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An optimization problem with volume constraint. (English) Zbl 0588.49005
The authors study the following optimization problem: To find \(u\in K_ 0\) such that J(u)\(\leq J(v)\), \(v\in K_ 0\) with \(J(v)=\int_{\Omega}| \nabla v|^ 2\) \((\Omega \subset {\mathbb{R}}^ n\) with bounded Lipschitz boundary \(\partial \Omega)\), \(K_ 0=\{v\in L^ 1_{loc}(\Omega)\), \(\nabla v\in L^ 2(\Omega)\), \(v=u_{0| \in \partial \Omega}\) and \({\mathcal L}^ n(\{v>0\})=w_ 0< {\mathcal L}^ n(\Omega)\}\) (\({\mathcal L}^ n\) denotes the n-dimensional Lebesgue measure).
To solve this problem, a penalization technique is used for the constraint \({\mathcal L}^ n(\{v>0\})=w_ 0\), by introducing the functional \(J_{\epsilon}(v)=\int_{\Omega}| \nabla v|^ 2+f_{\epsilon}({\mathcal L}^ n(\{v>0\}))\) with \(f_{\epsilon}(s)=(1/\epsilon)(s-w_ 0)\) for \(s\geq w_ 0\) and \(f_{\epsilon}(s)=\epsilon (s-w_ 0)\) for \(s\leq w_ 0.\)
Then the main results are:
1. the existence of a solution \(u_{\epsilon}\in C^{0,1}_{loc}(\Omega)\), 2. the existence of \(\lambda_{\epsilon}\) such that \(u_{\epsilon}\) is weak solution of the free boundary problem \(\partial \nu u_{\epsilon}=\lambda_{\epsilon}\), 3. for \(\epsilon\) small enough, \({\mathcal L}^ n(u_{\epsilon}>0)=w_ 0\). Then \(u_{\epsilon}\) minimizes the original functional J on \(K_ 0\).
Reviewer: Chr.Saguez

49J20 Existence theories for optimal control problems involving partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35R35 Free boundary problems for PDEs
49M30 Other numerical methods in calculus of variations (MSC2010)
35J20 Variational methods for second-order elliptic equations
93C20 Control/observation systems governed by partial differential equations
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