Abstract

Performing the shape derivative (Sokolowski and Zolesio, 1992) and
using the maximum principle, we show that the so-called Quadrature
Surfaces 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 and
only 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  a
necessary and sufficient condition of existence for the problem
$Q_S(f,k)$ where the term source $f$ is a uniform density supported
by a segment.

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