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The structure of the free boundary for lower dimensional obstacle problems. (English) Zbl 1185.35339
Consider the obstacle problem consisting in looking for the minimizer \(u(x)\) of the Dirichlet integral over the unit ball \(B_1\) in \(\mathbb{R}^n {(n\geq3)}\) among the elements of the closed convex set \[ K=\{v\in H^1(B_1), v=\phi\,\, {\text{on}}\quad \partial B_1, v| _{x_n=0}\geq0\} \] where \(\phi\) is a smooth function which is supposed to be positive on \(\partial B_1\), intersected with \(x_n=0\) and to assume also negative values. The coincidence set \(\lambda(u)\) is the subset of \(x_n=0\) where \(u\) vanishes and we are also interested in the free boundary \(F(u)\), which is the boundary of the set \(\{u\geq\varphi\}\bigcap\{x_n=0\}\). The paper is concerned with the study of \(F(u)\), which is shown to be a \(C^{1, \alpha}(n-2)\)-dimensional surface in \(\mathbb{R}^{n-1}\) near non-degenerate points.

MSC:
35R35 Free boundary problems for PDEs
35J20 Variational methods for second-order elliptic equations
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