On the equation $\operatorname {div}Y=f$ and application to control of phases

Abstract:

The main result is the following. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^{d}$, $d\geq 2$. Then for every $f\in L^{d}(\Omega )$ with $\int f =0$, there exists a solution $u\in C^{0}(\bar \Omega )\cap W^{1, d}(\Omega )$ of the equation div $u=f$ in $\Omega$, satisfying in addition $u=0$ on $\partial \Omega$ and the estimate \begin{equation*}\Vert u\Vert _{L^{\infty }}+\Vert u\Vert _{W^{1, d}}\leq C\Vert f\Vert _{L^{d}} \end{equation*} where $C$ depends only on $\Omega$. However one cannot choose $u$ depending linearly on $f$. Our proof is constructive, but nonlinear—which is quite surprising for such an elementary linear PDE. When $d=2$ there is a simpler proof by duality—hence nonconstructive.