www.ccsenet.org/jmr Journal of Mathematics Research Vol. 2, No. 4; November 2010 Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem Mohammed Barkatou (Corresponding author) Faculty of Sciences, University of Chouaib Doukkali PO box 20, 24000 El Jadida, Morocco E-mail: barkatoum@gmail.com 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 Q S ( f , k) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −Δu Ω = f in Ω u Ω = 0 on ∂Ω |∇u Ω | = k (constant) on ∂Ω. has a solution which contains strictly the support of f if and only if  C f ( x)dx > k  ∂C dσ. 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. Keywords: Dirichlet problem, Quadrature surfaces, Shape derivative, Shape optimization 1. Introduction Assuming throughout that: D ⊂ R N (N ≥ 2) is a bounded ball which contains all the domains we use. If ω is an open subset of D, let ν be the outward normal to ∂ω and let |∂ω| (respectively |ω|) be the perimeter (respectively the volume) of ω. Let k > 0 and let f be a positive function belonging to L 2 (R N ) and having a compact support K with nonempty interior. Denote by C the convex hull of K and consider the following free boundary problem. Find an open set Ω ⊂ D which contains strictly C and such that the following problem has a solution: Q S ( f , k) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −Δu Ω = f in Ω u Ω = 0 on ∂Ω |∇u Ω | = k on ∂Ω. Notice that since u Ω = 0 on ∂Ω then |∇u Ω | = − ∂u Ω ∂ν , where ν is the outward normal vector to ∂Ω. Imposing boundary conditions for both u Ω and |∇u Ω | on ∂Ω makes problem Q S ( f , k) overdetermined, so that in general without any assumptions on data this problem has no solution. The problem Q S ( f , k) is called the quadrature surfaces free boundary problem and arises in many areas of physics (free streamlines, jets, Hele-show flows, electromagnetic shaping, gravitational problems etc.) It has been intensively studied from different points of view, by several authors. For more details about the methods used for solving this problem see the (Gustafsson and Shahgholian, Introduction, 1996). Using the maximum principle together with the compatibility condition of the Neumann problem, the authors gave sufficient condition of existence for problem Q S ( f , k) (Barkatou and al., 2005). Thegoal of this paper is to prove the following Theorem 1.1 The problem Q S ( f , k) has a solution if and only if (NS )  C f ( x)dx > k|∂C|. This theorem says that the inequality (NS ) is a necessary and sufficient condition of existence for the quadrature surfaces free boundary problem. Published by Canadian Center of Science and Education 93