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On the equation \(\operatorname{div}Y=f\) and application to control of phases. (English) Zbl 1075.35006

Summary: 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 \[ \| u\|_{L^{\infty}}+\| u\|_{W^{1, d}}\leq C\| f\|_{L^{d}} \] 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.

MSC:

35F05 Linear first-order PDEs
35B10 Periodic solutions to PDEs
35F15 Boundary value problems for linear first-order PDEs
42B05 Fourier series and coefficients in several variables
47N20 Applications of operator theory to differential and integral equations
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