Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem
- Mohammed Barkatou
Abstract
Performing the shape derivative (Sokolowski and Zolesio, 1992) andusing 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.
- Full Text:
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- DOI:10.5539/jmr.v2n4p93
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