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On the porosity of free boundaries in degenerate variational inequalities. (English) Zbl 0956.35054

Let us define \[ K_{\theta}=\{ v\in W^{1,p}(\Omega): \;v-\theta \in W^{1,p}_{0}(\Omega), \;v\geq 0 \text{ a.e. in }\Omega \} \] for a given bounded domain \( \Omega \) and \(\theta \in W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). A function \(v\in K_{\theta}\) is a solution to the obstacle problem if \[ \int\limits_{\Omega}(|\nabla u|^{p-2}\nabla u (\nabla v-\nabla u)+f(v-u))dx\geq 0 \] whenever \(v\in K_{\theta}\). The authors obtained exact estimates for the solution to the obstacle problem near the free boundary and prove that the free boundary is porous and therefore its Hausdorff dimension is less than \(N\) and hence it is of Lebesgue measure zero.

MSC:

35J70 Degenerate elliptic equations
35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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References:

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