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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
##### Keywords:
Hausdorff measure estimates
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##### References:
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