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

Abstract

Performing the shape derivative (Sokolowski and Zolesio, 1992) and using the maximum principle, we show that the so-called Quadrature Surfaces free boundary problemhas a solution which contains strictly the support of f if and only ifWhere 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.

Symmetry result for some overdetermined boundary value problems

“…B being the unit ball in R 2 . Notice that in the special case where g ≡ k = const., the the condition above becomes a > 2k and it is necessary and sufficient condition of existence for QS(a, k) [3]. C Ω being the unique Cheeger set of Ω.…”

“…The method used by Gustafsson and Shahgholian goes back to K. Friedrichs [18], or even to T. Carleman [11], and was considerably developed by H. W. Alt and L. A. Caffarelli [1]. Recently, by combining the maximum principle to the compatibility condition of the Neumann problem, Barkatou et al [3] gave, |∇u C | > k on ∂C as a sufficient condition of existence for Q S (f, g). Later, Barkatou [2] showed that this problem admits a solution if and only if the condition…”

Symmetry Result for Some Overdetermined Value Problems

“…In [21], in order to get solutions to Problem QS(f, g), A. Henrot used the method of subsolutions and supersolutions introduced by A. Beurling [8]. By combining the maximum principle to the compatibility condition of the Neumann problem, Barkatou et al [3] gave, |∇u C f | > k on ∂C f as a sufficient condition of existence for QS(f, g). Later, Barkatou [2] showed that this problem admits a solution if and only if the condition…”