arXiv2025
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.