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

Mohammed Barkatou

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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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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