Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.

• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$ Can one find a solution in $W^{1,1}_0(\Omega)$? (I think the answer is no, but I do not know a counterexample). I know that $-\Delta u=f\in L^1$ has a unique solution in $W^{1,p}_0(\Omega)$ for all $1\leq p<n/(n-1)$, (see https://mathoverflow.net/a/298962/121665) but now the right hand side is much worse.
• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^p(\Omega,\mathbb{R}^n), 1<p<\infty? $$ If $1<p<\infty$ we can solve the quation $-\Delta U=F$, $U\in W^{2,p}(\Omega)$. Then $u={\rm div}\, U\in W^{1,p}$ is a solution to $-\Delta u={\rm div}\, F$, but I do not know if we can take $u\in W^{1,p}_0(\Omega)$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.

• What are the regularity results for solutions to $$ -\Delta u= \operatorname{div} F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$ Can one find a solution in $W^{1,1}_0(\Omega)$? (I think the answer is no, but I do not know a counterexample). I know that $-\Delta u=f\in L^1$ has a unique solution in $W^{1,p}_0(\Omega)$ for all $1\leq p<n/(n-1)$, (see https://mathoverflow.net/a/298962/121665) but now the right hand side is much worse.
• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^p(\Omega,\mathbb{R}^n), 1<p<\infty? $$ If $1<p<\infty$ we can solve the quation $-\Delta U=F$, $U\in W^{2,p}(\Omega)$. Then $u={\rm div}\, U\in W^{1,p}$ is a solution to $-\Delta u=\operatorname{div} F$, but I do not know if we can take $u\in W^{1,p}_0(\Omega)$.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.

• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$ Can one find a solution in $W^{1,1}_0(\Omega)$? (I think the answer is no, but I do not know w counterexample). I know that $-\Delta u=f\in L^1$ has a unique solution in $W^{1,p}(\Omega)$ for all $1\leq p<n/(n-1)$, but now the right hand side is much worse.
• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^p(\Omega,\mathbb{R}^n), 1<p<\infty? $$

I will write later today what I know about solutions to these problems.

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.

• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^1(\Omega,\mathbb{R}^n)? $$ Can one find a solution in $W^{1,1}_0(\Omega)$? (I think the answer is no, but I do not know a counterexample). I know that $-\Delta u=f\in L^1$ has a unique solution in $W^{1,p}_0(\Omega)$ for all $1\leq p<n/(n-1)$, (see https://mathoverflow.net/a/298962/121665) but now the right hand side is much worse.
• What are the regularity results for solutions to $$ -\Delta u= {\rm div }\, F, \qquad F\in L^p(\Omega,\mathbb{R}^n), 1<p<\infty? $$ If $1<p<\infty$ we can solve the quation $-\Delta U=F$, $U\in W^{2,p}(\Omega)$. Then $u={\rm div}\, U\in W^{1,p}$ is a solution to $-\Delta u={\rm div}\, F$, but I do not know if we can take $u\in W^{1,p}_0(\Omega)$.