Optimal regularity results in Sobolev-Lorentz spaces for linear elliptic equations with $L^1$- or measure data

TL;DR

This paper establishes optimal regularity results for solutions to linear elliptic equations with measure data in Sobolev-Lorentz and Besov spaces, extending known results to near second-order regularity.

Contribution

It provides new optimal regularity estimates in Sobolev-Lorentz and Besov spaces for solutions of elliptic equations with measure data, including general boundary conditions.

Findings

Solutions belong to specific Sobolev-Lorentz spaces with near second-order regularity.

Solutions also belong to corresponding Besov spaces, indicating refined regularity.

Regularity results hold for general linear elliptic equations with nonhomogeneous boundary data.

Abstract

It has been well known that if $\Omega$ is a bounded $C^1$-domain in $\R^n,\ n \ge 2$, then for every Radon measure $f$ on $\Omega$ with finite total variation, there exists a unique weak solution $u\in W_0^{1,1}(\Omega )$ of the Poisson equation $-\Delta u=f$ in $\Omega$ satisfying $\nabla u \in L^{n/(n-1),\infty}(\Omega;\R^n )$. In this paper, optimal regularity properties of the solution $u$ are established in Sobolev-Lorentz spaces $L_{\alpha}^{p,q}(\Omega )$ of order $ \alpha$ less than but arbitrarily close to $2$. More precisely, for any $0 \le \alpha<1$, we show that $u\in L_{\alpha+1}^{p(\alpha),\infty}(\Omega )$, where $p(\alpha )= n/(n-1+\alpha )$. Moreover, using an embedding result for Sobolev-Lorentz spaces $L_{\alpha}^{p,q}(\Omega )$ into classical Besov spaces $B_\alpha^{p,q}(\Omega )$, we deduce that $u\in B_{\alpha+1}^{p(\alpha),\infty}(\Omega )$. Indeed, these regularity results are proved for solutions of the Dirichlet problems for more general linear elliptic equations with nonhomogeneous boundary data. On the other hand, it is known that if $\Omega$ is of class $C^{1,1}$, then for each $G\in L^1 (\Omega ;\R^n )$ there exists a unique very weak solution $v\in L^{n/(n-1),\infty} (\Omega )$ of $-\Delta v= {\rm div}\, G$ in $\Omega$ satisfying the boundary condition $v=0$ in some sense. We prove that $v$ has the optimal regularity property, that is, $v\in L_{\alpha}^{p(\alpha),\infty}(\Omega )\cap B_{\alpha}^{p(\alpha),\infty}(\Omega )$ for every $0 \le \alpha < 1$. This regularity result is also proved for more general equations with nonhomogeneous boundary data.