A minimum problem with free boundary for a degenerate quasilinear operator

Abstract.In this paper we 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\left( |\nabla u|^p + \lambda^p\chi_{\{u>0\}}\right)dx\rightarrow \min\qquad (1

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