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Hausdorff measure estimates of a free boundary for a minimum problem. (English) Zbl 0555.35128
If \(G\subset R^ n\) is a bounded set with \(\partial G\) a Lipschitz graph, \(B_ r(x)\) the closed ball with radius r and center x, then for a non-negative function u defined on G, let \(\Lambda (u)=\{x\in G: u=0\},\) \(\Omega (u)=\{x\in G: u>0\},\) \(F(u)=\partial \Omega (u)\cap \partial \Lambda (u).\) The author gives a number of Hausdorff measure estimates for \(F(u)\cap B_ r\) where \(B_{4r}\in G\) and the function u is a minimum for the functional \(J(\nu)=\int_{G}(1/2| \nabla \nu |^ 2+| \nu |^{\gamma})dx\) in the convex set \(K=\{\nu \in H^ 1(G): \nu -u_ 0\in H^ 1_ 0(G)\},\) for a fixed \(u_ 0\in H^ 1_ 0\), \(u_ 0\geq 0\), \(0<\gamma <2\).
Reviewer: W.Rundell

MSC:
35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
49J20 Existence theories for optimal control problems involving partial differential equations
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