Isoperimetric inequalities for Bergman analytic content

TL;DR

This paper explores isoperimetric inequalities related to the Bergman p-analytic content of planar domains, linking it to torsional rigidity and characterizing cases of equality.

Contribution

It establishes new isoperimetric inequalities for Bergman p-analytic content and analyzes conditions for equality with bounds.

Findings

Derived inequalities connecting Bergman p-analytic content and torsional rigidity.

Characterized domains where equality holds in these inequalities.

Extended concepts to higher dimensions using harmonic vector fields.

Abstract

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $\Omega $ measures the $L^{p}(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^{p}(\Omega )$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

Equations

$$\sqrt{\rho (\Omega )}\le {\lambda }_{A^2}(\Omega )\le \frac{m(\Omega )}{\sqrt{2\pi }},$$

$${\lambda }_{A_p}(\Omega )=\inf \{{\Vert x-f\Vert }_{{\mathcal{L}}_p}:f\in A_p(\Omega )\}(1\le p<\infty ),$$

$${\mathcal{L}}_p={\mathcal{L}}_p(\Omega )={\left(L^p(\Omega )\right)}^N,\Vert f{\Vert }_{{\mathcal{L}}_p}={\left({\int }_{\Omega }{\Vert f\Vert }^{p}dm\right)}^{1/p}$$

$$\frac{\partial f_j}{\partial x_k}-\frac{\partial f_k}{\partial x_j}=0\text{ for all }j,k\in \{1,...,N\}\text{ on }\Omega .$$

$$Q_q(\Omega )=\underset{u\in W_0^{1,q}(\Omega )\backslash \{0\}}{\sup }\frac{N}{\Vert \nabla u{\Vert }_{{\mathcal{L}}_q}}{\int }_{\Omega }udm(1<q\le \infty ).$$

$$Q_q(\Omega )\le {\lambda }_{A_p}(\Omega ).$$

$${\lambda }_{A_p}(\Omega )\le Q_q(B(r_{\Omega })).$$

$$Q_q(B(r))={\left(\frac{N}{N+p}m(B)\right)}^{1/p}{r}^{1+N/p}(1\le p<\infty ).$$

$${\lambda }_{A_p}(\Omega )\le {\left(\frac{N}{N+p}m(B)\right)}^{1/p}{r}_{\Omega }^{1+N/p},$$

$$Q_q(\Omega )\le {\lambda }_{A_p}(\Omega )\le Q_q(B(r_{\Omega })).$$

$${\lambda }_{B_p}(\Omega )$$

$${\lambda }_{D_p}(\Omega )$$

$$B_p(\Omega )$$

$$D_p(\Omega )$$

$${\lambda }_{D_p}(\Omega )\le {\lambda }_{A_p}(\Omega )\le {\lambda }_{B_p}(\Omega ).$$

$$\left\{\begin{array}[]{cc}-\Delta_{q}w_{q}=1&\text{in }\Omega\\ w_{q}=0&\text{on }\partial\Omega\end{array}\right.,$$

$$Q_q(\Omega )=\frac{N}{\Vert \nabla u{\Vert }_{{\mathcal{L}}_q}}{\int }_{\Omega }udm.$$

$$Q_q(\Omega )$$

$$f(t)={\left(Q_q(\Omega )\right)}^q{\int }_{\Omega }\Vert \nabla (u+t\varphi ){\Vert }^{q}dm-{\left(N{\int }_{\Omega }(u+t\varphi )dm\right)}^q(t\in \mathbb{R}).$$

$${\left(Q_q(\Omega )\right)}^q{\int }_{\Omega }{\Vert \nabla u\Vert }^{q-2}\nabla u\cdot \nabla \varphi dm-N^q{\left({\int }_{\Omega }udm\right)}^{q-1}{\int }_{\Omega }\varphi dm=0\left(\varphi \in C_c^{\infty }(\Omega )\right),$$

$${\int }_{\Omega }\left({\left(Q_q(\Omega )\right)}^q{\Delta }_{q}u+N^q{\left({\int }_{\Omega }udm\right)}^{q-1}\right)\varphi dm=0\left(\varphi \in C_c^{\infty }(\Omega )\right).$$

$${\int }_{\Omega }{w}_{q}dm={\int }_{\Omega }{w}_q(-{\Delta }_q{w}_q)dm={\int }_{\Omega }{\Vert \nabla w_q\Vert }^{q-2}\nabla w_q\cdot \nabla w_{q}dm={\Vert \nabla w_q\Vert }_{{\mathcal{L}}_q}^q.$$

$$\Vert \nabla u{\Vert }_{{\mathcal{L}}_q}={\left(\frac{Nm(B)}{N+p}{r}^{N+p}\right)}^{1/q},{\int }_{\Omega }udm=\frac{m(B)}{N+p}{r}^{N+p}.$$

$$Q_q(B(r))=\frac{N}{\Vert \nabla u{\Vert }_{{\mathcal{L}}_q}}{\int }_{\Omega }udm={\left(\frac{N}{N+p}m(B)\right)}^{1/p}{r}^{1+N/p}.$$

$$D_p(\Omega {)}^{\perp }=\left\{g\in {\mathcal{L}}_q(\Omega ):{\int }_{\Omega }f\cdot gdm=0\text{ for all }f\in D_p(\Omega )\right\};$$

$${\lambda }_{D_p}(\Omega )=Q_q(\Omega )=\frac{-1}{\Vert \nabla w_q{\Vert }_{{\mathcal{L}}_q}}{\int }_{\Omega }(x-f)\cdot \nabla w_{q}dm(f\in D_p(\Omega )).$$

$${\int }_{\Omega }{\Vert x-f_0\Vert }^{p-2}(x-f_0)\cdot gdm=0.$$

$$f\in D_p(\Omega )\iff {\int }_{\Omega }f\cdot \nabla \varphi dm=0(\varphi \in C_c^{\infty }(\Omega )).$$

$$D_p(\Omega {)}^{\perp }={\overline{\{\nabla \varphi :\varphi \in C_c^{\infty }(\Omega )\}}}^{{\mathcal{L}}_q(\Omega )}$$

$${\int }_{\Omega }f\cdot gdm=1,{\int }_{\Omega }f\cdot \nabla \varphi dm=0(\varphi \in C_c^{\infty }(\Omega )),$$

$$D_p(\Omega {)}^{\perp }=\{\nabla u:u\in W_0^{1,q}(\Omega )\}.$$

Full text

1 Introduction

00footnotetext: 2010 *Mathematics Subject Classification * 31B05. Keywords: harmonic vector field, Bergman space, isoperimetric inequality, torsional rigidity

The Bergman $p$-analytic content ($1\leq p<\infty$) of a bounded planar domain $\Omega$ was introduced by Guadarrama and Khavinson [15]. It is defined by the formula $\lambda_{A^{p}}(\Omega)=\inf_{f\in A^{p}(\Omega)}\left\|\overline{z}-f\right\|_{p}$, where $\left\|\cdot\right\|_{p}$ is the usual $L^{p}(\Omega)$-norm and $A^{p}(\Omega)$ is the Bergman space of $L^{p}(\Omega)$-integrable holomorphic functions $f$ on $\Omega$. In the case where $p=2$, Fleeman and Khavinson [8] showed that, for any simply connected domain $\Omega$ with piecewise smooth boundary,

[TABLE]

where $\rho(\Omega)$ denotes the torsional rigidity of $\Omega$ and $m$ is Lebesgue measure. Subsequently, Fleeman and Lundberg [9] showed that the left hand inequality is actually an equality for any bounded simply connected domain, and this relationship has been further exploited by Fleeman and Simanek [10]. Bell, Ferguson and Lundberg [3] established related inequalities concerning torsional rigidity and the norm of the self-commutator of a Toeplitz operator. The limiting case of Bergman $p$-analytic content where $p=\infty$ is the notion of analytic content, which has been studied for many years: see, for example, [11], [4], [1] for the case of the plane, and [16], [12] for its extension to higher dimensions.

Rewriting $\lambda_{A^{p}}(\Omega)$ as $\inf_{\phi\in A^{p}(\Omega)}\left\|z-\overline{\phi}\right\|_{p}$, we see that a natural generalization to bounded domains $\Omega$ in Euclidean space $\mathbb{R}^{N}$ ($N\geq 2$) is given by

[TABLE]

where $A_{p}(\Omega)$ denotes the space of harmonic vector fields $f=(f_{1},...,f_{N})$ in $\mathcal{L}_{p}\cap C^{1}(\Omega)$,**

[TABLE]

and $\left\|\cdot\right\|$ is the usual Euclidean norm on $\mathbb{R}^{N}$. Thus $f$ satisfies $\mathop{\mathrm{d}iv}f=0$ and $\mathop{\mathrm{c}url}f=0$, where the latter condition means that

[TABLE]

The gradient of any harmonic function is a harmonic vector field, and the converse assertion is also true when $\Omega$ is simply connected. We will assume from now on that $\Omega$ is smoothly bounded.

The purpose of this paper is to investigate isoperimetric inequalities for $\lambda_{A_{p}}(\Omega)$ $(1\leq p<\infty)$ in all dimensions, and to examine the cases of equality with the upper and lower bounds (cf. Problem 3.4 of [5]). We denote by $q$ the dual exponent of $p$, whence $1/p+1/q=1$ (or $q=\infty$ if $p=1$), and note that the dual space $\mathcal{L}_{p}^{\ast}$ can be identified with $\mathcal{L}_{q}$. When $q<\infty$ we denote by $W_{0}^{1,q}(\Omega)$ the closure of $C_{c}^{\infty}(\Omega)$ in the Sobolev space $W^{1,q}(\Omega)$; these are the functions in $W^{1,q}(\Omega)$ that have trace zero on $\partial\Omega$ (see Section 5.5 of [7]). Since any function in $W^{1,\infty}(\Omega)$ has a Lipschitz representative, it is natural to denote by $W_{0}^{1,\infty}(\Omega)$ the subset of $W^{1,\infty}(\Omega)$ comprising those functions which vanish on $\partial\Omega$. We define

[TABLE]

When $q<\infty$, the quantity $\left(Q_{q}(\Omega)\right)^{q}$ is known as the St Venant $q$-functional of $\Omega$. Its relationship with the torsional rigidity $\rho(\Omega)$ will be discussed in Section 4.

We begin with the case $p=2$, where we can add the following to the results of [8] and [9].

Theorem 1

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain, then $\lambda_{A_{2}}(\Omega)=Q_{2}(\Omega)$. Further, $\lambda_{A_{2}}(\Omega)=\sqrt{\rho(\Omega)}$ if and only if $\mathbb{R}^{N}\backslash\Omega$ is connected.

Next, we establish a lower bound for $\lambda_{A_{p}}(\Omega)$ for all $p$.

Theorem 2

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain and $p\in[1,\infty)$, then

[TABLE]

*Further, equality holds if and only if either

(a) $p=2$, or

(b) $\Omega$ is a ball or an annular region.*

The case of equality above when $p\neq 2$ is a counterpart of a recent result of Abanov, Bénéteau, Khavinson and Teodorescu [1] concerning analytic content in the plane (that is, where $p=\infty$ and $N=2$).

It remains to establish an upper bound for $\lambda_{A_{p}}(\Omega)$. Let $B(r)$ denote the open ball in $\mathbb{R}^{N}$ of centre [math] and radius $r$, and let $B=B(1)$. Further, let $r_{\Omega}>0$ be chosen so that $m(B(r_{\Omega}))=m(\Omega)$. Then, by the generalized Faber-Krahn inequality (cf. [6]), we have $Q_{q}(\Omega)\leq Q_{q}(B(r_{\Omega}))$. The result below is new in all dimensions.

Theorem 3

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain and $p\in[1,2]$, then

[TABLE]

Further, equality holds if and only if $\Omega$ is a ball.

We will see later, in Proposition 5, that the upper bound in (3) is given explicitly by****

[TABLE]

Recent work of the authors [12] shows that there is a harmonic function $h$ on $\Omega$ satisfying $\sup_{\Omega}\left\|x-\nabla h\right\|\leq r_{\Omega}$, whence $\lambda_{A_{p}}(\Omega)\leq(m(B))^{1/p}r_{\Omega}^{1+N/p}$ for general $p$. We conjecture that balls are always the extremal domains for (3); that is, the sharper estimate of Theorem 3,

[TABLE]

remains valid for all $p\in[1,\infty)$.

Theorems 2 and 3 together yield the following isoperimetric inequality for Bergman $p$-analytic content.

Corollary 4

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain and $p\in[1,2]$, then

[TABLE]

The remainder of the paper is devoted to proving the above results.

2 Existence and uniqueness of extremal functions

In the course of proving our results concerning $\lambda_{A_{p}}(\Omega)$ and $Q_{q}(\Omega)$, we are led to consider the related domain constants

[TABLE]

where

[TABLE]

Since $B_{p}(\Omega)\subset A_{p}(\Omega)\subset D_{p}(\Omega),$ we see that

[TABLE]

In this section we will prove existence and uniqueness results concerning the extremal functions for $Q_{q}(\Omega)$, $\lambda_{D_{p}}(\Omega)$, $\lambda_{B_{p}}(\Omega)$ and $\lambda_{A_{p}}(\Omega)$.

Let $\Delta_{q}$ denote the $q$-Laplacian, given by $\Delta_{q}u=\nabla\cdot\left(\left\|\nabla u\right\|^{q-2}\nabla u\right)$, where $1<q<\infty$.  We define the $q$-torsion function $w_{q}$ on $\Omega$ to be the weak solution of

[TABLE]

and note from [19] that $w_{q}\in C^{1}(\overline{\Omega})$. Further, we define $w_{\infty}(x)=\mathrm{dist}(x,\partial\Omega)$.

Proposition 5

*Let $\Omega\subset\mathbb{R}^{N}$ be a smoothly bounded domain and $p\in[1,\infty)$.

(i) There exists $u\in W_{0}^{1,q}(\Omega)$ such that*

[TABLE]

*(ii) The functions $u\in W_{0}^{1,q}(\Omega)$ which satisfy (6) are precisely the positive multiples of $w_{q}$.

(iii) $Q_{q}(\Omega)=N\left(\int_{\Omega}w_{q}dm\right)^{1/p}$. Further, $Q_{q}(\Omega)=N\|\nabla w_{q}\|_{\mathcal{L}_{q}}^{q-1}$ if $p>1$.

(iv) $Q_{q}(B(r))=\left(\dfrac{N}{N+p}m(B)\right)^{1/p}r^{1+N/p}.$*

Proof. (i) We choose a maximizing sequence $(u_{j})$ for (1) such that $\|u_{j}\|_{W_{0}^{1,q}(\Omega)}=1$ for all $j$. (The quotient in (1) is unaffected when $u$ is multiplied by a positive constant.)

Firstly, we suppose that $p>1$. In view of the Banach-Alaoglu theorem we can arrange, by taking a subsequence, that $(u_{j})$ converges weakly to some non-zero function $u\in W_{0}^{1,q}(\Omega)$. Further, by the Rellich-Kondrachov theorem (see, for example, Section 5.7 in [7]), we can arrange that $u_{j}\rightarrow u$ strongly in $L^{1}(\Omega)$. Clearly $\int_{\Omega}u~{}dm>0$. By the weak lower semicontinuity of the $\mathcal{L}_{q}$-norm,

[TABLE]

and so equality holds throughout.

If $p=1$, whence $q=\infty$, then we instead appeal to the Arzelà-Ascoli theorem to see that there is a subsequence of $(u_{j})$ that converges uniformly on $\Omega$, and make use of the fact that each $u_{j}$ can be represented by a Lipschitz function.

(ii) Suppose firstly that $p>1$. For any $\phi\in C_{c}^{\infty}(\Omega)$ we define

[TABLE]

Since $u$ is a maximizer for $Q_{q}(\Omega)$, we see that $\int_{\Omega}u~{}dm>0$ and $f^{\prime}(0)=0$, whence

[TABLE]

and so

[TABLE]

Thus $\Delta_{q}u$ is a negative constant in $\Omega$, and so $u$ is a positive multiple of $w_{q}$.

Now let $p=1$, so that $q=\infty$. In the formula (1) we can normalize to consider only those functions $u$ such that $\left\|\nabla u\right\|_{\mathcal{L}_{\infty}}=1$, whence $u$ is majorized by the Lipschitz function$\ w_{\infty}$. Further, the supremum can only be attained among functions $u$ satisfying $\left\|\nabla u\right\|_{\mathcal{L}_{\infty}}=1$ by the function $w_{\infty}$. More generally, the supremum can only be attained by a positive multiple of $w_{\infty}$.

(iii) If $p>1$, then we see from (5) that

[TABLE]

Hence, by parts (i) and (ii), $Q_{q}(\Omega)=N\|\nabla w_{q}\|_{\mathcal{L}_{q}}^{q-1}=N\left(\int_{\Omega}w_{q}dm\right)^{1/p}$.

If $p=1$, then it is immediate that $Q_{\infty}(\Omega)=N\int_{\Omega}w_{\infty}dm$.

(iv) If $\Omega=B(r)$, then $w_{q}(x)$ is clearly a multiple of $x\mapsto r^{p}-\left\|x\right\|^{p}$. Letting $u(x)=\left(r^{p}-\left\|x\right\|^{p}\right)/p$, we have $\left\|\nabla u\right\|=\left\|x\right\|^{p-1}$ and

[TABLE]

Thus, by parts (i) and (ii),

[TABLE]

As usual, we define

[TABLE]

$B_{p}(\Omega)^{\perp}$ and $A_{p}(\Omega)^{\perp}$ are defined analogously.

Proposition 6

*Let $\Omega\subset\mathbb{R}^{N}$ be a smoothly bounded domain and $p\in(1,\infty)$.

(i) There exists $f_{0}\in D_{p}(\Omega)$ such that $\lambda_{D_{p}}(\Omega)=\left\|x-f_{0}\right\|_{\mathcal{L}_{p}}$.

(ii) This function $f_{0}$ satisfies $\left\|x-f_{0}\right\|^{p-2}(x-f_{0})\in D_{p}(\Omega)^{\perp}$.

(iii) There exists $u_{0}\in W_{0}^{1,q}(\Omega)$ such that $\nabla u_{0}=-\left\|x-f_{0}\right\|^{p-2}(x-f_{0})$.

(iv) The function $u_{0}$ is a positive multiple of $w_{q}$, and*

[TABLE]

Proof. (i) We choose a sequence $(f_{j})$ in $D_{p}(\Omega)$ such that $\left\|x-f_{j}\right\|_{\mathcal{L}_{p}}\rightarrow\lambda_{D_{p}}(\Omega)$. By weak compactness we can arrange, by choosing a suitable subsequence, that $(f_{j})$ is weakly convergent to some $f_{0}$ in $\mathcal{L}_{p}$. Further, $\mathop{\mathrm{d}iv}f_{0}=0$ on $\Omega$ in the sense of distributions, so $f_{0}\in D_{p}(\Omega)$. Finally, $\lambda_{D_{p}}(\Omega)=\left\|x-f_{0}\right\|_{\mathcal{L}_{p}}$ by the weak lower semicontinuity of the norm.

(ii) For any $g\in D_{p}(\Omega)$ we can differentiate the function $t\mapsto\int_{\Omega}\left\|x-f_{0}-tg\right\|^{p}dm$ and then set $t=0$ to see that

[TABLE]

(iii) If $f\in\mathcal{L}_{p}$, then by definition,

[TABLE]

Hence

[TABLE]

since, if $g\in\mathcal{L}_{q}(\Omega)$ does not belong to the above closure, the Hahn-Banach theorem would yield the existence of $f\in\mathcal{L}_{q}^{\ast}(\Omega)\equiv\mathcal{L}_{p}(\Omega)$ such that

[TABLE]

whence $f\in D_{p}(\Omega)$ and so $g\notin D_{p}(\Omega)^{\perp}$.

We claim next that

[TABLE]

Clearly the right hand side of (10) is contained in the right hand side of (9). To see the reverse inclusion, let $(\phi_{k})$ be a sequence in $C_{c}^{\infty}(\Omega)$ such that $(\nabla\phi_{k})$ converges in $\mathcal{L}_{q}(\Omega)$. Then $(\phi_{k})$ is Cauchy in $\mathcal{L}_{q}(\Omega)$, by Poincaré’s inequality for $W_{0}^{1,q}(\Omega)$. It follows that $(\phi_{k})$ converges in $W_{0}^{1,q}(\Omega)$ to some function $u$ and $\lim_{k\rightarrow\infty}\nabla\phi_{k}=\nabla u$. Hence (10) holds, and the desired conclusion now follows from part (ii).

(iv) By the divergence theorem, (10) and Hölder’s inequality,

[TABLE]

with equality precisely when $\nabla u$ is a negative multiple of $\left\|x-f_{0}\right\|^{p-2}(x-f_{0})$. It now follows from (1) and Proposition 5(ii) that $u_{0}$ is a positive multiple of $w_{q}$, and from part (i) and (10) that (7) holds.

The next result shows that (7) also holds when $p=1$. Inequality (2) will follow from (11) in view of (4).

Proposition 7

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain and $p\in[1,\infty)$, then

[TABLE]

Proof. We know from Theorem 1 of [17] that $w_{q}\rightarrow w_{\infty}$ uniformly on $\Omega$ as $q\rightarrow\infty$. Since the function $p\mapsto\left(m(\Omega)\right)^{-1/p}\lambda_{D_{p}}(\Omega)$ is increasing, we see from Propositions 6(iv) and 5(iii) that

[TABLE]

For large $k\in\mathbb{N}$ let $v_{k}$ be a mollification of $(w_{\infty}-k^{-1})^{+}$ that belongs to $C_{c}^{\infty}(\Omega)$. Since $\nabla v_{k}\in D_{1}(\Omega)^{\perp}$ by (8), and $(\nabla v_{k})$ is boundedly convergent almost everywhere to $\nabla w_{\infty}$, we see that $\nabla w_{\infty}\in D_{1}(\Omega)^{\perp}$. Thus, by the divergence theorem,

[TABLE]

Hence $Q_{\infty}(\Omega)\leq\lambda_{D_{1}}(\Omega)$, and (11) follows in view of (12).

We note that

[TABLE]

Proposition 8

*Let $p\in[1,\infty)$.

(i) There exists $f\in B_{p}(\Omega)$ such that $\lambda_{B_{p}}(\Omega)=\left\|x-f\right\|_{\mathcal{L}_{p}}$; equivalently, there exists $u_{0}\in W^{1,p}(\Omega)$ such that $\Delta u_{0}=Nm$ in $\Omega$ and $\lambda_{B_{p}}(\Omega)=\|\nabla u_{0}\|_{\mathcal{L}_{p}}$.

(ii) The function $u_{0}$ satisfies $\|\nabla u_{0}\|^{p-2}\nabla u_{0}\in B_{p}(\Omega)^{\perp}$.

(iii) The function $f_{0}=\|\nabla u_{0}\|^{p-2}\nabla u_{0}\in\mathcal{L}_{q}$ satisfies $\lambda_{B_{p}}(\Omega)=\left(\int_{\Omega}f_{0}\cdot x~{}dm\right)/\|f_{0}\|_{\mathcal{L}_{q}}.$

(iv) The function $u_{0}$ is unique up to an additive constant, and $\nabla u_{0}$ is uniquely determined by the properties*

[TABLE]

Proof. (i) We can choose a minimizing sequence $(u_{j})$ for (13), where $\int_{\Omega}u_{j}~{}dm=0$ for each $j$. By Poincaré’s inequality $(u_{j})$ is bounded in $W^{1,p}(\Omega)$, and by the Rellich-Kondrachov theorem we can arrange that $(u_{j})$ converges strongly in $L^{1}(\Omega)$ to a function $u_{0}$. Since the functions $\{u_{j}:j\geq 0\}$ all have distributional Laplacian equal to $Nm$, we can choose smooth representatives of these functions and arrange that $u_{j}\rightarrow u_{0}$ and $\partial u_{j}/\partial x_{i}\rightarrow\partial u_{0}/\partial x_{i}$ locally uniformly on $\Omega$ for each $i$. Now

[TABLE]

so we can let $j\rightarrow\infty$ and use the density of $C_{c}^{\infty}(\Omega)$ in $L^{q}(\Omega)$ to see that $\left\|\nabla u_{0}\right\|_{\mathcal{L}_{p}}\leq\lambda_{B_{p}}(\Omega)$. (When $p=1$ and so $q=\infty$, we instead use the fact that, for any $g\in\left(L^{\infty}(\Omega)\right)^{N}$, there is a sequence $(\phi_{n})$ in $\left(C_{c}^{\infty}(\Omega)\right)^{N}$ that converges pointwise almost everywhere to $g$ on $\Omega$ and satisfies $\sup_{\Omega}\left\|\phi_{n}\right\|\leq\mathrm{ess}\sup_{\Omega}\left\|g\right\|$ for all $n$.) Similarly, $u_{0}\in L^{p}(\Omega)$, so $u_{0}\in W^{1,p}(\Omega)$ and $u_{0}$ is a minimizer for (13).

(ii) Given any $h\in W^{1,p}(\Omega)\cap C^{2}(\Omega)$ such that $\Delta h=0$ on $\Omega$, we differentiate $\|\nabla(u_{0}+th)\|_{\mathcal{L}_{p}}^{p}$ with respect to $t$ and then put $t=0$ to see that

[TABLE]

(When $p=1$, we know that $m(\{\|\nabla u_{0}||=0\})=0$, and the above equation still follows by dominated convergence, since $\left|\left\|\nabla(u_{0}+th)\right\|-\left\|\nabla u_{0}\right\|\right|/t\leq\left\|\nabla h\right\|$.) Thus $\|\nabla u_{0}\|^{p-2}\nabla u_{0}\in B_{p}(\Omega)^{\perp}$.

(iii) If we take $h=u_{0}-\left\|x\right\|^{2}/2$ in (15), then we find that

[TABLE]

Since $\|f_{0}\|_{\mathcal{L}_{q}}=\|\nabla u_{0}\|_{\mathcal{L}_{p}}^{p/q}$, we obtain the desired equality.

(iv) In view of parts (i) and (ii) it only remains to check that (14) uniquely determines $u_{0}$ up to a constant. (When $p>1$, the uniqueness of the gradient $\nabla u_{0}$ also follows from the strict convexity of the $\mathcal{L}_{p}$-norm.) To see this, let $v$ be another such function and consider the harmonic function $v-u_{0}$. It follows from (15) that

[TABLE]

and

[TABLE]

Hölder’s inequality now shows that $\left\|\nabla u_{0}\right\|_{\mathcal{L}_{p}}=\left\|\nabla v\right\|_{\mathcal{L}_{p}}$, and we deduce that $\nabla u_{0}\equiv\nabla v$. (If $p=1$, then Hölder’s inequality is unnecessary.)

Proposition 9

*Let $p\in[1,\infty)$.

(i) There exists $f\in A_{p}(\Omega)$ such that $\lambda_{A_{p}}(\Omega)=\left\|x-f\right\|_{\mathcal{L}_{p}}$.

(ii) The function $f$ satisfies $\|x-f\|^{p-2}(x-f)\in A_{p}(\Omega)^{\perp}$.

(iii) The function $f_{0}=\|x-f\|^{p-2}(x-f)\in\mathcal{L}_{q}$ satisfies $\lambda_{A_{p}}(\Omega)=\left(\int_{\Omega}f_{0}\cdot x~{}dm\right)/\|f_{0}\|_{\mathcal{L}_{q}}.$

(iv) The function $f$ is uniquely determined by the properties*

[TABLE]

Proof. (i) We choose a sequence $(f^{(j)})$ in $A_{p}(\Omega)$ such that $\left\|x-f^{(j)}\right\|_{\mathcal{L}_{p}}\rightarrow\lambda_{A_{p}}(\Omega)$. Since $(\left\|f^{(j)}\right\|_{\mathcal{L}_{p}})$ is bounded and the functions $\left\|f^{(j)}\right\|$ are subharmonic (by Theorem 3.4.5 of [2]), the harmonic co-ordinate functions $f_{i}^{(j)}$ ($i=1,...,N$) are locally uniformly bounded. Thus, by taking a subsequence, we can arrange that $(f^{(j)})$ converges locally uniformly to some function $f$ satisfying $\mathop{\mathrm{d}iv}f=0$ and $\mathop{\mathrm{c}url}f=0$ on $\Omega$. Since

[TABLE]

we can let $j\rightarrow\infty$ and use the density of $C_{c}^{\infty}(\Omega)$ in $L^{q}(\Omega)$ to see that $\left\|x-f\right\|_{\mathcal{L}_{p}}\leq\lambda_{A_{p}}(\Omega)$. (When $p=1$ we make the same adjustments to this argument as in the proof of Proposition 8(i).) The reverse inequality is trivial.

(ii) - (iv) The arguments are analogous to those given for the previous proposition.

3 Proofs of Theorems 2 and 3

As noted previously, inequality (2) follows from (11) and (4). In this section we will complete the proofs of Theorem 2 (except where $p=2$) and Theorem 3. In view of (4) and Proposition 7, Theorem 3 is a consequence of the result below.

Theorem 10

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain and $p\in[1,2]$, then

[TABLE]

Further, equality holds if and only if $\Omega$ is a ball.

Proof. Let $u$ be the Green potential satisfying $\Delta u=N$ on $\Omega$ and $u=0$ on $\partial\Omega$. Next, let $w(x)=(\|x\|^{2}-r_{\Omega}^{2})/2$, so that $\Delta w=N$ in $B(r_{\Omega})$ and $w=0$ on $\partial B(r_{\Omega})$. We make use of a result of Talenti [23] concerning spherical rearrangements. Theorem 1(v) of that paper tells us that, provided $p\leq 2$, we have $\|\nabla u\|_{\mathcal{L}_{p}(\Omega)}\leq\|\nabla w\|_{\mathcal{L}_{p}(B(r_{\Omega}))}$. Hence

[TABLE]

by (13) and then Proposition 5(iv).

Finally, if $\|\nabla u\|_{\mathcal{L}_{p}(\Omega)}=\|\nabla w\|_{\mathcal{L}_{p}(B(r_{\Omega}))}$, then Propositions 3.2.1 and 3.2.2 of Kesavan [18] tell us that $\Omega$ must be a ball.

Lemma 11

Let $p\in[1,\infty)$. If $\Omega$ is either a ball or an annular region, then

[TABLE]

Proof. In view of (4) and Proposition 7 it is enough to show that $\lambda_{B_{p}}(\Omega)\leq Q_{q}(\Omega)$ when $\Omega$ is either a ball or an annular region. If $\Omega=B(r)$, then (cf. (17))

[TABLE]

Thus it remains to consider the case where $\Omega=B(R)\setminus\overline{B(r)}$ and $0<r<R$.

If $p>1$, then it follows from spherical symmetry that there exists $v\in C^{1}(\overline{\Omega})$ such that $\nabla v=\left\|\nabla w_{q}\right\|^{q-2}\nabla w_{q}$. Writing $w=-Nv$, we see that $\Delta w=-N\Delta_{q}w_{q}=N$ and so, by (13) and Proposition 5(iii),

[TABLE]

as required.

Now suppose that $p=1$. By Proposition 5(iii) again,

[TABLE]

If we define

[TABLE]

then

[TABLE]

by (18).

Proposition 12

Let $p\in[1,\infty)$. If there exists $f\in A_{p}(\Omega)$ satisfying $\left\|x-f\right\|_{\mathcal{L}_{p}}=Q_{q}(\Omega)$, then $f\in B_{p}(\Omega)$.

Proof. First suppose that $p>1$, so that $q<\infty$. By (4) and Proposition 7,

[TABLE]

Since

[TABLE]

the equality case of Hölder’s inequality implies that $x-f=c\left\|\nabla w_{q}\right\|^{q-2}\nabla w_{q}$ on $\Omega$ for some constant $c$. Hence $x-f$ has a continuous extension to $\overline{\Omega}$, and on $\partial\Omega$ it is normal to $\partial\Omega$. We now choose a function $v\in C^{1}(\mathbb{R}^{N}\backslash\Omega)$ such that $v=0$ and $\nabla v=x-f$ on $\partial\Omega$. (Such a function exists by [13], for example.) Thus we obtain a continuous extension of $f$ to $\mathbb{R}^{N}$ by defining it to be $x-\nabla v$ on $\mathbb{R}^{N}\backslash\Omega$.

We claim that this extended function, which we also denote by $f$, is curl-free in the sense of distributions. By using a partition of unity it is enough to show that, for some $\delta>0$,

[TABLE]

whenever $\phi\in C^{\infty}(\mathbb{R}^{N})$ and $\mathrm{diam}(\mathrm{supp}(\phi))<\delta$. This equation trivially holds when $\mathrm{supp}(\phi)\cap\partial\Omega=\emptyset$, so it is enough to consider the case where

[TABLE]

for some $y\in\partial\Omega$ and $r>0$.

Without loss of generality we may assume that

[TABLE]

for some smooth function $g$. If $i<N$ and the co-ordinates $x_{j}$ ($j\neq i,N$) are fixed, then

[TABLE]

where $D_{1},D_{2}$ are the components of $\{(x_{i},x_{N}):(x_{1},...,x_{N})\in K\backslash\partial\Omega\}$ and $A$ denotes two-dimensional measure. Two applications of Green’s theorem, together with the fact that $\partial f_{i}/\partial x_{N}=\partial f_{N}/\partial x_{i}$ on $\mathbb{R}^{N}\backslash\partial\Omega$, show that this latter integral expression reduces to self-cancelling terms along the common boundary curve of $D_{1},D_{2}$. Hence (20) holds when $j=N$. If $j\neq N$, we apply a small rotation in the $(x_{j},x_{N})$-plane to see similarly that

[TABLE]

whence (20) again follows.

We now use a rotationally invariant smoothing kernel $\psi_{\varepsilon}$ supported by a ball of radius $\varepsilon$ to obtain a mollification $f^{\varepsilon}$ of $f$, which is also curl-free since

[TABLE]

Further, since each component $f_{i}$ of $f$ is harmonic in $\Omega$, the functions $f^{\varepsilon}$ and $f$ are equal on the set $\{x:\mathrm{dist}(x,\mathbb{R}^{N}\backslash\Omega)>\varepsilon\}$. Hence line integrals of $f$ in $\Omega$ are path independent, so $f$ is of the form $\nabla v$, where $\Delta v=\nabla\cdot f=0$, and thus $f\in B_{p}(\Omega)$.

Finally, if $p=1$, then (19) still holds, and now shows that $x-f=-\left\|x-f\right\|\nabla w_{\infty}$ on $\Omega$. We can thus apply the above argument to $\Omega_{\eta}=\{x:\mathrm{dist}(x,\mathbb{R}^{N}\backslash\Omega)>\eta\}$ to deduce that $f\in B_{1}(\Omega_{\eta})$ for arbitrarily small $\eta>0$, and so $f\in B_{1}(\Omega)$.

We now consider the overdetermined problem

[TABLE]

where $n$ denotes the exterior unit normal, $a_{i},c_{i}\in\mathbb{R}$ $(i=0,...,j)$ and $\{\Gamma_{i}\}$ are the components of $\partial\Omega$. (We use $\Gamma_{0}$ for the outer boundary component.) The following theorem, which generalizes earlier work of Serrin [21], is contained in Theorem 2 of Sirakov [22].

Theorem 13

Let $c_{0}=0$, $a_{0}\geq 0$, and $c_{i}<0$, $a_{i}\leq 0$ $(i=1,...,j)$. Then there exists $v\in C^{2}(\overline{\Omega})$ satisfying (21) if and only if $\Omega$ is a ball or an annular region. In either case, $v$ is a radial function.

The case $p\neq 2$ of the equality statement in Theorem 2 is established in the next result. The case where $p=2$ will be addressed in Section 4.

Theorem 14

Let $p\in[1,\infty)$, where $p\neq 2$. Then $\lambda_{A_{p}}(\Omega)=Q_{q}(\Omega)$ if and only if $\Omega$ is either a ball or an annular region.

Proof. For the “if” part we refer to Lemma 11. For the “only if” part it is enough, given Propositions 9 and 12, to show that, if there exists $v\in W^{1,p}(\Omega)$ such that $\Delta v=Nm$ and $\left\|\nabla v\right\|_{\mathcal{L}_{p}}=Q_{q}(\Omega)$, then $\Omega$ is either a ball or an annular region.

If $p>1$, then we see from Proposition 5 that

[TABLE]

where the last equality can be justified using the facts that $w_{q}\in W_{0}^{1,q}(\Omega)$ and that $C_{c}^{\infty}(\Omega)$ is dense in $W_{0}^{1,q}(\Omega)$. By Hölder’s inequality,

[TABLE]

where equality occurs if and only if $\nabla w_{q},\nabla v$ are always parallel, $\nabla w_{q}\cdot\nabla v$ does not change sign, and $\left\|\nabla w_{q}\right\|^{q}=c\left\|\nabla v\right\|^{p}$ in $\Omega$ for some constant $c>0$. Further, $\nabla w_{q}\neq 0$ on $\partial\Omega$ by Hopf’s lemma (see Theorem 5.5.1 of [20]). Thus the equality $Q_{q}(\Omega)=\left\|\nabla v\right\|_{\mathcal{L}_{p}}$ implies that each component of any level surface of $v$ is also a component of a level surface of $w_{q}$. Hence, for each component $\Gamma_{i}$ of $\partial\Omega$, there is a function $g_{i}$ such that $v=g_{i}\circ w_{q}$ near $\Gamma_{i}$. Further, $\nabla v=g_{i}^{\prime}(w_{q})\nabla w_{q}$, so

[TABLE]

Since $p\neq 2$, we have $p\neq q$ and so $\left\|\nabla v\right\|=c^{1/(q-p)}\left|g_{i}^{\prime}(w_{q})\right|^{q/(q-p)}$. Thus $\left\|\nabla v\right\|$ is constant on each component of a level surface of $w_{q}$ (which is also a level surface of $v$).

Since $\nabla w_{q}\cdot n<0$ on $\partial\Omega$ and $\nabla w_{q}\cdot\nabla v$ does not change sign, we can apply the divergence theorem to $\nabla v$ to see that $\nabla w_{q}\cdot\nabla v<0$ near $\partial\Omega$ and hence $\nabla v\cdot n>0$ on $\Gamma_{0}$. Now let $\varepsilon>0$ be small and let $\Omega_{\varepsilon}$ be the component of $\{0<w_{q}<\varepsilon\}$ which has $\Gamma_{0}$ as a boundary component. Since $w_{q}\in C^{1}(\overline{\Omega})$, $\|\nabla w_{q}\|^{q}=c\|\nabla v\|^{p}$ and $\|\nabla v\|=c^{1/(q-p)}|g_{0}^{\prime}(w_{q})|^{q/(q-p)}$, it follows that $\left|g_{0}^{\prime}(0)\right|<\infty$ and certainly $\left|g_{0}(0)\right|<\infty$. Thus $v$ has a (finite) constant value on each component of $\partial\Omega_{\varepsilon}$. Since $\Delta v=Nm$ on $\Omega$ we conclude that $v\in C^{2}(\overline{\Omega}_{\varepsilon})$ (see Theorem 6.14 of [14]). Thus we can apply Theorem 13 to $(v-g_{0}(0))/N$ on $\Omega_{\varepsilon}$ to see that $\Gamma_{0}$ is a sphere and $v$ is a radial function. By the analyticity of $v$, any other boundary of component of $\Omega$ must be a concentric sphere. Thus $\Omega$ is either a ball or an annular region.

The argument for the case $p=1$ is mostly similar. Since $\left\|\nabla w_{\infty}\right\|=1$, we have

[TABLE]

and

[TABLE]

The equality $Q_{\infty}(\Omega)=\left\|\nabla v\right\|_{\mathcal{L}_{1}}$ implies that $\nabla w_{\infty}$ and $\nabla v$ are parallel. Thus, for each component $\Gamma_{i}$ of $\partial\Omega$, there is a function $g_{i}$ such that $v=g_{i}\circ w_{\infty}$ near $\Gamma_{i}$. We no longer claim that this equation holds on $\partial\Omega$. However, as in the proof of Proposition 12, we can work instead with $\Omega_{\eta}=\{x:\mathrm{dist}(x,\mathbb{R}^{N}\backslash\Omega)>\eta\}$ for small $\eta>0$ and now argue as before to conclude that $\Omega$ is either a ball or an annular region.

4 The case where $p=2$

It follows from Proposition 8(i) that there exist harmonic functions $h$ satisfying $\lambda_{B_{2}}(\Omega)=\|x-\nabla h\|_{\mathcal{L}_{2}}$. We will now identify all such functions. (This was already done in [8] in the case of planar domains.)

Theorem 15

The harmonic functions $h\in W^{1,2}({\Omega})$ which satisfy $\lambda_{B_{2}}(\Omega)=\|x-\nabla h\|_{\mathcal{L}_{2}}$ are precisely the functions of the form $H_{\|x\|^{2}/2}^{\Omega}+c\mathrm{,}$ where $H_{g}^{\Omega}$ is the solution to the Dirichlet problem on $\Omega$ with boundary data $g$, and $c\in\mathbb{R}$.

Proof. Let $h\in W^{1,2}(\Omega)$ be a harmonic function satisfying $\lambda_{B_{2}}(\Omega)=\|x-\nabla h\|_{\mathcal{L}_{2}}$, and let k\in C^{1}(\overline{\Omega})\be harmonic on $\Omega$. Since the function $t\mapsto\|x-\nabla(h+tk)\|_{2}^{2}$ has a minimum at $t=0$, we see that

[TABLE]

Hence, by the divergence theorem,

[TABLE]

where $\sigma$ denotes surface area measure. Since we can solve the Neumann problem

[TABLE]

for any smooth function $\phi$ satisfying $\int_{\partial\Omega}\phi d\sigma=0$, we see from (22) that $\left\|x\right\|^{2}/2-h(x)\mathrm{\ }$is constant on $\partial\Omega.$

The torsional rigidity of $\Omega$ is defined by

[TABLE]

where $v$ is the solution to the Dirichlet problem

[TABLE]

here $\Gamma_{0}$ is again the boundary of the unbounded component of $\mathbb{R}^{N}\backslash\Omega$, while $G_{1},G_{2},\ldots,G_{j}$ are the bounded components of $\Omega^{c}$ with boundaries $\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{j}$ and the constants $c_{i}$ are chosen so that

[TABLE]

From Proposition 5(iii) we see that

[TABLE]

when $\mathbb{R}^{N}\backslash\Omega$ has no bounded components.

Theorem 1 is contained in the result below.

Theorem 16

If $\Omega\subset\mathbb{R}^{N}$ is a smoothly bounded domain, then

[TABLE]

Further, these quantities are equal to $\sqrt{\rho(\Omega)}$ if and only if $\mathbb{R}^{N}\backslash\Omega$ is connected.

Proof of Theorem 16. Let $u(x)=$ $H_{\|x\|^{2}/2}^{\Omega}-\left\|x\right\|^{2}/2$. By Theorem 15,

[TABLE]

where for the last step we applied the divergence theorem and noted that $u=0$ on $\partial\Omega$. Hence

[TABLE]

Equation (26) now follows from Proposition 7 and (4).

We know from (25) that $Q_{2}(\Omega)=\sqrt{\rho(\Omega)}$ if $\mathbb{R}^{N}\backslash\Omega$ is connected. Conversely, suppose that $\mathbb{R}^{N}\backslash\Omega$ is not connected, and let $c_{k}=\min\{c_{1},...,c_{j}\}$. If $c_{k}\leq 0$, then the Hopf boundary point lemma (see Section 6.4.2 of [7]) would tell us that $\partial v/\partial n<0$ on $\Gamma_{k}$, which contradicts (24). Thus $c_{i}>0$ $(i=1,...,j)$ in (23), so $v$ cannot be a multiple of $w_{2}$, and it now follows from Proposition 5 that $Q_{2}(\Omega)>\sqrt{\rho(\Omega)}$.