On the equation 𝑑𝑖𝑣𝑌=𝑓 and application to control of phases

Abstract

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