A short proof of local regularity of distributional solutions of Poisson's equation

TL;DR

This paper presents a concise proof demonstrating local regularity of distributional solutions to Poisson's equation with L^p data, utilizing Weyl's lemma and Riesz-Fréchet theorem.

Contribution

It introduces a very short and elegant proof of local regularity for solutions of Poisson's equation with L^p data, simplifying previous approaches.

Findings

Established local regularity for distributional solutions with minimal assumptions.

Provided a concise proof leveraging classical functional analysis tools.

Enhanced understanding of regularity properties in elliptic PDEs.

Abstract

We prove a local regularity result for distributional solutions of the Poisson's equation with $L^p$ data. We use a very short argument based on Weyl's lemma and Riesz-Fr\'echet representation theorem.

Equations

$$\Vert D^{2}u{\Vert }_{L^2(\Omega )}^2$$

$$\Delta u=f,$$

$$-{\int }_{\Omega }\nabla u\cdot \nabla vdx={\int }_{\Omega }fvdx\forall v\in W_0^{1,2}(\Omega ).$$

$$-{\int }_{\Omega }\nabla u\cdot \nabla vdx={\int }_{\Omega }fvdx\forall v\in C_0^{\infty }(\Omega ).$$

$$-\mathrm{div}(A\nabla u)=f$$

$${\int }_{\Omega }u\Delta \phi dx={\int }_{\Omega }f\phi dx\forall \phi \in C_0^{\infty }(\Omega ).$$

$$\Vert u{\Vert }_{L^1(\Omega )}\le \Vert f\cdot \mathrm{dist}(x,\partial \Omega ){\Vert }_{L^1(\Omega )}.$$

$${\int }_{\Omega }u\Delta \phi dx=0\forall \phi \in C_0^{\infty }(\Omega ).$$

$$⟨u,\Delta \phi ⟩=⟨f,\phi ⟩\forall \phi \in C_0^{\infty }(\Omega ).$$

$$\Delta u\in W^{-1,2}(\Omega )⟹u\in W_{loc}^{1,2}(\Omega ).$$

$$-{\int }_{\Omega }\nabla u_F\cdot \nabla \phi =⟨F,\phi ⟩$$

Full text

1. Introduction

Following the pleasant introduction on the regularity theory of elliptic equations in [15], if $u\in C^{3}_{0}(\Omega)$, $\Omega$ being an open, bounded set in $\mathbb{R}^{n}$, $n\geq 2$, then, using integrations by parts and Schwarz’s theorem, and identifying the continuous, compactly supported functions with their corresponding elements in $L^{2}(\Omega)$,

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This means that if $u\in C^{3}_{0}(\Omega)$ solves, in the classical sense, the Poisson’s equation

[TABLE]

then the $L^{2}$ norm of the datum $f$ controls the $L^{2}$ norm of all second derivatives of $u$. This statement is a typical example of a result in the theory of elliptic regularity, whose main aim is to deduce this kind of results, but under weaker a priori hypotheses on the regularity of the solution $u$.

1.1. The notions of weak, very weak, and distributional solutions

For a given $f\in L^{2}(\Omega)$, it is natural to study equation (1) in the $weak$ $sense$. This amounts to interpret (1) as equality between elements of the dual space $W^{-1,2}(\Omega)$ of the Sobolev space $W_{0}^{1,2}(\Omega)$; their images, when tested on every element $v\in W_{0}^{1,2}(\Omega)$, must coincide. If one looks for functions $u$ in $W_{0}^{1,2}(\Omega)$ which satisfy (1) in the weak sense (i.e., for weak solutions), the requirement is that

[TABLE]

Since $W_{0}^{1,2}(\Omega)$ is a Hilbert space, by the Riesz-Fréchet representation theorem (see, e.g., [12, p. 118], [3, Theorem 5.5 p. 135]), one gets the existence and uniqueness of the weak solution. Note, however, that the weak formulation (2) relies on the apriori assumption that the solution $u$ has first derivatives with the same integrability property of the datum. For such solutions, one can prove the $W_{loc}^{2,2}(\Omega)$ regularity (see, e.g., [15, Theorem 8.2.1]). Also, we recall that under specific assumptions on the regularity of $\Omega$, one can get a better $global$ regularity for $u$, while under regularity assumptions on the datum, one can get a better local regularity result for the solution (see, e.g., [15, Theorem 8.2.2 and Corollary 8.2.1]).

By (2) one gets the following equivalent equation (the equivalence with (2) immediately follows from a standard density argument), where now the test functions $v$ are in $C_{0}^{\infty}(\Omega)$:

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When the problem is in this form, one can look for solutions of equation (1) in the space $W^{1,1}(\Omega)$, because the $L^{2}(\Omega)$ integrability of the gradient is not needed to give sense to the equation. Regularity results when the datum is in $L^{p}(\Omega)$, $1<p<\infty$, are classical, and rely upon the well known Calderón-Zygmund inequality from which one can get the $W_{loc}^{2,p}(\Omega)$ regularity (see, e.g., [15, Theorem 9.2.2, p. 248], or [10, Corollary 9.10, p. 235]). We mention here also the method of difference quotients introduced by Nirenberg (see, e.g., [7, 20], [5, Step 1, p. 121], and [15, Theorem 9.1.2 p.245]).

Equation (3) is a special case of a class of linear equations which can be written in the form

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for which it is known ([23]) that even in the case $f\equiv 0$, when $A$ is a matrix function whose entries are locally $L^{1}$, there exist weak solutions, assumed a priori in $W_{loc}^{1,1}(\Omega)$, which are not in $W_{loc}^{1,2}(\Omega)$. As soon as one assumes that a solution is in $W_{loc}^{1,2}(\Omega)$, much local regularity can be gained by the celebrated De Giorgi’s theorem (see e.g. [10, Chap. 8] and references therein).

We recall, in passing, that for $\Omega\subset\mathbb{R}^{2}$, it is possible to prove an existence and uniqueness theorem for weak solutions of (4) (and even for a nonlinear variant) in a space slightly larger than $W_{0}^{1,2}(\Omega)$, the so-called grand Sobolev space $W_{0}^{1,2)}(\Omega)$, when the datum is just in $L^{1}(\Omega)$ (see [9, Theorem A]). For an excellent survey about solutions of a number of elliptic equations, called very weak because the solutions are assumed a priori in Sobolev spaces with exponents below the natural one, the reader is referred to [11].

One can further weaken the notion of solution, and look for solutions of (1) in the space of regular distributions, that is, among elements of the dual space $\mathcal{D}^{\prime}(\Omega)$ of $C_{c}^{\infty}(\Omega)$ that can be identified with elements of $L^{1}_{loc}(\Omega)$. In other words, their images, when computed in every element $\varphi\in C_{c}^{\infty}(\Omega)$, must coincide:

[TABLE]

Solutions in $L^{1}_{loc}(\Omega)$ of equation (5) are called very weak solutions.

A condition on the datum $f$ ensuring existence and uniqueness in $L^{1}(\Omega)$ has been found in [4, Lemma 1], namely, if $f$ is in the weighted Lebesgue space where the weight is the distance function from the boundary of $\Omega$, there exists a unique solution $u\in L^{1}(\Omega)$ such that

[TABLE]

Differentiability results for very weak solutions are treated in a number of papers, see, e.g., [6, 21] and references therein (see also [8, Theorem 4.2]). However, all such references gain regularity from data in weighted Lebesgue spaces, where the distance to the boundary is involved in the weight and the domain $\Omega$ has itself some regularity assumptions.

When the datum $f$ is identically zero, the masterpiece theorem of regularity for very weak solutions has been proved by Hermann Weyl in [28, pp. 415/6]. It dates back to 1940, well before the introduction of Sobolev spaces [18, 16], and is nowadays referred to as Weyl’s Lemma:

Lemma 1** (H. Weyl, 1940).**

Let $\Omega\subset\mathbb{R}^{n}$ be an open set. Suppose that $u\in L^{1}_{loc}(\Omega)$ and

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Then there exists a unique $\widetilde{u}\in C^{\infty}(\Omega)$ such that $\Delta\widetilde{u}=0$ in $\Omega$ and $\widetilde{u}=u$ a.e. in $\Omega$.

The proof given by Weyl in [28] is elementary and clever. Modern rephrasing of the proof can be found in classical textbooks (see, e.g., [15, Corollary 1.2.1], [5, Theorem 4.7], [24, Appendix, n.2], [27]). A beautiful note devoted entirely on this result and its development is the paper by Strook [26], where Weyl’s lemma is stated under the weaker assumption that $u\in\mathcal{D}^{\prime}(\Omega)$. Indeed, one can go still further, and write (5) (in fact, (1)) in the form

[TABLE]

Any solution of equation (6), is called a distributional solution of the Poisson’s equation (1). The statement proved therein is the following (see also [22, 13, 31] for a more general result, valid for a broader class of differential operators).

Lemma 2** (Weyl’s lemma in $\mathcal{D}^{\prime}(\Omega)$).**

Let $\Omega\subset\mathbb{R}^{n}$ be an open set. Suppose that $u\in\mathcal{D}^{\prime}(\Omega)$ satisfies $\Delta u=f\in C^{\infty}(\Omega)$ in the sense of (6). Then $u\in C^{\infty}(\Omega)$.

The proof in [26] is very short and elegant. For our purposes, however, it is sufficient the particular case $f\equiv 0$, for which the proof in [26] further simplifies.

1.2. Contributions of present work

In this note, we are interested in regularity results for distributional solutions of (1), i.e., solutions satisfying (6). We prove a local regularity result for distributional solutions of the Poisson’s equation with $L^{p}$ data. We use a concise argument based on Weyl’s lemma and Riesz-Fréchet representation theorem. As a byproduct, we get the following classical result on very weak solutions

Theorem 1**.**

If $f\in L^{2}_{loc}(\Omega)$, then any solution $u\in L^{2}_{loc}(\Omega)$ of $\Delta u=f$ $($i.e., satisfying (5)$)$ belongs to $W_{loc}^{2,2}(\Omega)$.

Theorem 1, known since 1965 (see [1, Theorem 6.2 p. 58] for a more general result, proved for uniformly elliptic operators with Lipschitz continuous coefficients), is also quoted in the Brezis book [3, Remark 25 p. 306], where it is claimed the delicateness of the proof of interior regularity of very weak solutions, based on estimates for the difference quotient operator (see [1, Def. 3.3 p. 42]). In [2, Section 3 p. 92] the reader can find a modern proof, valid for a wide class of operators, which uses a precise estimate by Hörmander in combination with a spectral representation for hypoelliptic operators. We quote also [29, Theorem 1.3], where for general operators with locally Lipschitz continuous coefficients, in the case $f\equiv 0$, it is shown that very weak solutions in $L^{1}_{loc}(\Omega)$ are in fact in $W_{loc}^{2,p}(\Omega)$ for every $p\in[1,\infty)$; in [30, Proposition 1.1], the same authors, for general operators having locally Lipschitz continuous coefficients, in the case $f\in L^{p}_{loc}(\Omega)$, $1<p<\infty$, get that very weak solutions in $L^{1}_{loc}(\Omega)$ are in fact in $W_{loc}^{2,p}(\Omega)$.

For other results of regularity for very weak solutions of the Poisson’s equation, see, e.g., [17, Section 7.2 p. 223] and [19, Section 4.1 p. 198]. In particular, we mention here that, following Hilbert, one can ask whether a solution, being a distribution, is analytic in the case where the right-hand side $f$ is analytic: the answer is positive for equation (4) when $A$ is analytic, see [14].

2. Regularity of very weak solutions of Poisson’s equation in the $L^{2}$-setting

The main ingredient is stated in the following result which, remarkably, is essentially based on Weyl’s lemma.

Lemma 3**.**

Let $\Omega\subset\mathbb{R}^{n}$ be an open set, and let $u\in\mathcal{D}^{\prime}(\Omega)$. Then

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Proof.

Since $\Delta u\in W^{-1,2}(\Omega)$, by Riesz representation theorem, there exists $v\in W_{0}^{1,2}(\Omega)$ such that $\Delta v=\Delta u$ in $\mathcal{D}^{\prime}(\Omega)$. In particular, $\Delta(u-v)=0$ in $\mathcal{D}^{\prime}(\Omega)$. By Weyl’s lemma (Lemma 2 used with $f\equiv 0$), we know that $u-v\in C^{\infty}(\Omega)$. Hence $u=(u-v)+v\in C^{\infty}(\Omega)+W_{0}^{1,2}(\Omega)\subset W_{loc}^{1,2}(\Omega)$. ∎

We remark that solutions of Dirichlet problems by Hilbert spaces methods are a classic matter for $weak$ solutions, see, e.g., [12, p. 117]. Again, for weak solutions, we quote [25, Lemma 2.1 p.48], where from the assumption of being locally in a Sobolev space, the authors get a better local regularity, still in Sobolev spaces.

Theorem 2**.**

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open set, and let $u\in\mathcal{D}^{\prime}(\Omega)$. If $\Delta u\in L^{2}_{loc}(\Omega)$, then $\nabla u\in W_{loc}^{1,2}(\Omega)$. If, in addition, $u\in L^{2}_{loc}(\Omega)$, then $u\in W_{loc}^{2,2}(\Omega)$.

Proof.

Due to the local character of the result, we can assume $\Delta u\in L^{2}(\Omega)$. Therefore, it is sufficient to note that if $\Delta u=f$ with $f\in L^{2}(\Omega)$ then, for any distributional partial derivative of $u$, we have $\Delta(\nabla u)=\nabla f$ with $\nabla f\in W^{-1,2}(\Omega)$. By the previous lemma, we get $\nabla u\in W_{loc}^{1,2}(\Omega)$. Thus, $u\in W_{loc}^{2,2}(\Omega)$ if we assume $u\in L^{2}_{loc}(\Omega)$. ∎

3. Regularity of very weak solutions of Poisson’s Equation in the $L^{p}$-setting

We point out that the same argument shows that if $f\in L^{p}(\Omega)$, $1<p<\infty$, then $u\in W^{2,p}_{\operatorname{loc}}(\Omega)$. Precisely, the following result holds:

Theorem 3**.**

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open set, and let $u\in\mathcal{D}^{\prime}(\Omega)$. If $\Delta u\in L^{p}_{loc}(\Omega)$, then $\nabla u\in W_{loc}^{1,p}(\Omega)$. If, in addition, $u\in L^{p}_{loc}(\Omega)$, then $u\in W_{loc}^{2,p}(\Omega)$.

Proof.

Indeed (see, e.g., [25, pp. 10-11]), if $1/p+1/q=1$ and $F\in W^{-1,q^{\prime}}(\Omega)$ then there exists a function $u_{F}\in W^{1,p}(\Omega)$ such that

[TABLE]

for every $\varphi\in W^{1,q}_{0}(\Omega)$. Note that this can be considered as the $q$-exponent version of the Riesz representation theorem. Now, (8) implies that for any $F\in W^{-1,q^{\prime}}(\Omega)$ there exists a distribution in $v_{F}\in W^{1,p}(\Omega)$ such that $\Delta v_{F}=F$ in $\mathcal{D}^{\prime}(\Omega)$. After that, assume that $f\in L^{p}(\Omega)$ and $u\in\mathcal{D}^{\prime}(\Omega)$ is a distributional solution of (6). Then $\nabla u$ satisfies $\Delta(\nabla u)=\nabla f$ with $\nabla f\in W^{-1,q^{\prime}}(\Omega)$. Therefore, as in Lemma 3, $\nabla u\in W^{1,p}_{\operatorname{loc}}(\Omega)$, and we conclude. ∎

4. Acknowledgments

The first author acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65).