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A minimum problem with free boundary for a degenerate quasilinear operator. (English) Zbl 1068.35187

The authors prove \(C^{1,\alpha}\) regularity (near flat points) of the free boundary \(\partial\{u>0\}\cap\Omega\) in the Alt-Caffarelli type minimum problem for the \(p\)-Laplace operator: \[ J(u) = \int_{\Omega} (|\nabla u|^{p} + \lambda^{p}\chi_{\{u>0\}}) \,dx \to \min \quad (1<p<\infty). \]

MSC:

35R35 Free boundary problems for PDEs
49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
35J40 Boundary value problems for higher-order elliptic equations
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