Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem

Abstract

Performing the shape derivative (Sokolowski and Zolesio, 1992) andusing the maximum principle, we show that the so-called QuadratureSurfaces free boundary problem\begin{equation*}Q_S(f,k) \left\{\begin{array}{l}-\Delta u_{\Omega}=f \quad \text{in }\Omega\\u_{\Omega}=0\text{ on }\partial \Omega\\ \left|\nabla u_{\Omega }\right|=k \;(\text{constant})\text{ on }\partial \Omega.\end{array}\right.\end{equation*}has a solution which contains strictly the support of $f$ if andonly if$$\int_Cf(x)dx>k\int_{\partial C}d\sigma.$$ Where $C$ is the convex hull of the support of $f$. We also give anecessary and sufficient condition of existence for the problem$Q_S(f,k)$ where the term source $f$ is a uniform density supportedby a segment.