commute. Rellich inequalities are equivalent to spectral inequalities for $A$ and, moreover, the description of the domain of $A$ allows us to identify precise classes where Rellich inequalities hold.

Let us briefly describe the content of the sections. In Section 2 we present the basic ideas and some explicit counterexamples which serve as a guide for the rest of the paper. We reduce Rellich inequalities to a spectral problem for an operator with singular coefficients $A=|x|^{2}\Delta+cx\cdot\nabla$ which is therefore analysed in detail in Section 3, which is the core of the paper. Rellich inequalities for the ball and for the whole space are easily deduced in Section 4 from the analysis of Section 3. The case of general domains, without any rotational symmetry, is studied in Section 5: here we need $1<p<\infty$ and $D\geq 0$, a condition which is known to be equivalent to the existence of positive solutions for elliptic and parabolic problems related to $L$. When $L=\Delta-b|x|^{-2}$, this condition reduces to the classical one $b+(N-2)^{2}/4\geq 0$. The main tool to pass from the ball to a general $\Omega$ is a pointwise estimate of the Green function of $-L$ which follows from precise bounds of the heat kernel. Rellich inequalities in exterior domains not containing the origin are easily treated via the Kelvin transform. In Section 6 we show that, when Rellich inequalities fail, modified inequalities which include logarithmic terms are still valid. The situation is similar to Hardy inequality, when the classical one fails. In Section 7, we analyse the remainder term in Rellich inequalities when (6) is satisfied.

Notation. We denote by $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ the natural numbers including 0. If $\Omega$ is an open subset of $\mathbb{R}^{N}$, $C_{b}(\Omega)$ is the Banach space of all continuous and bounded functions in $\Omega$, endowed with the sup-norm, $C_{0}(\overline{\Omega})$ its subspace consisting of functions vanishing at the boundary and $C_{0}^{0}(\overline{\Omega})$ its subspace consisting of functions vanishing at the origin and at the boundary, when $0\in\Omega$. $C_{c}^{\infty}(\Omega)$ denotes the space of infinitely continuously differentiable functions with compact support in $\Omega$. The unit sphere $\{\|x\|=1\}$ in $\mathbb{R}^{N}$ is denoted by $S^{N-1}$; $\Delta_{0}$ is its Laplace-Beltrami operator. We adopt standard notation for $L^{p}$ and Sobolev spaces when $1\leq p<\infty$ but we use $L^{\infty}(\Omega)$ for $C_{b}(\Omega)$ to unify the notation. $B_{r}$ is the ball of center [math] and radius $r$, $B_{r}^{c}=\mathbb{R}^{N}\setminus B$. We write $B$ for $B_{1}$. For $V\subseteq\mathbb{R}^{N}$, we denote by $\overset{\mathrm{o}}{V}$ the interior part of $V$. When $L$ is a closed operator $\sigma(L)$, $P\sigma(L)$, $A\sigma(L)$, $R\sigma(L)$, denote the spectrum, the point-spectrum, the approximate point spectrum and the residual spectrum, respectively. Definitions and the relevant properties are listed in the Appendix.

2 Basic results and methods

Let $L$ be as in (1) and let $\Omega$ be an open, bounded, connected subset of $\mathbb{R}^{N}$ containing the origin and with a smooth boundary, or $\Omega=\mathbb{R}^{N}$. For $1\leq p\leq\infty$, $\alpha\in\mathbb{R}$ we define

[TABLE]

$Lu$ is understood as a distribution in $\Omega\setminus\{0\}$. Since the coefficients of $L$ are $C^{\infty}$ away from the origin, by local elliptic regularity it follows that, if $u\in D_{p,\alpha}(\Omega)$, then $u\in W^{2,p}_{loc}(\mathbb{R}^{N}\setminus\{0\})$ when $\Omega=\mathbb{R}^{N}$ and $u\in W^{2,p}(\Omega\setminus B_{\varepsilon})$ for every $\varepsilon>0$, when $\Omega$ is bounded. This clearly holds for $1<p<\infty$; when $p=\infty$, the same is true for any $q<\infty$.

Note that, when $\Omega$ is bounded, also the class

[TABLE]

could be considered. However, since every function $u\in D_{p,\alpha,0}(\Omega)$, extended by [math] to $\mathbb{R}^{N}$, belongs to $D_{p,\alpha}(\mathbb{R}^{N})$, the problem is then reduced to the case of the whole space. A scaling argument, moreover, shows that Rellich inequalities (2) hold in $D_{p,\alpha,0}(\Omega)$ if and only if they hold in $D_{p,\alpha}(\mathbb{R}^{N})$.

Defining

[TABLE]

it is straightforward to compute that $|x|^{\alpha}Lu=Av-\mu v$, where

[TABLE]

Then Rellich inequalities (2) are equivalent to the spectral estimates

[TABLE]

where

[TABLE]

and $Au$ is understood as a distribution as above. Moreover, the constants $C$ in (2) and (8) are the same.

Inequalities (8) hold precisely when $\mu$ does not belong to the approximate point spectrum of $A$. This explains why a large part of this paper is devoted to the study of the operator $A$ and of the fine structure of its spectrum.

In the next proposition we state the above reduction, for further reference, and prove a density result using the same method. We refer to Section 8.2 for basic definitions and results from spectral theory.

Proposition 2.1

Let $L$ be as in (1) and let $\Omega$ be an open, bounded, connected subset of $\mathbb{R}^{N}$ containing the origin and with a $C^{2,\beta}$ boundary, or $\Omega=\mathbb{R}^{N}$. Then

• (i)

Rellich inequalities (2) hold if and only if $\mu=b-(2-\alpha)(N-\alpha+c)$ does not belong to the approximate point spectrum of $(A,D_{p,max}(\Omega))$.

• (ii)

Rellich inequalities (2) hold for functions in $D_{p,\alpha}(\Omega)$ if and only if they hold for $C^{2}$ - functions vanishing in a neighbourhood of the origin and on $\partial\Omega$, when $\Omega$ is bounded, or also in a neighbourhood of infinity, when $\Omega=\mathbb{R}^{N}$.

Proof. The discussion above shows that Rellich inequalities hold if and only the spectral inequalities (8) are valid in $D_{p,max}(\Omega)$, hence when $\mu$ does not belong to the approximate point spectrum of $A$, by Proposition 8.7. This proves (i). To prove (ii) it is sufficient to note that the transformation $v(x)=|x|^{\alpha-2}u(x)$ preserves the class of functions defined in (ii) and that, by Lemma 3.20 and Proposition 3.28, these functions constitute a core of $(A,D_{p,max}(\Omega))$.

The interplay between the operators $A$ and $L$ allows to give simple proofs of Rellich inequalities in special cases where best constants can be computed.

Proposition 2.2

Let $\Omega$ be an open, bounded, connected subset of $\mathbb{R}^{N}$ with a $C^{1}$ boundary, or $\Omega=\mathbb{R}^{N}$. Assume that $1\leq p\leq\infty$, that $D:=b+\left(\frac{N-2+c}{2}\right)^{2}>0$ and that

[TABLE]

Then Rellich inequalities (2) hold in $D_{p,\alpha}(\Omega)$ with C:=b+\Bigl{(}\frac{N}{p}-2+\alpha\Bigr{)}\Bigl{(}\frac{N}{p^{\prime}}-\alpha+c\Bigr{)}. The constant $C$ is optimal when $\Omega$ contains the origin.

Proof. We have to show that (8) holds, with the constant $C$ above, for $A$ and $\mu$ defined in (7). This is proved in Theorem 3.24, using only integration by parts and Hardy inequality (change $c$ with $c+4-2\alpha$ and $\lambda-\omega_{p}$ with $\mu$, therein). We note that $C>0$ is equivalent to (9).

To prove the optimality of $C$, when $0\in\Omega$, we observe that Rellich inequalities are invariant under dilations. If $C_{\Omega}$ is the best constant in $\Omega$, then $C_{r\Omega}=C_{\Omega}$ for any $r>0$. Letting $r\to\infty$ we see that $C_{\mathbb{R}^{N}}\leq C_{\Omega}$. However, C_{\mathbb{R}^{N}}=b+\Bigl{(}\frac{N}{p}-2+\alpha\Bigr{)}\Bigl{(}\frac{N}{p^{\prime}}-\alpha+c\Bigr{)}, by [24, Theorem 3.1].

Note that when $L=\Delta$, then $D=(N-2)^{2}/4$ and (9) reduces to $2-N/p<\alpha<N/p^{\prime}$ and C=\Bigl{(}\frac{N}{p}-2+\alpha\Bigr{)}\Bigl{(}\frac{N}{p^{\prime}}-\alpha\Bigr{)}. If $\Omega$ does not contain the origin the constant $C$ above is not optimal, in general, see again [24, Section 6] for the case of the half space.

Next, we show explicit counterexamples to Rellich inequalities already appeared in [28] when $L=\Delta$. We distinguish between free counterexamples depending on the singularity at zero, which appear in any set $\Omega$ containing the origin and counterexamples where the boundary $\partial\Omega$ is involved, appearing only when $\Omega$ is bounded in addition to the preceding ones. We confine here only to the case of the unit ball $B$; the general case will be treated in Section 5.

We employ spherical coordinates on $\mathbb{R}^{N}\setminus\{0\}$ and write $x=r\omega$, where $r:=|x|$, $\omega:=x/|x|\in{\mathbb{S}}^{N-1}$. Then

[TABLE]

where $D_{rr}$, $D_{r}$ denote radial derivatives and $\Delta_{0}$ is the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{N-1}$. Let $P$ be a spherical harmonics of order $n\in N_{0}$, with $\Delta_{0}P=-\lambda_{n}P$, $\lambda_{n}=n(N+n-2)$. If $u(r\omega)=v(r)P(\omega)$ then

[TABLE]

The equation $Lu=0$ has solutions $|x|^{-s_{1}^{n}}P$, $|x|^{-s_{2}^{n}}P$ where the function $r^{-s_{1}^{n}}$, $r^{-s_{2}^{n}}$ solve

[TABLE]

$s_{1}^{n},s_{2}^{n}$ are the roots of the indicial equation $f(s)=-s^{2}+(N-2+c)s+b+\lambda_{n}=0$ given by

[TABLE]

where

[TABLE]

The following Examples shows that, due to the singularity of $L$ at [math], Rellich inequalities always fail when $\alpha$ equals one of the values

[TABLE]

Example 2.3

Let $1\leq p\leq\infty$ and let $\Omega\subseteq\mathbb{R}^{N}$ be an open subset of $\mathbb{R}^{N}$ such that $0\in\Omega$. If $\alpha=\alpha_{n}^{\pm}$, then Rellich inequalities (2) do not hold in $D_{p,\alpha}(\Omega)$.

Proof. Suppose, for example, that $\alpha=\alpha_{n}^{-}$. Let $s_{1}^{n}$ be defined in (10) and $\gamma=-\textrm{\emph{Re}\,}s_{1}^{n}$. We fix $R>0$ such that $B_{R}\subseteq\Omega$ and take $P$ a spherical harmonics of order $n$. The function

[TABLE]

satisfies $Lu=0$ but $|x|^{\alpha_{n}^{-}-2}u\notin L^{p}\left(B_{r}\right)$ since

[TABLE]

Let $\varphi\in C^{\infty}(\mathbb{R})$ such that $\mbox{supp}\,\varphi\subseteq[\frac{1}{4},\frac{1}{2}]$ and $\varphi_{\epsilon}(r):=\varphi(r^{\epsilon})$. By construction $u_{\epsilon}:=u\varphi_{\epsilon}$ has support in $[\left(\frac{1}{4}\right)^{\frac{1}{\epsilon}},\left(\frac{1}{2}\right)^{\frac{1}{\epsilon}}]$, lies in $D_{p,\alpha}\left(\Omega\right)$ and satisfies

[TABLE]

If $1\leq p<\infty$ and $\bar{r}>0$ such that $\mbox{supp}\,\varphi_{\epsilon}\subseteq B_{\bar{r}}$ we get

[TABLE]

where $C=\int_{\mathbb{S}^{N-1}}|P(\omega)|^{p}\,d\omega$. On the other hand

[TABLE]

It follows, from the previous equalities, that

[TABLE]

which tends to [math] as $\epsilon\to 0$, hence Rellich inequalities do not hold in $D_{p,\alpha}(\Omega)$ for $1\leq p<\infty$.

If $p=\infty$, then $\alpha_{n}^{-}-2+\gamma=0$ and an analogous computation yields

[TABLE]

This implies

[TABLE]

which tends to [math] as $\epsilon\to 0$. The proof for $\alpha=\alpha_{n}^{+}$ is similar, choosing $\gamma=-\textrm{\emph{Re}\,}s_{2}^{n}$.

Next we consider the case where $\Omega=B$ and show that, due to the Dirichlet boundary condition at $\partial B$, new counterexamples appear, in addition to the previous ones. The same result is proved in Section 5 for general bounded domains.

Proposition 2.4

If $1\leq p\leq\infty$ and \alpha>N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}+\textrm{\emph{Re}\,}\sqrt{D}, then the Rellich inequalities (2) cannot hold in $D_{p,\alpha}(B)$.

Proof. Let \alpha>N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}+\textrm{\emph{Re}\,}\sqrt{D} and let $s_{1,2}$ be defined in (10) with $n=0$. The function

[TABLE]

satisfies $Lu=0$ and $|x|^{\alpha-2}u\in L^{p}(B)$, since $\alpha-2+{\rm Re\ }s_{1,2}>-N/p$. Furthermore $u=0$ on $\partial B$, hence $u\in D_{p,\alpha}(B)$ and, since $Lu=0$, Rellich inequalities fail.

3 The operator $A=|x|^{2}\Delta+cx\cdot\nabla$

Let $c\in\mathbb{R}$ and

[TABLE]

This section is devoted to the analysis of $A$ acting on $L^{p}(\Omega)$ for $1\leq p\leq\infty$, where $\Omega=\mathbb{R}^{N}$ or a bounded domain, endowed with Dirichlet boundary conditions in this last case. The operator is degenerate both at [math] and at $\infty$. Employing spherical coordinates on $\mathbb{R}^{N}\setminus\{0\}$ we write $x=r\omega$, where $r:=|x|$, $\omega:=x/|x|\in{\mathbb{S}}^{N-1}$ and

[TABLE]

where $D_{rr}$, $D_{r}$ denote radial derivatives and $\Delta_{0}$ is the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{N-1}$. Thus we obtain

[TABLE]

Defining

[TABLE]

the operators $\Gamma$ and $\Delta_{0}$ act on independent variables and therefore, when $\Omega$ is spherically symmetric, generation and spectral properties of $A$ can be proved through tensor products methods.

We start by analysing $\Gamma$ and $\Delta_{0}$ separately and then we deduce properties of $A$ on $L^{p}(\Omega)$ when $\Omega=\mathbb{R}^{N}$ and $\Omega=B$. This method has the advantage to apply also on more general subspaces defined as tensor products of radial functions and spherical harmonics. Finally, we study $A$ in a general open set $\Omega$.

3.1 The Laplace-Beltrami operator $\Delta_{0}$ on $L^{p}_{J}(S^{N-1})$

We summarize in the next proposition some well known results about $\Delta_{0}$ referring, for example, to [18, 27, 35] for further details. We recall that a spherical harmonic $P^{n}$ of order $n$ is the restriction to $\mathbb{S}^{N-1}$ of a homogeneous harmonic polynomial of degree $n$. We write $L^{\infty}(S^{N-1})$ for $C(S^{N-1})$.

Proposition 3.1

The Laplace-Beltrami operator $\Delta_{0}$ generates an analytic semigroup $(T_{S^{N-1}}(t))_{t\geq 0}$ in $L^{p}(S^{N-1})$ (with respect to the surface measure $d\sigma$) for every $1\leq p\leq\infty$ . If $1<p<\infty$, its domain $D_{p}(\Delta_{0})$ coincides with $W^{2,p}(S^{N-1},d\sigma)$ . The spectrum of the operator $(\Delta_{0},D_{p}(\Delta_{0}))$ is independent of $1\leq p\leq\infty$ and consists of eigenvalues $-\lambda_{n}:=-n(n+N-2)$, $n\in\mathbb{N}_{0}$. The eigenspace corresponding to $-\lambda_{n}$ consist of all spherical harmonics of degree $n$ and has dimension $a_{n}$ where $a_{0}=1$, $a_{1}=N$ and for $n\geq 2$

[TABLE]

The linear span of spherical harmonics coincides with the set of all polynomials and it is dense in $C(\mathbb{S}^{N-1})$, hence in $L^{p}(\mathbb{S}^{N-1})$ for every $1\leq p<\infty$.

Proof. The generation and spectral properties of the Laplace Beltrami operator $\Delta_{0}$ are classic result about Heat operators on compact manifolds. If $1<p<\infty$, $D_{p}(\Delta_{0})=W^{2,p}(S^{N-1},d\sigma)$ by elliptic regularity. The analyticity of the semigroup as well as the invariance of the spectrum follows, for example, from the Gaussian estimates of the heat kernel of $\Delta_{0}$ (see e.g. [10, Theorem 5.2.1, Theorem 5.5.1]) using [32, Corollary 7.5, Theorem 7.10]. The main properties of spherical harmonics can be found in [27, Chapter II] and [35, Chapter IV.2].

Accordingly to the latter proposition let

[TABLE]

be the spectrum of $(-\Delta_{0},D_{p}(\Delta_{0}))$ and let us write $\{P_{j},\}_{j\in\mathbb{N}_{0}}$ and $\{\lambda(P_{j})\}_{j\in\mathbb{N}_{0}}$ to denote the sequences of the ($L^{2}$-orthonormal) eigenfunctions and their respectively eigenvalues repeated according to the relative multiplicity. With this notation $P_{j}$ is a spherical harmonics whose eigenvalue is $\lambda(P_{j})=n(n+N-2)$ and $n=\mbox{deg}(P_{j})$.

We extend the analysis of $\Delta_{0}$ on more general subspaces defined by spherical harmonics.

Definition 3.2

For a given $J\subseteq\mathbb{N}_{0}$ we define

[TABLE]

where the closure is taken in $L^{p}(S^{N-1})$, $1\leq p\leq\infty$.

It is clear that $L^{p}_{J}(S^{N-1})$ is $\Delta_{0}$-invariant and that the domain of ${\Delta_{0}}_{|L^{p}_{J}(S^{N-1})}$ is given by $D_{p}(\Delta_{0})\cap L^{p}_{J}(S^{N-1})$. The following lemma is elementary and proved in [24, Lemma 5.8].

Lemma 3.3

Let $1\leq p\leq\infty$ and $J\subseteq\mathbb{N}_{0}$. Then ${\Delta_{0}}_{|L^{p}_{J}(S^{N-1})}$ generates in $L^{p}_{J}(S^{N-1})$ the analytic semigroup

[TABLE]

Moreover ${span}\{P_{j}:j\in J\}$ is a core for $\Delta_{0}$ in $L^{p}_{J}(S^{N-1})$ and

[TABLE]

where $\lambda(P_{j})$ is the eigenvalue whose eigenfunction is $P_{j}$.

Note that, since each eigenvalue can have more than one eigenfunction, different set of indexes leads to different spaces but not necessarily to different spectra.

The asymptotic behaviour of $\left(T_{S^{N-1}}(t)_{|L^{p}_{J}(S^{N-1})}\right)_{t\geq 0}$ in $L^{p}_{J}(S^{N-1})$ is determined by the first eigenvalue. However we need a better estimate near $t=0$ which relies on a Poincaré-type inequality.

Lemma 3.4

([20, Lemma 2.7]) Let $1<p<\infty$ and $J\subseteq\mathbb{N}_{0}$ such that $n:=\mbox{min}J\geq 1$. Let $\widetilde{C}_{p,n}$ be the best constant for which

[TABLE]

Then $\widetilde{C}_{p,n}$ are finite, decreasing and satisfy $\widetilde{C}_{p,n}\to 0$ as $n\to\infty$.

In the next Proposition we assume that the numbers $\lambda(P_{j})$ are listed in the increasing order.

Proposition 3.5

Let $J\subseteq\mathbb{N}_{0}$ and let $n$ be the smallest integer in $J$. There exists $M$ (depending on $n$ but not on $p$) such that for every $1\leq p\leq\infty$

[TABLE]

Furthermore $M=1$ when $n=0$. If $1<p<\infty$ then

[TABLE]

where $\tilde{C}_{p,n}$ is the best constant of Lemma 3.4.

Proof. The first statement is proved in [24, Lemma 5.9]. To prove the second it is enough to show the dissipativity of $\Delta_{0}+\frac{p-1}{\tilde{C}_{p,n}}$ on $L^{p}_{J}(S^{N-1})$ or equivalently that, for every $u\in C^{\infty}(S^{N-1})\cap L^{p}_{J}(S^{N-1})$,

[TABLE]

Consider first the case $2\leq p<\infty$. Setting $u^{\star}=u|u|^{p-2}$ we multiply $\Delta_{0}u$ by $u^{\star}$ and integrate over $S^{N-1}$. Integrating by parts and using Lemma 3.4 we get

[TABLE]

For $1<p<2$ it is sufficient to replace $u^{\star}$ by $u(u^{2}+\delta)^{\frac{p}{2}-1}$, $\delta>0$; and then let $\delta$ to [math] to obtain the same inequality.

3.2 The operator $\Gamma$ on $L^{p}(I,r^{N-1}\,dr)$

In this section we summarize the main results about generation and spectral properties for the operator

[TABLE]

acting, for $1\leq p<\infty$, on $L^{p}(I,r^{N-1}dr)$, where $I=]0,\infty[$ or $I=]0,1[$. When $p=\infty$, $L^{\infty}(I,r^{N-1}\,dr)$ stands for the space $C_{0}^{0}(I)$ of all the continuous functions defined on $I$ vanishing at both endpoints.

For $1\leq p\leq\infty$ we define $\Gamma_{p}$ as the operator $\Gamma$ endowed with the domain $D(\Gamma_{p})$ defined, when $I=]0,\infty[$, as

[TABLE]

and for $I=]0,1[$

[TABLE]

In the next Theorem we show that $\Gamma_{p}$ always generates an analytic semigroup in $L^{p}(I,r^{N-1}\,dr)$; the spectral analysis is more subtle since the spectrum and the approximate point spectrum of $\Gamma_{p}$ drastically change accordingly to $I$ being bounded or not and to the sign of $N\left(1-\frac{2}{p}\right)-2+c$.

Let us introduce some notation: for $1\leq p\leq\infty$ (limiting values are taken for $p=\infty$), let us set

[TABLE]

and

[TABLE]

where

[TABLE]

${\cal P}_{p}$ is a parabola having vertex $-\omega_{p}$, symmetric with respect to the $x$ axis whereas ${\cal Q}_{p}$ is the region enclosed inside ${\cal P}_{p}$. Obviously ${\cal P}_{p}$ coincides with the boundary of ${\cal Q}_{p}$ and, when $N\left(1-\frac{2}{p}\right)-2+c=0$, both reduce to the half line $(-\infty,-\omega_{p}]$.

Theorem 3.6

Let $1\leq p\leq\infty$. Then the operator $\Gamma_{p}$ generates a strongly continuous analytic semigroup $(S(t))_{t\geq 0}$ in $L^{p}(I,r^{N-1}dr)$ which satisfies the estimate

[TABLE]

If $I=]0,\infty[$ we have

[TABLE]

If $I=]0,1[$, then

[TABLE]

Moreover

• (i)

if $N\left(1-\frac{2}{p}\right)-2+c<0$, then  $\sigma(\Gamma_{p})=A\sigma(\Gamma_{p})={\cal Q}_{p}$,  $P\sigma(\Gamma_{p})\supset\overset{\mathrm{o}}{\cal Q}_{p}$;

• (ii)

if $N\left(1-\frac{2}{p}\right)-2+c=0$, then  $\sigma(\Gamma_{p})=A\sigma(\Gamma_{p})=(-\infty,-\omega_{p}]$;

• (iii)

if $N\left(1-\frac{2}{p}\right)-2+c>0$, then  $A\sigma(\Gamma_{p})={\cal P}_{p}$,  $\overset{\mathrm{o}}{\cal Q}_{p}=R\sigma(\Gamma_{p})\setminus A\sigma(\Gamma_{p})$.

Proof. Assume first that $I=]0,1[$. Let $J=]-\infty,0[$ and consider the isometry $S$ defined, for $1\leq p<\infty$, by

[TABLE]

and, for $p=\infty$, by

[TABLE]

It follows that

[TABLE]

By classical results, $S^{-1}\Gamma S$, endowed with domain $D_{p}(S^{-1}\Gamma S)$

[TABLE]

generates a strongly continuous analytic semigroup in $L^{p}\left(J\right)$ whose norm is bounded by $e^{-\omega_{p}t}$.

It is elementary to check that

[TABLE]

It follows that $\Gamma_{p}$ generates a strongly continuous and analytic semigroup $(S(t))_{t\geq 0}$ in the space $L^{p}(]0,1[,{r}^{N-1}d{r})$ which satisfies $\|S(t)\|_{p}\leq e^{-\omega_{p}t}$. The case $I=]0,\infty[$ is similar and proved in [24, Proposition 5.1] by considering $S$ with $J=\mathbb{R}$.

Concerning the second part of the statement we observe that the spectra of $\Gamma_{p}$ and $S^{-1}\Gamma_{p}S$ coincide.

When $I=]0,\infty[$, the operator $S^{-1}\Gamma_{p}S$ is uniformly elliptic in $L^{p}(\mathbb{R},ds)$, hence its spectrum is independent of $p$ and coincides with the spectrum in $L^{2}(\mathbb{R},ds)$ which is ${\cal P}_{p}$, using the Fourier transform. Furthermore, since ${\cal P}_{p}$ coincides with its boundary, it follows, from Proposition 8.8, that $\sigma(\Gamma_{p})=A\sigma(\Gamma_{p})={\cal P}_{p}$.

When $I=]0,1[$ we use Lemma 8.11 to see that the spectrum of $S^{-1}\Gamma_{p}S$, hence of $\Gamma_{p}$, coincides with the region ${\cal Q}_{p}$. Moreover, for the same reason, the approximate point spectrum $A\sigma(\Gamma_{p})$ coincides with ${\cal Q}_{p}$ if $N\left(1-\frac{2}{p}\right)-2+c<0$ (and in this case $P\sigma(\Gamma_{p})\supset\overset{\mathrm{o}}{\cal Q}_{p}$), with the boundary ${\cal P}_{p}$ if $N\left(1-\frac{2}{p}\right)-2+c>0$ (and in this case $\overset{\mathrm{o}}{\cal Q}_{p}=R\sigma(\Gamma_{p})\setminus A\sigma(\Gamma_{p})$) and with the half line $(-\infty,-\omega_{p}]$ when $N\left(1-\frac{2}{p}\right)-2+c=0$.

Remark 3.7

Since the domain $D_{p}(S^{-1}\Gamma S)$ coincides with its maximal one

[TABLE]

as it easily follows from the classical interpolative inequalities $\|u^{\prime}\|_{p}\leq\epsilon\|u^{\prime\prime}|_{p}+\frac{C}{\epsilon}\|u\|_{p},$ it follows that

[TABLE]

3.3 The operator $A=|x|^{2}\Delta+cx\cdot\nabla$ on $L^{p}_{J}(\mathbb{R}^{N})$ and $L^{p}_{J}(B)$

In this section we use tensor arguments to combine the previous results on $\Gamma$ and $\Delta_{0}$ and deduce generation and spectral properties of

[TABLE]

on $L^{p}(\Omega)$ when $\Omega=\mathbb{R}^{N}$ and $\Omega=B$. We extend the analysis also on more general subspaces defined by tensor products of radial functions and spherical harmonics.

If $X,Y$ are function spaces over $G_{1},G_{2}$ we denote by $X\otimes Y$ the algebraic tensor product of $X,Y$, that is the set of all functions $u(x,y)=\sum_{i=1}^{n}f_{i}(x)g_{i}(y)$ where $f_{i}\in X,g_{i}\in Y$ and $x\in G_{1},y\in G_{2}$. If $T,S$ are linear operators on $X,Y$ we denote by $T\otimes S$ the operator on $X\otimes Y$ defined by

[TABLE]

Let us fix a complete orthonormal system of spherical harmonics $\{P_{j},\}_{j\in\mathbb{N}_{0}}$ $L^{2}(S^{N-1})$ and let $\{\lambda(P_{j})\}_{j\in\mathbb{N}_{0}}$ be the sequence of the corresponding eigenvalues repeated according to their multiplicity. With this notation $-\Delta_{0}(P_{j})=\lambda(P_{j})P_{j}$ and $\lambda(P_{j})=n(n+N-2)$, where $n=\mbox{deg}(P_{j})$.

Unless otherwise specified $\Omega$ denotes $\mathbb{R}^{N}$ or $B$, $I$ stands for $]0,\infty[$, $]0,1[$, respectively. As usual we write $L^{\infty}(\Omega)$ for $C_{0}^{0}(\Omega)$.

Definition 3.8

Let $1\leq p\leq\infty$ and let $J\subseteq\mathbb{N}_{0}$. We define

[TABLE]

where the closure is taken in $L^{p}(\Omega)$. Fixing $n\in\mathbb{N}_{0}$ we write $L^{p}_{\geq n}(\Omega),L^{p}_{n}(\Omega),L^{p}_{<n}(\Omega)$ when $J$ identifies all spherical harmonics of order $\geq n$, $n$ and $<n$ respectively. The spaces $L^{p}_{>n}(\Omega),L^{p}_{\leq n}(\Omega),L^{p}_{\neq n}(\Omega)$ are defined similarly.

Note that $L^{p}_{J}(\Omega)=L^{p}(\Omega)$ if $J=\mathbb{N}_{0}$.

The next lemma clarifies the structure of the spaces $L^{p}_{J}(\Omega)$.

Lemma 3.9

Assume that the $L^{2}$ orthogonal projection $P:L^{2}(S^{N-1})\to L^{2}_{J}(S^{N-1})$ extends to a bounded projection $P$ in $L^{p}(S^{N-1})$. Then

[TABLE]

and

[TABLE]

When $J$ is finite

[TABLE]

and the projection $I\otimes P:L^{p}(\Omega)\to L^{p}_{J}(\Omega)$ is given by

[TABLE]

where

[TABLE]

Proof. When $\Omega=\mathbb{R}^{N}$ we refer to [24, Lemma 5.11]. The proof for $\Omega=B$ is identical.

Remark 3.10

(i) The equality

[TABLE]

*holds without assuming the boundedness of the projection $P$ (see [25, Proposition 2.8]).

(ii) $L^{p}_{0}(\Omega)$ consists of radial functions and $L^{p}(\Omega)=L^{p}_{\leq n}(\Omega)\oplus L^{p}_{>n}(\Omega)$.*

The following result follows from well-known and elementary facts about Tensor Product Semigroups, see [29, AI, Section 3.7]. A proof is provided in [24, Proposition 5.14] when $\Omega=\mathbb{R}^{N}$, the case of the ball is similar.

Proposition 3.11

For $1\leq p\leq\infty$, let $D(\Gamma_{p})$ and $D({\Delta_{0}}_{|L^{p}_{J}(S^{N-1})})$ be the domains of $\Gamma_{p}$ and ${\Delta_{0}}_{|L^{p}_{J}(S^{N-1})}$ introduced in the previous subsection. Then the closure of the operator

[TABLE]

generates a strongly continuous analytic semigroup $(T_{p,J}(t))_{t\geq 0}$ in $L^{p}_{J}(\Omega)$. Let $n$ be the smallest integer in $J$. Then there exists $M$ (depending on $n$ but not on $p$) such that for every $1\leq p\leq\infty$

[TABLE]

where $\omega_{p}$ is defined in (19) and $M$ is the constant in (13) which satisfies $M=1$ when $n=0$. Moreover, if $1<p<\infty$, then

[TABLE]

where $\tilde{C}_{p,n}$ is the best constant of Lemma 3.4.

Definition 3.12

We denote by $A_{p,J}$ the closure of $(A,D(\Gamma_{p})\otimes D({\Delta_{0}}_{|L^{p}_{J}(S^{N-1})}))$ in $L^{p}_{J}(B)$. When $J=\mathbb{N}_{0}$ we write $A_{p}$ for $A_{p,J}$ and $T_{p}(t)$ for $T_{p,J}(t)$.

The proof of the following corollary is immediate.

Corollary 3.13

$T_{p,J}(t)$* is the restriction of $T_{p}(t)$ to $L^{p}_{J}(B)$ and its generator $A_{p,J}$ is the part of $A_{p}$ in $L^{p}_{J}(B)$.*

As in [24, Proposition 5.16], we prove that the smooth functions are a core for $A_{p,J}$.

Proposition 3.14

Let $1\leq p\leq\infty$. The set

[TABLE]

is a core for $A_{p,J}$ when $\Omega=B$. When $\Omega=\mathbb{R}^{N}$, $C^{\infty}_{c}\left(\mathbb{R}^{N}\setminus\{0\}\right)$ is a core for $A_{p,J}$.

Proof. Let us suppose that $\Omega=B$. Recalling the proof of Theorem 3.6, we observe that, since by Proposition 8.1 the set

[TABLE]

is dense in $D_{p}(S^{-1}\Gamma_{p}S)$, then

[TABLE]

is dense in $D(\Gamma_{p})$. Moreover $\mbox{span}\{P_{j}:j\in J\}$ is dense in $D({\Delta_{0}}_{|L^{p}_{J}(S^{N-1})})$. Since by construction $D(\Gamma_{p})\otimes D({\Delta_{0}}_{|L^{p}_{J}(S^{N-1})})$ is a core for $A_{p,J}$, it follows that

[TABLE]

is dense in $D(A_{p,J})$. Observing that

[TABLE]

we get the thesis. The proof for $\Omega=\mathbb{R}^{N}$ is similar.

In order to prove the main result of this section, namely

[TABLE]

we need two preliminary lemmas. The first provides some regularity properties of the projection defined in (23) and is proved in [20, Lemma 2.15] when $\Omega=\mathbb{R}^{N}$.

Lemma 3.15

Let $J\subseteq\mathbb{N}_{0}$ and let $j_{0}\in J$. Let us consider the operator $T_{j_{0}}:L^{p}_{J}(\Omega)\to L^{p}(I,\ r^{N-1}dr)$ defined by

[TABLE]

and the projection

[TABLE]

given, for $u\in L^{p}_{J}(\Omega)$, $r\in I$, $\omega\in S^{N-1}$, by

[TABLE]

Then $T_{j_{0}}$, $I\otimes P_{j_{0}}$ are well defined and bounded operator. Furthermore $T_{j_{0}}$ maps $D(A_{p,J})$ onto $D(\Gamma_{p})$ and one has

[TABLE]

The next lemma relates the spectra of $\Gamma_{p}$ and $A_{p,J}$.

Lemma 3.16

Let $1\leq p\leq\infty$, $J\subseteq\mathbb{N}_{0}$ and $j_{0}\in J$. Let $\Omega$ stand for $\mathbb{R}^{N}$ or $B$ and let $A_{p,J}$ be the operator defined in Definition 3.12. The following properties hold.

• (i)

If $\lambda\in P\sigma(\Gamma_{p})$ then $\lambda-\lambda(P_{j_{0}})\in P\sigma(A_{p,J})$;

• (ii)

If $\lambda\in A\sigma(\Gamma_{p})$ then $\lambda-\lambda(P_{j_{0}})\in A\sigma(A_{p,J})$;

• (iii)

If $\lambda\in R\sigma(\Gamma_{p})$ then $\lambda-\lambda(P_{j_{0}})\in R\sigma(A_{p,J})$;

Proof. Let $\lambda\in P\sigma(\Gamma_{p})$ and let $0\neq u\in D(\Gamma_{p})$ be such that $\Gamma u=\lambda u$. Then it is immediate to see that the function $f=uP_{j_{0}}$ satisfies $f\in D(A_{p,J})$ and $Af=\left(\lambda-\lambda(P_{j_{0}})\right)f$. This proves (i).

Assertion (ii) follows similarly by using Lemma 8.6.

Let us now consider (iii) and let $\lambda\in R\sigma(\Gamma_{p})$. Recalling Definition 8.5 we have to show that $\mbox{rg}\left(\lambda-\lambda(P_{j_{0}})-A_{p,J}\right)$ is not dense in $L^{p}(A_{p,J})$. Since $\lambda\in R\sigma(\Gamma_{p})$, $\mbox{rg}(\lambda-\Gamma_{p})$ is not dense in $L^{p}(I,r^{N-1}dr)$ and therefore there exists a linear form $0\neq G$ in the dual space $\left(L^{p}(I,r^{N-1}dr)\right)^{\prime}$ which vanishes over $\mbox{rg}(\lambda-\Gamma_{p})$. Let us consider the projection

[TABLE]

Using Lemma 3.15 we see that $0\neq T=G\circ T_{j_{0}}$ belongs to the dual space $\left(L^{p}_{j}(\Omega)\right)^{\prime}$ and satisfies for $u\in D(A_{p,J})$,

[TABLE]

This implies that $T$ vanishes over $\mbox{rg}(\lambda-\lambda(P_{j_{0}})-A_{p,J})$ and proves (iii).

We can finally describe in detail the spectrum of $A_{p,j}$. We are mainly interested in the computation of the complement of the approximate point spectrum, that is the set of all $\lambda$ such that the inequality

[TABLE]

holds, since it is equivalent to Rellich inequalities. Observe that the situation is more complicate in the case where $N\left(1-\frac{2}{p}\right)-2+c>0$ since residual spectra appear.

We recall that ${\cal P}_{p}$ and ${\cal Q}_{p}$ are defined in (17) and (18).

Theorem 3.17

Let $1\leq p\leq\infty$, $J\subseteq\mathbb{N}_{0}$ and $j_{0}:=\min\{j\in J\}$. The following properties hold

If $\Omega=\mathbb{R}^{N}$, the spectrum of $A_{p,J}$ in $L^{p}_{J}(\mathbb{R}^{N})$ is given by

[TABLE]

and reduces to $]-\infty,-\omega_{p}-\lambda(P_{j_{o}})]$ when $N\left(1-\frac{2}{p}\right)-2+c=0$.

• 2.

If $\Omega=B$, the spectrum of $A_{p,J}$ in $L^{p}_{J}(B)$ is given by

[TABLE]

and reduces to $]-\infty,-\omega_{p}-\lambda(P_{j_{o}})]$ when $N\left(1-\frac{2}{p}\right)-2+c=0$. In particular we have

• (i)

If $N\left(1-\frac{2}{p}\right)-2+c<0$, then

[TABLE]

• (ii)

If $N\left(1-\frac{2}{p}\right)-2+c=0$, then

[TABLE]

• (iii)

If $N\left(1-\frac{2}{p}\right)-2+c>0$, then

[TABLE]

Proof. We give a proof only when $\Omega=B$, since the case $\Omega=\mathbb{R}^{N}$ is similar and proved in [24, Theorem 5.17]. Let us prove first the inclusion

[TABLE]

Let $\lambda\not\in{\cal Q}_{p}-\lambda(P_{j_{0}})$ and fix $n\in\mathbb{N}_{0}$ such that

[TABLE]

According to Lemma 3.9 we write $L^{p}_{J}(B)=L^{p}_{J_{n}}(B)\oplus L^{p}_{J\setminus J_{n}}(B)$, where $J_{n}=J\cap\{0,1,\dots,n\}$ (note that if $J_{n}=\emptyset$ then $L^{p}_{J_{n}}(B)={0}$ and $L^{p}_{J}(B)\subseteq L^{p}_{>n}(B)$). Since both $L^{p}_{J_{n}}(B)$ and $L^{p}_{J\setminus J_{n}}(B)$ are $A_{p,J}$ invariant, then $\lambda\in\rho(A_{p,J})$ if and only if $\lambda\in\rho(A_{p,J_{n}})$ and $\lambda\in\rho(A_{p,J\setminus J_{n}})$. The second inclusion follows immediately from (24) with $J\setminus J_{n}$ instead of $J$, since $\rm{Re}\,\lambda$ is greater than the growth bound of $(T_{p,J\setminus J_{n}})_{t\geq 0}$, by (27). Concerning the first inclusion let us suppose that $J_{n}\neq\emptyset$ and, without loss of generality, let us assume $J_{n}=\{0,1,\dots,n\}$. We note that

[TABLE]

and that each $L^{p}_{i}(B)$ is $A_{p,J}$ invariant. Moreover, $\lambda-A_{p,J}$ coincides with $\left(\lambda+\lambda(P_{i})-\Gamma_{p}\right)\otimes I$ on $L^{p}_{i}(B)$, hence it is invertible on it, since $\lambda+\lambda(P_{i})\not\in{\cal Q}_{p}=\sigma(\Gamma_{p})$ by assumption. This shows that $\lambda\in\rho(A_{p,J})$, hence

[TABLE]

Let us prove the opposite inclusion. Using the description of the spectrum of $\Gamma_{p}$ proved in Theorem 3.6 and Lemma 3.16, we get immediately the reverse inclusion and (i) and (ii).

In the case $N\left(1-\frac{2}{p}\right)-2+c>0$, Lemma 3.16 only shows that

[TABLE]

To end the proof we need to show that, if $\lambda\in\left(\overset{\mathrm{o}}{\cal Q}_{p}-\lambda(P_{j_{0}})\right)\setminus\bigcup\limits_{j\in J}({\cal P}_{p}-\lambda(P_{j}))$, then $\lambda\notin A\sigma(A_{p,J})$. Recalling Proposition 8.7 this is equivalent to the validity, for some $C>0$, of the inequality

[TABLE]

Let us fix $\lambda\in\left(\overset{\mathrm{o}}{\cal Q}_{p}-\lambda(P_{j_{0}})\right)\setminus\bigcup\limits_{j\in N_{0}}({\cal P}_{p}-\lambda(P_{j}))$ and let $\overline{n}\in N_{0}$ sufficiently large such that $\lambda\notin{\cal Q}_{p}-\lambda(P_{\overline{n}})$. Then, by (28), $\lambda$ belongs to the resolvent of the operator $A_{p,>\overline{n}}$ in $L^{p}_{>\overline{n}}(B)$. It follows that (29) is true in $L^{p}_{>\overline{n}}(B)$.

Since from (20), $L^{p}(B)=L^{p}_{\leq\overline{n}}(B)\oplus L^{p}_{>\overline{n}}(B)$, it remains to proves (29) for any $v\in D(A_{p,J})\cap L^{p}_{\leq\overline{n}}(B)$. Recalling (22) and Lemma 3.15, one has

[TABLE]

for some $c_{i}\in D(\Gamma_{p})$. Then

[TABLE]

where in the last equality we have used spherical coordinates to evaluate the integrals.

By the assumption on $\lambda$ and recalling (iii) in Theorem 3.6, one has $\lambda+\lambda(P_{i})\notin{\cal P}_{p}=A\sigma(\Gamma_{p})$, which implies, for a possibly different constant $C>0$,

[TABLE]

This proves (29) in the remaining case.

Remark 3.18

The inclusion

[TABLE]

follows also from the more general result [4, Theorem 7.3] since the semigroups generated by $\Gamma$ and ${\Delta_{0}}_{|L^{p}_{J}(S^{N-1})}$ are analytic and commute. **

Corollary 3.19

Let $\Omega$ be equal to $\mathbb{R}^{N}$ or $B$ and assume that $\lambda+\omega_{p}>0$. Then the best constant for which the inequality

[TABLE]

holds is given by

[TABLE]

Proof. If $\lambda+\omega_{p}>0$, then $\lambda\in\rho(A_{p})$, by the preceding theorem, and then the optimal constant in (30) is $\|R(\lambda,A_{p})\|_{p}$. Recalling (70) we have

[TABLE]

Using the contractivity estimates (24) and writing the resolvent as the Laplace transform of the semigroup we see that also the reverse inequality

[TABLE]

holds.

3.4 The operator $A=|x|^{2}\Delta+cx\cdot\nabla$ on $L^{p}(\Omega)$

In this section we complete the study of the operator $A$ in $\mathbb{R}^{N}$ and $B$ by providing a complete description of the domain. Then we use the results in the whole space to extend our results to bounded sets containing the origin. In particular we prove that the domain of the operator coincides with the maximal one, see Proposition 3.28. This allows to state the precise class of functions where Rellich inequalities hold. Note that $A$ is singular both at [math] and at $\infty$.

Let $\beta\in(0,1]$. In what follows we assume $\Omega$ to be $\mathbb{R}^{N}$ or a bounded open connected subset of $\mathbb{R}^{N}$ whose boundary $\partial\Omega$ is $C^{2,\beta}$ and such that $0\notin\partial\Omega$. For any $p\in]1,\infty[$ we define $A_{p}$ by $A_{p}=Au$ in

[TABLE]

for $p=1$ we define $\left(A_{1},D_{1}(\Omega)\right)$ as

[TABLE]

When $\Omega=\mathbb{R}^{N}$ and, correspondingly, $\partial\Omega=\emptyset$, the requirement ”$u=0$ on $\partial\Omega$” must be disregarded. When $\Omega$ is bounded the Dirichlet boundary condition $u(x)=0$ for $x\in\partial\Omega$ makes sense in the sense of traces since $u$ has first derivatives in $L^{p}$ in a neighbourhood of the boundary $\partial\Omega$. The case $0\notin\Omega$ is classical since the term $|x|$ is negligible and, for $1<p<\infty$, $D_{p}(\Omega)$ becomes $W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)$.

For $p=\infty$, we also consider the operator $A_{\infty}$ endowed with the domain

[TABLE]

where $C^{0}(\overline{\Omega})$ denotes the space of bounded and continuous functions defined in $\overline{\Omega}$ and vanishing at the origin, if $0\in\Omega$; $C_{0}^{0}(\overline{\Omega})$ is its subspace consisting of functions vanishing also at $\infty$ when $\Omega=\mathbb{R}^{N}$ and at the boundary $\partial\Omega$, otherwise.

When $\Omega$ is bounded we use Proposition 8.2 to fix $\delta>0$ such that the subsets

[TABLE]

have $C^{2,\beta}$ boundary. Furthermore we can write $\overline{\Omega}=\overline{\Omega}_{\delta}\cup\Omega_{0}$ where $\Omega_{0}$ is an open subset $\Omega_{0}\subset\subset\Omega$ and we fix a partition of unity $\{\eta_{\delta}^{2},\eta_{0}^{2}\}$ such that

[TABLE]

In order to identify a core for $A_{p}$ we define

[TABLE]

Lemma 3.20

The space $C_{c,0}^{2}\left(\Omega\right)$ is dense in $D_{p}(\Omega)$, endowed with the norm

[TABLE]

When $\Omega=\mathbb{R}^{N}$, $C^{\infty}_{c}\left(\mathbb{R}^{N}\setminus\{0\}\right)$ is is dense in $D_{p}(\Omega)$.

Proof. Let us consider, preliminarily, $\Omega=\mathbb{R}^{N}$.

Let $u\in D_{p}(\mathbb{R}^{N})$; we approximate $u$ with functions in $D_{p}(\mathbb{R}^{N})$ having compact support in $\mathbb{R}^{N}\setminus\{0\}$. Let

[TABLE]

where $\phi$ is a classical mollifier supported in $B_{1}$, with $\int_{\mathbb{R}^{N}}\phi=1$ and $\phi_{\frac{1}{n}}(x)=n^{N}\phi\left(nx\right)$. It is easy to check that $\xi_{n}(x)=1$ for $x\in\Omega_{n}$, $\xi_{n}$ is supported in $\mathbb{R}^{N}\setminus\{0\}$ and that $0\leq\xi_{n}\leq 1$, $|\nabla\xi_{n}|\leq Cn$, $|D^{2}\xi_{n}|\leq Cn^{2}$. Consider also a smooth function $\eta$ such that $\chi_{B_{1}}\leq\eta\leq\chi_{B_{2}}$ and, for every $n\in\mathbb{N}$, define $\eta_{n}(x)=\eta\left(\frac{x}{n}\right)$. Set $u_{n}=\xi_{n}\eta_{n}u$. It is immediate to check, using Lebesgue’s Theorem, that $u_{n}$ tends to $u$ in $L^{p}(\mathbb{R}^{N})$. Concerning the gradient term, we have

[TABLE]

The last inequality implies

[TABLE]

which tends to 0 by dominated convergence. Using a similar argument one shows that, if $1<p<\infty$, $|x|^{2}D^{2}u_{n}$ tends to $|x|^{2}D^{2}u$ in $L^{p}(\mathbb{R}^{N})$ and that, if $p=1,\infty$, $|x|^{2}\Delta u_{n}$ tends to $|x|^{2}\Delta u$ in $L^{p}(\mathbb{R}^{N})$. This proves that $u_{n}$ tends to $u$ in $D_{p}(\mathbb{R}^{N})$; we also note that, by construction, $\mbox{supp\,}u_{n}\subseteq\mbox{supp\,}u$. Finally we can use a standard convolution argument to approximate in $D_{p}(\mathbb{R}^{N})$ functions having compact support in $\mathbb{R}^{N}\setminus\{0\}$ with $C_{c}^{\infty}\left(\mathbb{R}^{N}\setminus\{0\}\right)$ functions.

Let us consider, now, a bounded set $\Omega\subset\mathbb{R}^{N}$ and let $u\in D_{p}(\Omega)$. We use the partition of unity defined in (34) to write

[TABLE]

The function $u_{0}$ satisfies $\mbox{supp\,}u_{0}=\Omega_{0}\subset\subset\Omega$: the same proof as before shows that we can approximate $u_{0}$ in $D_{p}(\Omega)$ with $C_{c}^{\infty}(\Omega\setminus\{0\})$ functions.

On the other hand the function $u_{\delta}$ satisfies $u_{\delta}\in D_{p}(\Omega_{\delta})$ since $u=0$ on $\partial\Omega$ and $\mbox{supp}\,\eta_{\delta}\subseteq K_{\delta}$. Since no singularity appears in $\Omega_{\delta}$, the approximation problem is a classical one: Proposition 8.1 then proves that $u_{\delta}$ can be approximated in $D_{p}(\Omega)$ with functions in $C_{c,0}^{2}\left(\Omega\right)$.

The previous Lemma shows that $C_{c,0}^{2}\left(\Omega\right)$ is a core for $A_{p}$. When $\Omega=\mathbb{R}^{N}$ or $\Omega=B$, Proposition 3.14 states that $C_{c,0}^{2}\left(\Omega\right)$ is also a core for the operator $A_{p,J}$ of Definition 3.12. We have therefore proved the following result which provides a description of the operators introduced in the previous subsection.

Proposition 3.21

Let $1\leq p\leq\infty$ and $\Omega=\mathbb{R}^{N}$ or $\Omega=B$. Then the operator $A_{p}$ coincides with that of Definition 3.12 for $J=\mathbb{N}_{0}$.

In the next lemma we state some interpolative and a-priori estimates.

Lemma 3.22

Let $1\leq p\leq\infty$. Then there exist $\varepsilon_{0},\ C>0$ depending only on $c,N,\Omega$ such that for every $0<\varepsilon<\varepsilon_{0}$ and $u\in D_{p}(\Omega)$ one has

[TABLE]

Moreover, if $1<p<\infty$,

[TABLE]

Proof. In view of Lemma 3.20, it is enough to prove these estimates for $u\in C_{c,0}^{2}\left(\Omega\right)$. The proof of (35) follows as in [14, Lemma 2.4] with minor modifications (in particular, one intersects the balls $B(x_{0},\rho)$ with $\Omega$). To prove (36) for $1<p<\infty$, it is sufficient to apply the classical elliptic estimate $\|D^{2}u\|_{p}\leq C\|\Delta u\|_{p}$ (which holds both in $\mathbb{R}^{N}$ as well as in a bounded $\Omega$ if $u$ vanishes at the boundary) to $|x|^{2}u$ and then to interpolate the terms containing $\nabla u$, by (35).

In the next Propositions we prove dissipativity properties for $A_{p}$ through Hardy type inequalities. In the spirit of Section 4, this is equivalent to the fact that the Rellich inequalities (2) for the operator $L$, when $b$ is sufficiently large, can be proved using integration by parts and Hardy inequalities (37). We begin by the recalling the following result.

Proposition 3.23

(see [24, Proposition 8.3]). Let $1<p<\infty$, $\beta\in\mathbb{R}$. Then, if $N-2+\beta\neq 0$, for every $u\in C_{c}^{\infty}(\mathbb{R}^{N}\setminus\{0\})$,

[TABLE]

We prove now that $A_{p}$ is quasi-dissipative.

Theorem 3.24

Let $1\leq p\leq\infty$ and set $\omega_{p}=\frac{N}{p^{2}}\left[p(N-2+c)-N\right]$. Then, for every $u\in D_{p}(\Omega)$, $\lambda>0$,

[TABLE]

Proof. We consider, preliminarily, $1<p<\infty$ and prove the inequality

[TABLE]

Let $2\leq p<\infty$. By Proposition 3.20, we may assume that $u\in C_{c,0}^{2}\left(\Omega\right)$. Setting $u^{\star}=u|u|^{p-2}$ we multiply $Au$ by $u^{\star}$ and integrate over $\Omega$. Integrating by parts we get

[TABLE]

By Hardy inequality (37) with $\beta=2$,

[TABLE]

and therefore

[TABLE]

For $1<p<2$ the integration by parts is not straightforward (but still allowed, see [22]) since $|u|^{p-2}$ becomes singular near the zeros of $u$. In this case it is sufficient to replace $u^{\star}$ by $u(u^{2}+\delta)^{\frac{p}{2}-1}$ where $\delta$ is a positive parameter and then let $\delta$ to [math] obtaining the required estimates also in this case.

It is clear that (39) implies (38) which is therefore proved for $1<p<\infty$. Letting $p\to 1,\infty$, we see that (38) holds in all cases.

Remark 3.25

• (i)

$\omega_{\infty}=0$ and $\omega_{1}=(c-2)N$;

• (ii)

$\omega_{p}\geq 0$ iff $p\geq\frac{N}{N-2+c}$. Moreover $\omega_{p}$ attains its maximum value at $\overline{p}=\frac{2N}{N-2+c}$ and $\omega_{\overline{p}}=\left(\frac{N-2+c}{2}\right)^{2}$.

The previous theorem, combined with Lemma 3.22, allows us to deduce the following result.

Corollary 3.26

Let $1\leq p\leq\infty$. There exist two constants $\Lambda>0$ and $C>0$ such that, for every $u\in D_{p}(\Omega)$ and every $\rm{Re}\lambda\geq\Lambda_{p}$

[TABLE]

If $1<p<\infty$, we have also

[TABLE]

Proof. The estimate

[TABLE]

is nothing but sectoriality. The gradient estimate follows from it, using (35) with $\varepsilon=|\lambda|^{-\frac{1}{2}}$. The Hessian estimate for $1<p<\infty$ follows from (36).

The next theorem shows that $A_{p}$ is the generator of a contractive analytic semigroup in $L^{p}(\Omega)$.

Theorem 3.27

For any $1\leq p\leq\infty$, the operator $(A_{p}+\omega_{p},D_{p}(\Omega))$ generates a contractive analytic semigroup in $L^{p}(\Omega)$.

Proof. To distinguish, we write $\tilde{A}_{p}$ for $A_{p}$ when $\Omega=\mathbb{R}^{N}$. Observe that, by Proposition 3.11, $\tilde{A}_{p}$ generates an analytic semigroup, hence its resolvent contains a sector

[TABLE]

with $\theta>\pi/2$ where the following resolvent estimate holds

[TABLE]

Let $\Omega\subset\mathbb{R}^{N}$ and define $\eta_{\delta}$ and $\eta_{0}$ as in (34). For $\lambda\in\Sigma_{\theta,\rho}$, $f\in L^{p}(\Omega)$, set $R_{0}(\lambda)f=\eta_{0}(\lambda-\tilde{A}_{p})^{-1}(\eta_{0}f)\in D_{p}(\mathbb{R}^{N})$, $R_{\delta}(\lambda)f=\eta_{\delta}(\lambda-A_{\delta})^{-1}(\eta_{\delta}f)\in W^{2,p}(K_{\delta})\cap W_{0}^{1,p}(K_{\delta})$ where $A_{\delta}$ is the operator $A$ in $K_{\delta}$ with Dirichlet boundary conditions. We have

[TABLE]

where

[TABLE]

is a first order operator supported on $K_{\delta}$. Using Corollary 3.26 (and disregarding $|x|$ which is bounded above and below from 0 in $K_{\delta}$) we see that

[TABLE]

for $\lambda\in\Sigma_{\theta,\rho}$ and with $c_{1}$ depending only on $\delta$. In similar way we get

[TABLE]

with

[TABLE]

for $\lambda\in\Sigma_{\theta,\rho}$ and with $c_{1}$ depending only on $\delta$, by classical results, since $A_{\delta}$ is uniformly elliptic in $K_{\delta}$. Then setting

[TABLE]

we have

[TABLE]

Choosing $|\lambda|>\rho_{1}$ large enough, we find $\|S(\lambda)\|_{p}\leq\frac{1}{2}$ and then the operator $I+S(\lambda)$ is invertible in $L^{p}(\Omega)$. Setting $V(\lambda)=(I+S(\lambda))^{-1}$ we have

[TABLE]

and hence the operator $R(\lambda)V(\lambda)$, which maps $L^{p}(\Omega)$ into $D_{p}(\Omega)$, is a right inverse of $\lambda-A$. Since both $\|R_{0}(\lambda)\|_{p},\|R_{\delta}(\lambda)\|_{p}\leq M|\lambda|^{-1}$ and $\|V(\lambda)\|_{p}\leq 2$, then

[TABLE]

for $\lambda\in\Sigma_{\theta,\rho_{1}}$. Clearly, $R(\lambda)V(\lambda)$ coincides with $(\lambda-A_{p})^{-1}$ whenever this last is injective, in particular for $\lambda>-\omega_{p}$, By Proposition 3.24. Then $(-\omega_{p},\infty)\subset\rho(A_{p})$, the a-priori estimates (40) shows that the norm of the resolvent cannot blow up in $\Sigma_{\theta,\rho_{1}}$, hence $\Sigma_{\theta,\rho_{1}}\subset\rho(A_{p})$ and the proof is complete.

In the next proposition we prove that the domain $D_{p}(\Omega)$ coincides with the maximal one. In what follows, $Au$ is understood in the sense of distributions in $\Omega\setminus\{0\}$. Since the coefficients of $A$ are $C^{\infty}$ away from the origin, by local elliptic regularity it follows that $u\in W^{2,p}_{loc}(\mathbb{R}^{N}\setminus\{0\})$ when $\Omega=\mathbb{R}^{N}$ and that $u\in W^{2,p}(\Omega\setminus B_{\varepsilon})$ for every $\varepsilon>0$, when $\Omega$ is bounded. This clearly holds for $1<p<\infty$; when $p=\infty$, the same is true for any $q<\infty$.

Proposition 3.28

Let $1\leq p\leq\infty$. The domain $D_{p}(\Omega)$ defined in (31) coincides with the maximal domain

[TABLE]

Proof. The inclusion $D_{p}(\Omega)\subset D_{p,max}(\Omega)$ is obvious. Conversely, let $u\in D_{p,max}(\Omega)$ and $\lambda>0$ be in the resolvent set of $(A_{p},D_{p}(\Omega))$. Set $f=\lambda u-A_{p}u$ and $v=u-R(\lambda,A_{p})f$. Then $v$ belongs to $D_{p,max}(\Omega)$ and satisfies $\lambda v-A_{p}v=0$. We prove that $v\equiv 0$ if $\lambda$ is large enough. Let us consider for large $n$

[TABLE]

where $\phi$ is a classical mollifier supported in $B_{1}$, with $\int_{\mathbb{R}^{N}}\phi=1$ and $\phi_{\frac{1}{n}}(x)=n^{N}\phi\left(nx\right)$. It is easy to check that $\xi_{n}(x)=1$ for $x\in\Omega_{\frac{n}{3}}$, $\xi_{n}$ is supported in $\Omega_{n}$ and that $0\leq\xi_{n}\leq 1$, $|\nabla\xi_{n}|\leq Cn$, $|D^{2}\xi_{n}|\leq Cn^{2}$. Consider also a smooth function $\eta$ such that $\chi_{B_{1}}\leq\eta\leq\chi_{B_{2}}$ and set $\eta_{n}(x)=\eta\left(\frac{x}{n}\right)$, $\zeta_{n}=\xi_{n}\eta_{n}$. Since $|\nabla\xi_{n}|\leq Cn\chi_{(\Omega_{n}\setminus\Omega_{\frac{n}{3}})}$ and $|\nabla\eta_{n}|\leq Cn^{-1}\chi_{(B_{2n}\setminus B_{n})}$, it follows that the function $\nabla\zeta_{n}$ has support in $F_{n}:=\left(\Omega_{n}\setminus\Omega_{\frac{n}{3}}\right)\cup\left(B_{2n}\setminus B_{n}\right)$ and satisfies $|x|^{2}|\nabla\zeta_{n}|^{2}\leq C$, with $C$ independent of $n$.

Let us consider, first, the case where $p\geq 2$. Integrating by parts the identity

[TABLE]

we obtain

[TABLE]

Using Hölder’s inequality we obtain

[TABLE]

Similarly

[TABLE]

Combining the last inequalities we obtain, up to slightly changing the constants,

[TABLE]

Finally, choosing $3\varepsilon<p-1$ and letting $n$ to infinity, we obtain

[TABLE]

which implies $v\equiv 0$, if $\lambda$ is large enough. For $1<p<2$ the integration by parts is not straightforward since $|v|^{p-2}$ becomes singular near the zeros of $v$, but still allowed ( see [22]) and one concludes as before (or, more simply, notice that $v$ is a smooth function, by elliptic regularity, replace $v|v|^{p-2}$ by $v(v^{2}+\delta)^{\frac{p}{2}-1}$ and then let $\delta\to 0$).

For $p=1$, we notice that $v$ is a smooth function away from the origin, by elliptic regularity, and consider a sequence of smooth functions $h_{n}:\mathbb{R}\rightarrow\mathbb{R}$ such that $|h_{n}|\leq 1,$ $h_{n}^{\prime}(s)\geq 0$ and $h_{n}(s)\rightarrow\mathop{\rm sign}(s)$ for every $s\in\mathbb{R}$. Integrating by parts the identity

[TABLE]

the proof follows as before.

For $p=\infty$ we note that $v$ vanishes at [math] and at $\partial\Omega$ when $\Omega$ is bounded or at $\infty$ if $\Omega=\mathbb{R}^{N}$. Moreover, by elliptic regularity, $v$ is a smooth function out of the origin. If $v$ is not identically zero, then it has a positive maximum point (or a negative minimum point ) at some $x_{0}\in\Omega$. The classical maximum principle yields $Av(x_{0})\leq 0$, hence $\lambda v(x_{0})\leq 0$, which is a contradiction for $\lambda>0$.

Finally, we consider the domain of the operator $A_{p,J}$ of Subsection 3.3.

Corollary 3.29

If $\Omega=\mathbb{R}^{N}$ or $\Omega=B$, then the domain $D_{p,J}(\Omega)$ of $A_{p,J}$ is given by

[TABLE]

Proof. By Corollary 3.13, the domain of $A_{p,J}$ is the intersection of the domain of $A_{p}$ with $L^{p}_{J}$ and the thesis follows from Propositions 3.21, 3.28.

4 Rellich inequalities in $\mathbb{R}^{N}$ and in $B$

In this section we prove weighted Rellich inequalities for the operator

[TABLE]

on $L^{p}(\Omega)$ when $\Omega=\mathbb{R}^{N}$ and $\Omega=B$. For $1\leq p\leq\infty$, $\alpha\in\mathbb{R}$ and $J\subset\mathbb{N}_{0}$ we define

[TABLE]

When $J=\mathbb{N}_{0}$ we write $D_{p,\alpha}(\Omega)$ in place of $D_{p,\alpha,\mathbb{N}_{0}}(\Omega)$. As in the previous section $Lu$ is understood as a distribution in $\Omega\setminus\{0\}$. Since the coefficients of $L$ are $C^{\infty}$ away from the origin, by local elliptic regularity it follows that, if $u\in D_{p,\alpha}(\Omega)$, then $u\in W^{2,p}_{loc}(\mathbb{R}^{N}\setminus\{0\})$ when $\Omega=\mathbb{R}^{N}$ and $u\in W^{2,p}(\Omega\setminus B_{\varepsilon})$ for every $\varepsilon>0$, when $\Omega$ is bounded. This clearly holds for $1<p<\infty$; when $p=\infty$, the same is true for any $q<\infty$.

Defining

[TABLE]

we have seen in Section 2 that

[TABLE]

where $A$ is the operator of Section 3 with $c+4-2\alpha$ in place of $c$,

[TABLE]

By construction $\Phi\left(D_{p,\alpha,J}(\Omega)\right)$ coincides with the domain $D_{p,J}(\Omega)=D_{p,max}(\Omega)\cap L^{p}_{J}(\Omega)$, see Corollary 3.29. In particular Rellich inequalities

[TABLE]

are equivalent to the spectral estimates

[TABLE]

which, recalling Proposition 8.7, hold precisely when $\mu\notin A\sigma(A_{p,J})$. The results of this section are then immediate consequences of Theorem 3.17 and Corollary 3.19.

Let us define

[TABLE]

and

[TABLE]

In what follows we refer to $D$ as the discriminant of $L$; in [20, 23] the authors show that $D$ takes a fundamental role in generation properties of $L$. We recall that ${\cal Q}_{p}$, ${\cal P}_{p}$, $\omega_{p}$ have been defined in (17), (18), (19). For clarity sake, we rewrite them in the present situation where $c+4-2\alpha$ takes the place of $c$:

[TABLE]

Note that, when N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1-\alpha+\frac{c}{2}=0, then

[TABLE]

In the following lemma we denote with $\sqrt{z}$ a complex square root of $z$ having non negative real part.

Lemma 4.1

Let $1\leq p\leq\infty$, $j\in\mathbb{N}_{0}$ and $\mu:=b-(2-\alpha)(N-\alpha+c)$. Then the following properties are equivalent

• (i)

$\mu\notin{\cal Q}_{p}-\lambda(P_{j})$;

• (ii)

$b+\gamma_{p}(\alpha,c)+\lambda(P_{j})>0$;

• (iii)

\left|N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}-\alpha\right|<\sqrt{D+\lambda(P_{j})}* and $D+\lambda(P_{j})>0$;*

• (iv)

\left|N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}-\alpha\right|<\textrm{\emph{Re}\,}\sqrt{D+\lambda(P_{j})}.

Proof. The proof follows from elementary calculations after noticing that

[TABLE]

Since $\mu\in\mathbb{R}$, the conditions $\mu\notin{\cal{P}}_{p}-\lambda(P_{j})$, $\mu\notin{\cal{Q}}_{p}-\lambda(P_{j})$ become $b+\gamma_{p}(\alpha,c)+\lambda(P_{j})\neq 0$, $b+\gamma_{p}(\alpha,c)+\lambda(P_{j})>0$, respectively.

The following is the main result of this section. Part 1 has been already proved in [24].

Theorem 4.2

Let $1\leq p\leq\infty$, $\alpha,\ b,\ c\in\mathbb{R}$ and $J\subseteq N_{0}$ with $j_{0}:=\min\{j\in J\}$.

If $\Omega=\mathbb{R}^{N}$, Rellich inequalities

[TABLE]

hold if and only if

[TABLE]

or equivalently when  $b+\gamma_{p}(\alpha,c)+\lambda(P_{j})\neq 0$ for every $j\in J$.

• 2.

If $\Omega=B$, Rellich inequalities

[TABLE]

hold if and only if

[TABLE]

In particular the latter conditions are verified

• (i)

when $\alpha\geq N\left(\frac{1}{2}-\frac{1}{p}\right)+1+\frac{c}{2}$, if and only if  $b+\gamma_{p}(\alpha,c)+\lambda(P_{j_{0}})>0$,

• (ii)

when $\alpha<N\left(\frac{1}{2}-\frac{1}{p}\right)+1+\frac{c}{2}$, if and only if  $b+\gamma_{p}(\alpha,c)+\lambda(P_{j})\neq 0$ for every $j\in J$.

If $J=\mathbb{N}_{0}$ and $b+\gamma_{p}(\alpha,c)>0$, that is

[TABLE]

then the optimal constant is given by $C=b+\gamma_{p}(\alpha,c)$.

Proof. Consider ${\cal Q}_{p}$, ${\cal P}_{p}$ and $\omega_{p}$ defined before Lemma 4.1 and let $\mu=b-(2-\alpha)(N-\alpha+c)$. Then Rellich inequalities hold if and only if $\mu\notin A\sigma(A_{p,J})$. The proof of the required claims follows then easily by combining Lemma 4.1, Theorem 3.17 and Corollary 3.19.

Remark 4.3

For a fixed $\alpha$, Rellich inequalities are always true in $L^{p}_{\geq n}(\Omega)$, for a sufficiently large $n\in\mathbb{N}_{0}$, even though they fail in the whole $L^{p}(\Omega)$. This phenomenon appears also in the extreme cases $p=1,\infty$. The failure of Rellich inequalities for some values of $\alpha$ is, therefore, always determined by subspaces defined by spherical harmonics of low order.

When $b=c=0$, the operator reduces to the Laplace operator $L=\Delta$. In this case

[TABLE]

Rellich inequalities in bounded domains for the Laplace operator have already been investigated in [28] where their validity is proved for $N\geq 3$, $1<p<\infty$ and

[TABLE]

This range coincides with the values of $\alpha$ for which Rellich inequalities can be proved using integration by parts and the Hardy inequalities (37) (see Theorem 3.24). The following corollary characterizes their validity in the ball.

Corollary 4.4

Let $1\leq p\leq\infty$, $\alpha\in\mathbb{R}$. If $\Omega=B$, Rellich inequalities

[TABLE]

hold if and only if

[TABLE]

5 Rellich inequalities in general domains

Let $\Omega$ be an open bounded and connected subset of $\mathbb{R}^{N}$ whose boundary $\partial\Omega$ is $C^{2,\beta}$ and such that $0\in\Omega$. In this section we show that Rellich inequalities for the operator $L$ hold in $\Omega$ if and only if they hold in the ball $B$. In terms of the auxiliary operator $A$, this means that its approximate point spectrum is independent of the bounded set $\Omega$. We have no direct proof of this fact which does not seem to be evident. We write $L$ in the symmetric form

[TABLE]

and we always assume $1<p<\infty$ and that

[TABLE]

This condition is crucial for the solvability of some elliptic problems related to $L$ which will be studied in the following subsection in a auxiliary weighted $L^{2}$ space.

5.1 The operator $L$ in $L^{2}(\Omega,d\mu)$

We need some preliminary facts concerning the operator $L$ in a weighted space and here we suppose $\Omega$ as above or $\Omega=\mathbb{R}^{N}$. We consider the weighted space $L^{2}(\Omega,d\mu)$, $d\mu=|x|^{c}dx$, and the symmetric form

[TABLE]

Using (43), we see that for $u,v\in C_{c,0}^{2}(\Omega)$

[TABLE]

To prove that $\mathfrak{a}$ is non-negative, we make different change of variables according to $D>0$ or $D=0$. When $D>0$ we write $u=u_{1}|x|^{-\frac{c}{2}}$ and $v=v_{1}|x|^{-\frac{c}{2}}$ to obtain, after integration by parts

[TABLE]

Then we use the classical Hardy inequality. When $D=0$ we are in the critical case of Hardy inequality and it is convenient to use the transformation (which is the basis of the proof of Hardy inequality) $u=u_{1}|x|^{-\frac{N-2+c}{2}}$ and $v=v_{1}|x|^{-\frac{N-2+c}{2}}$. Integrating by parts we get

[TABLE]

To identify the domain of the closure of $\mathfrak{a}$ we use the classical Sobolev space $H^{1}_{0}(\Omega)$ and also $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$ defined as the closure of $C_{c}^{2}(\Omega)$ with respect to the norm

[TABLE]

Note that we use $C^{2}_{c}(\Omega)$ and not $C_{c,0}^{2}(\Omega)$, that is we do not assume that the functions vanish in a neighbourhood of [math]. However, the above definition would not change using the smaller space. Let us recall, in fact, that, since $N\geq 2$, $C_{c,0}^{2}(\Omega)$ is dense in $H^{1}_{0}(\Omega)$ and the same is true for $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$, as we show below.

Lemma 5.1

$C_{c,0}^{2}(\Omega)$* is dense in $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$.*

Proof. Let us assume, for example that $\Omega=\mathbb{R}^{N}$ and let $f\in C_{c}^{2}(\mathbb{R}^{N})$. We approximate $f$ in the norm of $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$ with functions belonging to $C_{c,0}^{2}(\mathbb{R}^{N})$.

Let $\varphi\in C^{\infty}(\mathbb{R}^{+})$ such that $\varphi(r)=0$ if $0\leq r\leq\frac{1}{4}$ and $\varphi(r)=1$ if $r\geq\frac{1}{2}$ and set $\varphi_{\epsilon}(x):=\varphi(|x|^{\epsilon})$. By construction $f\varphi_{\epsilon}\in C_{c,0}^{2}(\mathbb{R}^{N})$ and, as $\epsilon\to 0^{+}$, $f\varphi_{\epsilon}$, $\partial_{i}f\varphi_{\epsilon}$ converge in $L^{2}\left(\Omega,|x|^{2-N}dx\right)$ to $f$, $\partial_{i}f$, respectively, by dominated convergence.

It remains to show that $f\partial_{i}\varphi_{\epsilon}$ converges to [math] in $L^{2}\left(\Omega,|x|^{2-N}dx\right)$. This is true since

[TABLE]

To prove the main properties of $\mathfrak{a}$ we may therefore use $C_{c,0}^{2}(\Omega)$.

Lemma 5.2

*Let $D\geq 0$. The form $\mathfrak{a}$ is non-negative and symmetric in $L^{2}(\Omega,d\mu)$. For $u\in C_{c,0}^{2}\left(\Omega\right)$, let $||u||_{\mathfrak{a}}:=\sqrt{\mathfrak{a}(u,u)+||u||^{2}_{L_{\mu}^{2}}}$. Then $||u||_{\mathfrak{a}}$ is equivalent to $\||x|^{\frac{c}{2}}\,u\|_{H_{0}^{1}\left(\Omega\right)}$, if $D>0$, and to $\||x|^{\frac{N-2+c}{2}}\,u\|_{H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)}$, if $D=0$. *

Proof. If $D>0$ we set $u=v|x|^{-\frac{c}{2}}$. We choose $\varepsilon$ small enough such that $D-\varepsilon\frac{(N-2)^{2}}{4}>0$. Using (45) and Hardy inequality

[TABLE]

we obtain

[TABLE]

On the other hand, by Hardy inequality again,

[TABLE]

This proves that $||u||_{\mathfrak{a}}$ and $||v||_{H_{0}^{1}\left(\Omega\right)}$ are equivalent norms. If $D=0$, setting $u=v|x|^{-\frac{N-2+c}{2}}$, we obtain from (46)

[TABLE]

Since also the norms of $u$ in $L^{2}(\Omega,d\mu)$ and $v$ in $L^{2}(\Omega,|x|^{2-N}\,dx)$ coincide, we see that the norms $||u||_{\mathfrak{a}}$ and $||v||_{H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)}$ are equivalent.

Using the density of $C_{c,0}^{2}(\Omega)$ in $H^{1}_{0}(\Omega)$ and in $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$, we extend the form ${\mathfrak{a}}$ to the domain

[TABLE]

thus obtaining a closed form.

Note that both the norms of $u|x|^{\frac{c}{2}}$ and $u|x|^{\frac{N-2+c}{2}}$ in the corresponding spaces equal the norm of $u$ in $L^{2}(\Omega,d\mu)$. The transformation $u=u_{1}|x|^{\frac{N-2+c}{2}}$ can be performed also in the case $D>0$. However it leads to the extra term $D(u_{1}v_{1})/|x|^{2}$ in the integral (46) which cannot be dominated by the norm of $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$.

Let $-L$ be the operator associated to ${\mathfrak{a}}$, that is

[TABLE]

Clearly, $L$ is given by (43) when $u\in C^{2}_{c,0}(\Omega)$. In the next lemma we prove the simplest inequality useful to prove compactness when $D=0$. Note that Hardy inequality fails with respect to the weight $|x|^{2-N}$.

Lemma 5.3

Let $\Omega$ be bounded and let $R(\Omega):=\max_{x\in\Omega}|x|$. Then, for every $u\in C_{c,0}^{2}(\Omega)$,

[TABLE]

*In particular the immersion $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)\hookrightarrow L^{2}\left(\Omega,|x|^{2-N}dx\right)$ is compact. *

Proof. Let us fix $u\in C_{c,0}^{2}(\Omega)$. Integrating by parts we have

[TABLE]

This implies, using the Cauchy-Schwarz inequality,

[TABLE]

and the inequality follows. To prove the compactness of the embedding, we take $u$ in the unit ball $\mathcal{B}$ of $H_{0}^{1}\left(\Omega,|x|^{2-N}dx\right)$ and fix $\epsilon>0$. Then

[TABLE]

Since $L^{2}\left(\Omega\cap B^{c}_{\epsilon},\,|x|^{2-N}dx\right)=L^{2}\left(\Omega\cap B^{c}_{\epsilon},\,dx\right)$, the compactness of $\mathcal{B}_{|\Omega\cap B^{c}_{\epsilon}}$ in $L^{2}\left(\Omega\cap B^{c}_{\epsilon},\,|x|^{2-N}dx\right)$ is classical. This fact and and the above estimate show that $\mathcal{B}$ is totally bounded.

In the next Proposition we collect the main properties of $L$ in $L^{2}(\Omega,d\mu)$..

Proposition 5.4

The operator $-L$ defined in (48) is non-negative and self-adjoint in $L^{2}(\Omega,d\mu)$. The generated semigroup $T_{\Omega}(t)$ is positivity preserving in $L^{2}(\Omega,d\mu)$. Moreover, $C_{c,0}^{2}(\Omega)\hookrightarrow D(L)$ and for every $u\in C_{c,0}^{2}(\Omega)$

[TABLE]

If $\Omega$ is bounded then $L$ has compact resolvent and is invertible in $L^{2}(\Omega,d\mu)$.

Proof. Non-negativity and self-adjointness of $-L$ follow by construction. The positivity of $T_{\Omega}(t)$ follows from that of the resolvent which is a consequence of the Beurling-Deny conditions.

Let us suppose, now, $\Omega$ be bounded and let us prove that $D({\mathfrak{a}})$ is compactly embedded in $L^{2}(\Omega,d\mu)$. To this aim let $\mathcal{U}$ be a bounded subset of $D({\mathfrak{a}})$. Assume $D>0$; then the set $\mathcal{U}^{\prime}=\{u|x|^{\frac{c}{2}}:\,u\in\mathcal{U}\}$ is a bounded subset of $H_{0}^{1}(\Omega)$, hence totally bounded in $L^{2}(\Omega)$, by the compactness of the embedding of $H^{1}_{0}(\Omega)$ into $L^{2}(\Omega)$. It is then immediate to check that $\mathcal{U}$ is totally bounded in $L^{2}(\Omega,d\mu)$, which proves the claim. The case $D=0$ follows similarly from Lemma 5.3.

In both cases $L$ has compact resolvent; its spectrum consists, therefore, of eigenvalues and, being injective by (46), (47), $L$ is invertible.

Next we need a maximum principle for the solution of an homogeneous problem related to $L$. Note that no singularity appears, since $0\not\in V$ below. However, comparison is not obvious since the coefficient $b$ can be negative even though $D\geq 0$.

Lemma 5.5

Let $V$ be an open bounded and connected subset of $\mathbb{R}^{N}$ whose boundary $\partial V$ is $C^{2,\beta}$ and such that $0\notin V$. For every $\varphi\in C^{2}(\partial V)$ the problem

[TABLE]

admits a unique solution $v\in C^{2}\left(V\right)\cap C\left(\bar{V}\right)$. Moreover $v$ satisfies $\inf_{\partial V}\varphi\leq v(x)\leq\sup_{\partial V}\varphi$ for every $x\in V$ .

Proof. The transformation $Su(x)=|x|^{-\frac{N-2+c}{2}}v(x)$ turns $L$ into

[TABLE]

which is uniformly elliptic with smooth coefficients and non-positive potential. Then the proof follows, immediately, by classical results.

In order to prove Rellich inequalities in domains, we need estimates for the Green function of $-L$ in $\Omega$, that is for the integral kernel expressing $(-L)^{-1}$ with respect to the Lebesgue measure. We start with the case $D>0$ where can use the results of [23] and compare the Green function in $\Omega$ with that in $\mathbb{R}^{N}$.

Proposition 5.6

Let $D>0$ and let $G(x,y)$, $x,y\in\Omega\times\Omega$ be the Green function of the operator $L$, written with respect to the Lebesgue measure. Then

[TABLE]

where if $N>2$

[TABLE]

and if $N=2$

[TABLE]

Proof. Let $T_{\Omega}(t)$, $T(t)$ be the semigroups generated by $L$ in $L^{2}(\Omega,d\mu)$ and $L^{2}(\mathbb{R}^{N},d\mu)$, respectively. From [32, Sections 2.3, 2.6, Proposition 4.23] it follows that $0\leq T_{\Omega}(t)f\leq T(t)f$ whenever $0\leq f\in L^{2}(\Omega,d\mu)$. Furthermore from [8, Corollary 4.6] $T(t)$ is an integral operator whose kernel $p(t,x,y)$, expressed with respect to the Lebesgue measure, satisfies, for every $\epsilon>0$ and some constant $C_{\epsilon}>0$,

[TABLE]

Using [5, Theorem 1.5], it follows that also $T_{\Omega}(t)$ is an integral operator whose kernel $p_{\Omega}$ satisfies the same estimate above. By [23, Theorem 7.1], since $D>0$, we have

[TABLE]

hence

[TABLE]

Remark 5.7

The inequality between the semigroups above easily follows from the the corresponding one for the resolvents. Let $\lambda>0$, $0\leq f\in L^{2}(\Omega,d\mu)$ and set $u=R(\lambda,L_{\Omega})f$, $w=R(\lambda,L_{\mathbb{R}^{N}})f$. Then $0\leq u\in D({\mathfrak{a}}_{\Omega})$ and $0\leq w\in D({\mathfrak{a}}_{\mathbb{R}^{N}})$; furthermore $\lambda u-Lu=\lambda w-Lw$ and, for every $v\in D({\mathfrak{a}}_{\Omega})$ one has

[TABLE]

Choosing $v=(u-w)^{+}\in D({\mathfrak{a}}_{\Omega})$ we get

[TABLE]

*which implies $(u-w)^{+}=0$ that is $u\leq w$.  

The case $D=0$ is more involved since, in this case, the integral in (52) is divergent near $\infty$. To overcome this problem, we use the boundedness of $\Omega$ to improve the decay of $p_{\Omega}$ as $t\to\infty$. We estimate directly $p_{\Omega}$ without comparing with the kernel in the whole space, by adapting to our case the arguments of [8].

We use the change of variable leading to (46) to get rid of the potential term $b|x|^{-2}$ and introduce the Hilbert space $L^{2}(\Omega,|x|^{-2s_{1}}\,d\mu)=L^{2}(\Omega,|x|^{2-N}\,dx)$, where $s_{1}=\frac{N-2+c}{2}$. Then (46) reads

[TABLE]

By construction $\mathfrak{b}$ is the inner product in $H_{0}^{1}\left(\Omega,|x|^{2-N}\,dx\right)$ , and $u\in L^{2}(\Omega,|x|^{2-N}\,dx)\mapsto|x|^{-s_{1}}u\in L^{2}(\Omega,\,d\mu)$ is an isometry which maps $D(\mathfrak{b})$ onto $D(\mathfrak{a})$. The operator $-\tilde{L}$ associated to $\mathfrak{b}$ then satisfies

[TABLE]

hence

[TABLE]

Clearly $-\tilde{L}$ is non-negative and self-adjoint in $L^{2}\left(\Omega,|x|^{2-N}\,dx\right)$. The semigroup $\left(e^{z\tilde{L}}\right)_{z\in\mathbb{C}_{+}}$ is analytic, submarkovian and satisfies

[TABLE]

where $\lambda_{1}>0$ is the first eigenvalue of $-\tilde{L}$, which is positive since $-\tilde{L}$ is non-negative and invertible, by the similarity with $-L$.

The following lemma is a special case of Caffarelli-Kohn-Nirenberg inequalities and we refer to [21, Lemma 3.2] for a short proof. It is used to prove the $L^{1}$-$L^{\infty}$ bound of the semigroup.

Lemma 5.8

Let $\sigma\in\mathbb{R}\setminus\{-N\}$. Then for every $q\in(2,\infty)$ satisfying $\frac{1}{q}\geq\frac{1}{2}-\frac{1}{N}$, there exists $C_{q}>0$ such that for every $u\in C^{2}_{c,0}(\Omega)$,

[TABLE]

In particular, when $\Omega$ is bounded and $\sigma\leq 0$, then

[TABLE]

Proposition 5.9

Let $D=0$ and $\Omega$ be bounded. Then the semigroup $T_{\Omega}(t)$ generated by $L$ in $L^{2}(\Omega,d\mu)$ has an heat kernel $p(t,x,y)$, with respect to the Lebesgue measure, which satisfies, for every $\epsilon>0$ and some constant $C_{\epsilon}>0$

[TABLE]

The Green function $G(x,y)$ of $L$, again written with respect to the Lebesgue measure, satisfies for some constant $C,k>0$,

[TABLE]

where if $N>2$

[TABLE]

and if $N=2$

[TABLE]

Proof. We make use of the results and methods of [8, Sections 3,4], pointing out the appropriate changes due to the boundedness of $\Omega$. The $L^{p}$ norms used here refer to the measure $|x|^{2-N}\,dx$.

The ultracontractivity estimate for $t\geq 0$

[TABLE]

follows from Lemma 5.8 with $\sigma=2-N\leq 0$ and any fixed $q$ as in its statement, using [32, Theorem 6.2].

Since $\tilde{L}$ is self-adjoint we have also $\|e^{it\tilde{L}}\|_{2\to 2}\leq 1$ for $t\in\mathbb{R}$. Using $\|T^{*}T\|_{1\to\infty}=\|T\|_{1\to 2}^{2}$ and recalling (54), we obtain for $t>0,\,s\in\mathbb{R}$,

[TABLE]

This proves

[TABLE]

The Dunford-Pettis criterion yields the existence of a kernel $\tilde{p}$ such that, for $z\in\mathbb{C}_{+}$,

[TABLE]

and

[TABLE]

By classic results, see e.g. [18, Theorem 7.20, page 208], $\tilde{p}$ is a continuous function of $(z,x,y)\in\mathbb{C}_{+}\times\Omega\setminus\{0\}\times\Omega\setminus\{0\}$, it is symmetric in $x,y$ and it is holomorphic in $z$.

Furthermore, the same argument as in [8, Theorem 4.4] proves that the family $\{e^{t\tilde{L}}:\ t\geq 0\}$ satisfies the Davies-Gaffney estimate in $L^{2}\left(\Omega,|x|^{2-N}\,dx\right)$ that is

[TABLE]

for all $t>0$, $U_{1}$, $U_{2}$ open subsets of $\Omega\setminus\{0\}$, $f_{i}$ in $L^{2}\left(U_{i},|x|^{2-N}\,dx\right)$ and $r:=d(U_{1},U_{2})$. Applying [9, Theorem 4.1] to the operator $-\frac{\lambda_{1}}{3}-\tilde{L}$ we get, for every $z\in\mathbb{C}_{+}$, $x,y\in\Omega\setminus\{0\}$ (here the joint continuity of $\tilde{p}(t,\cdot,\cdot)$ is used)

[TABLE]

Recalling (53), the heat kernel $p$ of $L$, taken with respect the Lebesgue measure, satisfies

[TABLE]

and (55) follows.

To prove the second statement we observe that

[TABLE]

where we put $h(t)=t^{-\frac{N}{2}}e^{-\frac{\lambda_{1}}{3}t}\exp\left(-\dfrac{|x-y|^{2})}{(4+\epsilon)t}\right)$. Using [13, Formula (29), page 146], we have

[TABLE]

where the $K_{\nu}$ is the modified Bessel function and satisfies the following asymptotics, see e.g., [1, 9.6 and 9.7].

[TABLE]

Inserting this relations into (57) we get if $N>2$

[TABLE]

and if $N=2$

[TABLE]

5.2 Main result

As in the cases $\Omega=B$ or $\Omega=\mathbb{R}^{N}$, we define

[TABLE]

Our main result is the following

Theorem 5.10

Let $N\geq 2$, $1<p<\infty$ and assume that (44) holds. Rellich inequalities

[TABLE]

hold if and only if

[TABLE]

Proof. We first prove that, if $\alpha$ is as in the assumptions, Rellich inequalities are true. Let $B_{R}\subseteq\mathbb{R}^{N}$ be such that $\Omega\subset B_{R}$, $R>0$. Without loss of generality we may assume that $R=1$. For a sufficiently small $\delta>0$, set

[TABLE]

We take a linear extension operators $E:W^{2,p}(\Omega)\to W^{2,p}_{0}(B)$ such that

[TABLE]

and let $u\in C_{c,0}^{2}(\Omega)$. By Theorem 4.2 and since all coefficients are bounded in $B\setminus\Omega$, we have

[TABLE]

By the interior estimates for elliptic equations (see [19, Theorem 1, Sec. 4, Ch.9])

[TABLE]

To conclude the proof we show that $\|u\|_{p,\Omega_{2\delta}}\leq C\||x|^{\alpha}Lu\|_{p,\Omega}$.

Set $f=-|x|^{\alpha}Lu$. Since $u\in C_{c,0}^{2}(\Omega)\subset D(L)$ and $L$ is invertible, by Proposition 5.4, then $u=(-L)^{-1}\frac{f}{|x|^{\alpha}}$. Using the estimates proved in Section 5.1, the Green function $G$ of $L$ in $\Omega$ satisfies

[TABLE]

where $G_{0}$ is defined in Proposition 5.6 when $D>0$ and in Proposition 5.9 when $D=0$.

Let us suppose preliminarily that $D>0$. Then, for $x\in\Omega_{2\delta}$,

[TABLE]

Since $x\in\Omega_{2\delta}$, there exists $a>0$ such that $|x|\geq a>0$. Consider first $u_{1}(x)$.

[TABLE]

For the first term $I_{1}(x)$ we get

[TABLE]

which therefore implies $\|I_{1}\|_{p,\Omega_{2\delta}}\leq C\|f\|_{p,\Omega}$. For $I_{2}(x)$, observe that $|x-y|\geq\frac{a}{2}$, therefore $|x||y|\geq|x-y|^{2}\geq\frac{a^{2}}{4}$ and, recalling that $\Omega\subset B$, $|y|\geq\frac{a^{2}}{4|x|^{2}}\geq\frac{a^{2}}{4}$. It follows that

[TABLE]

and $\|I_{2}\|_{p,\Omega_{2\delta}}\leq C\|f\|_{p,\Omega}$. Then $\|u_{1}\|_{p,\Omega_{2\delta}}\leq C\|f\|_{p,\Omega}$. Consider now $u_{2}(x)$; since $|x|\geq a$,

[TABLE]

As before, consider separately

[TABLE]

and

[TABLE]

Concerning $J_{1}$, we have $\frac{(|x||y|)^{\sqrt{D}}}{|x-y|^{2\sqrt{D}}}\leq 1$ and $(|x||y|)^{-\frac{N-2}{2}+\frac{c}{2}-\alpha}\leq C$, therefore

[TABLE]

and $\|J_{1}\|_{p,\Omega_{2\delta}}\leq C\|f\|_{p,\Omega}$. Finally, for $J_{2}$ we have $|x-y|\geq\frac{a}{2}$ and

[TABLE]

The last norm is finite if and only if

[TABLE]

which is equivalent to \alpha<N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}+\sqrt{D}, our assumption.

Let us suppose now that $D=0$. Then, similarly, we write for $x\in\Omega_{2\delta}$,

[TABLE]

Concerning $u_{1}$ we get

[TABLE]

which implies $\|u_{1}\|_{p,\Omega_{2\delta}}\leq C\|f\|_{p,\Omega}$ as before. Finally, for $u_{2}$ we have

[TABLE]

which is finite if and only if \alpha<N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}, our assumption when $D=0$ (note that in this case $c-s_{1}=-\frac{N-2}{2}+\frac{c}{2}$).

Let us now show that the conditions on $\alpha$ are also necessary and here we do not need to distinguish between $D>0$ and $D=0$.

When \alpha=N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}-\sqrt{D+\lambda_{n}},\ n\in\mathbb{N}_{0}, Rellich inequalities fail, by Example 2.3. Let \alpha>N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}+\sqrt{D} and assume, as above, that $\Omega\subseteq B$. Let $s_{2}$ be defined in (10) and

[TABLE]

Then (see Proposition 2.4), $Lu=0$ and $|x|^{\alpha-2}u\in L^{p}\left(B\right)$ since $\alpha-2-s_{2}>-\frac{N}{p}$. On the other hand $u\notin D_{p,\alpha}(\Omega)$ since $u$ does not vanish on $\partial\Omega$. We use Lemma 5.5 and, for $0<\epsilon<1$, let $v_{\epsilon}\in C^{2}\left(\Omega\setminus\bar{B_{\epsilon}}\right)\cap C\left(\bar{\Omega}\setminus B_{\epsilon}\right)$ satisfy

[TABLE]

Since $s_{2}>s_{1}$, one has, by construction, $v_{\epsilon}(x)\leq|x|^{-s_{1}}$ for every $x\in\partial\Omega\cup\partial B_{\epsilon}$. It follows from Lemma 5.5 that $0\leq v_{\epsilon}(x)\leq|x|^{-s_{1}}$ in $\Omega\setminus B_{\epsilon}$. Using local elliptic regularity and a standard diagonal argument, we prove that $v_{\epsilon}$ converges, up to subsequences, to a function $v$ in $W^{2,p}_{loc}\left(\Omega\setminus\{0\}\right)$. By construction $v$ satisfies $v=u$ in $\partial\Omega$ and $Lv=0$, $0\leq v\leq|x|^{-s_{1}}$ in $\Omega\setminus\{0\}$; in particular $|x|^{\alpha-2}v\in L^{p}(\Omega)$, since $\alpha-2-s_{1}>-\frac{N}{p}$. Then the function $w:=u-v$ satisfies $w=0$ in $\partial\Omega$ and $Lw=0$ in $\Omega\setminus\{0\}$. In particular $w\in D_{p,\alpha}(\Omega)$ but Rellich inequalities (2) fail for $w$.

5.3 Rellich inequalities in exterior domains

Let $V\subseteq\mathbb{R}^{N}$ be an exterior domain (that is the complement of a bounded set) which is also open, connected and does not not containing the origin. We also assume that $\partial V$ is $C^{2,\beta}$. As before, we define

[TABLE]

Proposition 5.11

Let $N\geq 2$, $1<p<\infty$ and assume that (44) holds. Rellich inequalities

[TABLE]

hold if and only if

[TABLE]

*When $V=B_{r}^{c}$ the same result holds when $D<0$ (replacing the square roots with their real parts) and for $p=1,\infty$. *

Proof. For $u\in D_{p,\alpha}(V)$, we use the Kelvin transform $u(x)=|x|^{2-N}v\left(\frac{x}{|x|^{2}}\right)$ where $v$ is defined in the bounded domain $\Omega=\left\{x\in\mathbb{R}^{N}:x/|x|^{2}\in V\right\}$, which contains the origin. Then by elementary computation

[TABLE]

where

[TABLE]

In particular its discriminant $\tilde{D}$ satisfies $\tilde{D}=D$. Setting $y=x/|x|^{2}$, $dx=|y|^{-2N}dy$ we see that the inequality

[TABLE]

is equivalent to

[TABLE]

with the same constant $C$ and $\tilde{\alpha}:=-\alpha+N+2-2N/p$. The statements then follow from Theorems 4.2 and 5.10.

6 Critical cases in $L^{p}(\mathbb{R}^{N})$

In this section we assume that $\Omega$ coincides with $\mathbb{R}^{N}$ and prove that, when Rellich inequalities fail, modified inequalities which include logarithmic terms are still valid. The situation is similar to Hardy inequality, when the classical one fails. By Theorem 4.2 Rellich inequalities fail in $\mathbb{R}^{N}$ if and only if

[TABLE]

or equivalently when

[TABLE]

To study these cases we need an unweighted one dimensional result for a general second order operator on the half line.

Proposition 6.1

Consider the operator with real constant coefficients

[TABLE]

in $(0,\infty)$ and fix $a>0$. If $\beta\neq 0$, then for every $v\in C_{c}^{2}(\mathbb{R}_{+})$,

[TABLE]

for $1<p\leq\infty$ and

[TABLE]

for $\varepsilon>0$. The weaker inequalities

[TABLE]

and

[TABLE]

hold when $\beta=0$.

In the proof we need the following lemma.

Lemma 6.2

Let $v\in C_{c}^{2}(\mathbb{R}_{+})$ and $f=\Gamma v$, with $\beta\neq 0$. Then

[TABLE]

Moreover, one has

[TABLE]

Proof. Identity (65) holds since $1,e^{\beta s}$ are solution of the adjoint $\Gamma^{*}=D^{2}-\beta D$. If $w$ is the right hand side of (64), by the variation of constants formula, $\Gamma w=f$ and, by (65) and since $f$ has a compact support, $w$ has a compact support, too. On the other hand, $\Gamma(v-w)=0$, hence $v-w=c_{1}+c_{2}e^{\beta s}$. Since both have a compact support in $(0,\infty)$, then $c_{1}=c_{2}=0$.

Proof. (Proposition 6.1) Let $f:=\Gamma v$ and assume first that $\beta=0$. Then

[TABLE]

and (62) follows from Hardy inequality. When $p=1$ we write

[TABLE]

and (63) is immediate.

We assume now that $\beta\neq 0$ and use (64)

[TABLE]

Since (65) holds, then

[TABLE]

if $1<p\leq\infty$, by Hardy inequality. When $p=1$, then $|v_{2}(s)|\leq C\|f\|_{1}$. This shows that (60), (61) hold for $v_{2}$. Since by (65)

[TABLE]

the estimate

[TABLE]

follows from Young’s inequality for every $1\leq p\leq\infty$ and concludes the proof.

In the following theorem we concentrate on the singularity at 0, hence we consider only $C^{2}$-functions vanishing on a neighbourhood of the origin and with a fixed common support which can be assumed to be $B_{R/2}$. We set

[TABLE]

Theorem 6.3

Assume that

[TABLE]

for some $n\in\mathbb{N}_{0}$ or, equivalently, that (59) holds.

Then for $1<p\leq\infty$ there exists a positive constant $C$, independent of $R$, such that for every $u\in{\cal D}_{R}$

[TABLE]

[TABLE]

When $p=1$, inequalities (66) and (67) hold with $|\log|R^{-1}x||^{-2}$ and $|\log|R^{-1}x||^{-1}$ replaced by $|\log|R^{-1}x||^{-2-\varepsilon}$ and $|\log|R^{-1}x||^{-1-\varepsilon}$, respectively.

Proof. By scaling we may assume that $R=1$. By Theorem 4.2, Rellich inequalities hold in $D_{p,\alpha}(\mathbb{R}^{N})\cap L^{p}_{\neq n}$. Then (67) hold in ${\cal D}_{1}\cap L^{p}_{\neq n}$, since the singularity at [math] is weaker and $u$ has support in $B_{1/2}$. Since, by Lemma 3.9

[TABLE]

and $L$ preserves both $L^{p}_{n}(\mathbb{R}^{N})$ and $L^{p}_{\neq n}(\mathbb{R}^{N})$, we have to show that (66) or (67) or their variants for $p=1$ hold in ${\cal D}_{1}\cap L^{p}_{n}(\mathbb{R}^{N})$.

Let $u(\rho,\omega)=c(\rho)P(\omega)$, where $P$ is a fixed spherical harmonic of order $n$. Using the transformation $c(\rho)=\rho^{-\alpha+2-\frac{N}{p}}v(-\log\rho)$ we have

[TABLE]

since $\gamma_{p}(\alpha,c)+b+\lambda_{n}=0$. At this point we apply Proposition 6.1 with $\beta=2\alpha-2-N+\frac{2N}{p}-c$ after noticing that

[TABLE]

Observe that, since \alpha=N\Bigl{(}\frac{1}{2}-\frac{1}{p}\Bigr{)}+1+\frac{c}{2}\pm\textrm{\emph{Re}\,}\sqrt{D+\lambda_{n}}, then $\beta\neq 0$ if and only if $D+\lambda_{n}>0$.

7 Best constants and remainder terms

When $D:=b+\left(\frac{N-2+c}{2}\right)^{2}>0$ and

[TABLE]

we have seen in Proposition 2.2 that Rellich inequalities (2) hold in $D_{p,\alpha}(\Omega)$ with the best constant

[TABLE]

As usual, $\Omega$ is an open bounded and connected set containing [math] and with a smooth boundary, or $\Omega=\mathbb{R}^{N}$. Best constants are not known in other cases, except for $p=2$ or in special subspaces, see [24].

A direct proof that, in the above range, the constant $C$ is optimal can be achieved by truncating the function $u(x)=|x|^{2-\alpha-N/p}$ as in Example (2.3).

Lemma 7.1

Assume $1<p<\infty$. Under the above assumption on $\alpha$, Rellich inequalities hold in $D_{p,\alpha}(\mathbb{R}^{N})\cap L^{p}_{\geq 1}(\mathbb{R}^{N})$ with a constant $C_{1}>C$.

Proof. According to equation (8) of Section 2, we have to show that the inequality

[TABLE]

holds for a suitable $C_{1}>C$. We revisit Theorem 3.24 where, we recall, $\mu=\lambda-\omega_{p}$ and $\lambda=\mu+\omega_{p}=b+\gamma_{p}(\alpha,c)=C$ (see also Lemma 4.1). Theorem 38 holds in $D_{p,max}(\mathbb{R}^{N})$ with a suitable $\omega_{p}^{1}>\omega$, by the results in Section 2 of [20], see in particular Proposition 2.8 and Remark 2.9. with $\beta=0$ therein. It follows that $\mu=\lambda-\omega=\lambda_{1}-\omega_{p}^{1}$ and $\lambda_{1}>\lambda$. Then estimate (38) holds with $\lambda_{1}>\lambda=C$.

The remainder term can arise, therefore, when considering radial functions. To deal with them, we need the following auxiliary result.

Lemma 7.2

Let $1<p<\infty$ and $\Gamma=D^{2}+\beta D-\lambda$ be an operator with real constant coefficients on $(0,\infty)$. Then for every $v\in C_{c}^{2}(0,\infty)$ and $\lambda>0$

[TABLE]

Proof. We have

[TABLE]

Since $v^{\prime}v|v|^{p-2}$ is the derivative of $p^{-1}|v|^{p}$, the last integral vanishes. By Hardy inequality of Proposition 3.23, with $N=1,\beta=0$ we have

[TABLE]

and therefore

[TABLE]

Let

[TABLE]

then from $\lambda A^{p}+B^{p}\leq CA^{p-1}$ we get $\lambda A\leq C$ and

[TABLE]

The main result of this section is stated below. As in the previous section we formulate it for functions belonging to

[TABLE]

Theorem 7.3

Let $1<p<\infty$, $D:=b+\left(\frac{N-2+c}{2}\right)^{2}>0$ and

[TABLE]

If $C$ is the best constant defined in (68), then there exists $c>0$, independent of $R$, such that for every $u\in{\cal D}_{R}$

[TABLE]

Proof. By scaling we may assume that $R=1$. If $u\in{\cal D}_{1}$, we split $u=u_{0}+u_{1}$, where $u_{0}$ is radial and $u_{1}\in L^{p}_{\geq 1}(\mathbb{R}^{N})$. By Lemma 7.1, inequality (69) holds for $u_{1}$.

For $u_{0}$ we proceed as in Theorem 6.3 writing $u_{0}(\rho)=\rho^{-\alpha+2-\frac{N}{p}}v(-\log\rho)$. Then

[TABLE]

Next we use Lemma 7.2 with $\lambda=\gamma(\alpha,c)+b=C$ to obtain

[TABLE]

The general case now follows, since $L_{0}(\mathbb{R}^{N}),L_{\geq 1}(\mathbb{R}^{N})$ are invariant under $L$ and under multiplication by radial weights and since $|u|^{p}_{p}:=\|u_{0}\|_{p}^{p}+\|u_{1}\|_{p}^{p}$ is an equivalent norm on $L^{p}(\mathbb{R}^{N})$.

8 Appendix

8.1 Approximation on Sobolev spaces on domains

Let $V$ be a $C^{2,\beta}$ bounded connected open subset of $\mathbb{R}^{N}$ and let $A$ be a uniformly elliptic operator $A=\mbox{tr}(A(x)D^{2})+c(x)\cdot\nabla-b(x)$, with $C^{\beta}$ coefficients, endowed with Dirichlet boundary conditions. We recall that for $1<p<\infty$

[TABLE]

whereas for $p=1$

[TABLE]

and for $p=\infty$

[TABLE]

both endowed with the graph norm.

Proposition 8.1

Under the above assumptions the set

[TABLE]

is dense in $D_{p}(\Omega)$ for every $1\leq p\leq\infty$.

Proof. Let $\lambda>0$ such that $\lambda-A$ is invertible from $D_{p}(\Omega)$ to $L^{p}(\Omega)$. If $u\in D_{p}(\Omega)$, $f=\lambda u-Au$ and $(f_{n})\subset C^{\beta}(\Omega)$ tends to $f$ in $L^{p}(\Omega)$, then $u_{n}=(\lambda-A)^{-1}f_{n}$ belongs to $C^{2,\beta}(\Omega)$, by the Schauder theory, vanishes at $\partial\Omega$ and approximates $u$ in the graph norm.

The following partition of unity of $\Omega$ has been used several times.

Proposition 8.2

Let $0\leq\beta\leq 1$ and let $\Omega$ be a bounded connected open subset of $\mathbb{R}^{N}$ whose boundary $\partial\Omega$ is of class $C^{2,\beta}$. Then there exist $\delta>0$ such that the distance function $x\mapsto\mbox{dist}(x,\partial\Omega)$ is $C^{2,\beta}$ over the set

[TABLE]

In particular $K_{\delta}$ and the subset

[TABLE]

have $C^{2,\beta}$ boundary. Furthermore there exists an open subset $\Omega_{0}\subset\subset\Omega$ for which $\overline{\Omega}=\overline{\Omega}_{\delta}\cup\Omega_{0}$ and there exists a partition of unity $\{\eta_{\delta}^{2},\eta_{0}^{2}\}$ such that

• (i)

$\eta_{\delta}\in C_{c}^{\infty}(K_{\delta})$, $0\leq\eta_{\delta}\leq 1$, $\eta_{\delta}=1$ in $\overline{\Omega}_{\frac{\delta}{2}}$;

• (ii)

$\eta_{0}\in C_{c}^{\infty}(\Omega_{0})$, $0\leq\eta_{\delta}\leq 1$;

• (iii)

$\eta_{\delta}^{2}+\eta_{0}^{2}=1$* in $\overline{\Omega}$.*

Proof. [16, Lemma 14.16] proves the case $\beta=0$ and that, for sufficiently small $\delta$, for every point $x\in K_{\delta}$ there exist a unique $y\in\partial\Omega$ such that $|x-y|=d(x,\partial\Omega)$. The result for $\beta>0$ then follows by [30]. The existence of such a partition of unity is a standard result.

8.2 Some results on spectral theory

We collect some definitions and results from spectral theory which are used throughout the paper. Let $X$ be a Banach space and let $A$ be a closed operator $A:D(A)\subseteq X\to X$. The spectrum of $A$ is denoted by $\sigma(A)$ and the resolvent set $\mathbb{C}\setminus\sigma(A)$ by $\rho(A)$.

Definition 8.3

The set

[TABLE]

is called the point spectrum of $A$. Moreover each $\lambda\in P\sigma(A)$ is called an eigenvalue and each $0\neq x\in D(A)$ satisfying $(\lambda-A)x=0$ is an eigenvector of $A$ (corresponding to $\lambda$).

Definition 8.4

The set

[TABLE]

is called the approximate point spectrum of $A$. Obviously $P\sigma(A)\subseteq A\sigma(A)$.

Definition 8.5

The set

[TABLE]

is called the residual spectrum of $A$.

Note that $P\sigma(A)\subset A\sigma(A)$, that $P\sigma(A)$ and $R\sigma(A)$, as well as $A\sigma(A)$ and $R\sigma(A)$ may overlap and that $\sigma(A)=A\sigma(A)\cup R\sigma(A)$.

Lemma 8.6

([12, Lemma 1.9, Chapter IV]) A number $\lambda\in\mathbb{C}$ belongs to $A\sigma(A)$ if and only if there exists a sequence $(x_{n})_{n\in\mathbb{N}}\subset D(A)$, called an approximate eigenvector, such that $\|x_{n}\|=1$ and $\lim_{n\to\infty}\|Ax_{n}-\lambda x_{n}\|=0$.

The following result is an elementary consequence of the previous Lemma.

Proposition 8.7

The following properties are equivalent

• (i)

There exists $C>0$ such that

[TABLE]

• (ii)

$\lambda$* does not belong to the approximate point spectrum of $A$.*

The next Proposition implies that $A\sigma(A)$ is never empty.

Proposition 8.8

[12, Proposition 1.10, Chapter IV]** The topological boundary of the spectrum is contained in the approximate point spectrum.

8.3 Spectrum of a second order ordinary differential operator

We present the following elementary result on the spectrum of the second order ordinary differential operator $B=D^{2}+\beta D$ in $L^{p}([0,\infty[)$, endowed with Dirichlet boundary condition at 0, that is

[TABLE]

As usually $L^{\infty}([0,+\infty))$ stands for $C_{0}^{0}([0,+\infty)$. Here $\beta\in\mathbb{R}$ and we recall that

[TABLE]

Note that

[TABLE]

and that

[TABLE]

Observe that the spectrum of $B$ in $L^{p}(\mathbb{R})$ is given by $\cal P$ and consists of approximate eigenvalues. This can be seen by noticing that the spectrum is independent of $p$ and using the Fourier transform in $L^{2}(\mathbb{R})$.

For $\lambda\in\mathbb{C}$, we consider the solutions of the homogeneous equation $\lambda u-Bu=0$ given by $e^{\mu_{i}t}$, $i=1,2$ where

[TABLE]

When $\lambda=-\beta^{2}/4$ then $\mu_{1}=\mu_{2}=-b/2$ and we substitute $e^{\mu_{2}t}$ with $te^{-\frac{\beta}{2}t}$.

Lemma 8.9

The inequality ${\rm Re}\sqrt{\beta^{2}+4\lambda}<|\beta|$ holds if and only if $\lambda\in\overset{\mathrm{o}}{\cal Q}$. Similarly, ${\rm Re}\sqrt{\beta^{2}+4\lambda}>|\beta|$ if and only if $\lambda\not\in{\cal Q}$ and ${\rm Re}\sqrt{\beta^{2}+4\lambda}=|\beta|$ if and only if $\lambda\in{\cal P}$. Here $\sqrt{z}$ denotes any square root of $z$ with non negative real part.

Proof. If $\sqrt{\beta^{2}+4\lambda}=x+iy$, with $x\geq 0$, then $4\lambda=(x^{2}-y^{2}-\beta^{2})+2ixy$ and $x=|\beta|$ if and only if $({\rm Im}\lambda)^{2}=-\beta^{2}{\rm Re}\lambda$. The other cases are similar.

Proposition 8.10

The spectrum of $B=D^{2}+\beta D$ in $L^{p}([0,+\infty))$, with Dirichlet boundary condition at 0, is given by $\sigma(B)={\cal Q}$. More specifically we have

• (i)

if $\beta>0$, then  $\sigma(B)=A\sigma(B)={\cal Q}$, $P\sigma(B)\supset\overset{\mathrm{o}}{\cal Q}$;

• (ii)

if $\beta=0$, then  $\sigma(B)=A\sigma(B)=(-\infty,0]$;

• (iii)

if $\beta<0$, then  $A\sigma(B)={\cal P}$,  $R\sigma(B)\setminus A\sigma(B)=\overset{\mathrm{o}}{\cal Q}$.

Proof. Let us prove preliminarily that $\cal{Q}^{\mbox{c}}\subseteq\rho(B)$ in all cases. If $\lambda\notin\cal{Q}$ by the lemma above ${\rm Re}\sqrt{\beta^{2}+4\lambda}>|\beta|$, hence ${\rm Re}\,\mu_{1}<0<{\rm Re}\,\mu_{2}$. It is then easy to see that $\lambda-B$ is invertible and that its inverse is given by the Green function

[TABLE]

where $u_{1}(t)=e^{\mu_{2}t}-e^{\mu_{1}t}$, $u_{2}(t)=e^{\mu_{1}t}$ and $W(t)=(\mu_{1}-\mu_{2})e^{(\mu_{1}+\mu_{2})t}=(\mu_{1}-\mu_{2})e^{-\beta t}$ is their Wronskian.

Let us suppose now that $\lambda\in\overset{\mathrm{o}}{\cal Q}$ and assume first $\beta>0$. Then ${\rm Re}\sqrt{\beta^{2}+4\lambda}<\beta$ and ${\rm Re}\mu_{1}\leq{\rm Re}\mu_{2}<0$. It follows that $\lambda$ is an eigenvalue with eigenfunction $u(t)=e^{\mu_{1}t}-e^{\mu_{2}t}$ (or $te^{-\frac{\beta}{2}t}$ when $\lambda=-\beta^{2}/4$). This proves that $\overset{\mathrm{o}}{\cal Q}\subseteq P\sigma(B)$ and case (i) is done, since the boundary of the spectrum is always contained in the approximate point spectrum, see Proposition 8.8.

Assume now $\beta<0$ and still that $\lambda\in\overset{\mathrm{o}}{\cal Q}$. Then ${\rm Re}\sqrt{\beta^{2}+4\lambda}<-\beta$ and $0<{\rm Re}\,\mu_{1}\leq{\rm Re}\,\mu_{2}$, hence $\lambda-B$ is injective. Moreover, $\lambda-B$ is invertible with a continuous inverse from its domain onto the closed subspace

[TABLE]

(with the usual change here and in what follows if $\lambda=-\beta^{2}/4$).

Indeed if $u\in D(B)$ set $f=(\lambda-B)u$ and $B^{*}u=u^{\prime\prime}-\beta u^{\prime}$. Since $(e^{-\mu_{1}s}-e^{-\mu_{2}s})(0)=0$ and $(\lambda-B^{*})(e^{-\mu_{1}s}-e^{-\mu_{2}s})=0$ , one has

[TABLE]

On the other hand, if $f\in L^{p}([0,+\infty[)$ satisfies $\int_{0}^{\infty}f(e^{-\mu_{1}s}-e^{-\mu_{2}s})ds=0$, by the variation of constants method, one finds that

[TABLE]

satisfies $u(0)=0$, $u\in D(B)$ and $(\lambda-B)u=f$.

This proves that $\lambda-B$ is injective and that $rg(\lambda-B)=X\subset L^{p}([0,\infty[)$ which, recalling Definitions 8.4, 8.5, gives $\overset{\mathrm{o}}{\cal Q}\subseteq R\sigma(B)\setminus A\sigma(B)$. Using again Proposition 8.8, (iii) is proved.

When $\beta=0$ one sees that $A\sigma(B)\supset(-\infty,0]$ by truncating the functions $\sin(\sqrt{-\lambda}\,t)$.

An analogous result can be obviously proved in $L^{p}(]-\infty,0])$ using the isometry

[TABLE]

Proposition 8.11

The spectrum of $B=D^{2}+\beta D$ in $L^{p}(]-\infty,0])$, with Dirichlet boundary condition at 0, is given by $\sigma(B)={\cal Q}$. More specifically we have

• (i)

if $\beta<0$, then  $\sigma(B)=A\sigma(B)={\cal Q}$, $P\sigma(B)\supset\overset{\mathrm{o}}{\cal Q}$;

• (ii)

if $\beta=0$, then  $\sigma(B)=A\sigma(B)=(-\infty,0]$;

• (iii)

if $\beta>0$, then  $A\sigma(B)={\cal P}$,  $R\sigma(B)\setminus A\sigma(B)=\overset{\mathrm{o}}{\cal Q}$.