We obtain density estimates for the free boundaries of minimizers $ u \ge 0 $ of the Alt-Phillips functional involving negative power potentials
$ \int_\Omega \left(|\nabla u|^2 + u^{-\gamma} \chi_{\{u>0\}}\right) \, dx, \quad \quad \gamma \in (0, 2). $
These estimates remain uniform as the parameter $ \gamma \to 2 $. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $ \gamma \to 2 $. The results are based on the $ \Gamma $-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional
$ \int_{\Omega} |\nabla u|^2 dx + Per_{\Omega}(\{ u = 0\}), $
considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.
Citation: Daniela De Silva, Ovidiu Savin. Uniform density estimates and $ \Gamma $-convergence for the Alt-Phillips functional of negative powers[J]. Mathematics in Engineering, 2023, 5(5): 1-27. doi: 10.3934/mine.2023086
We obtain density estimates for the free boundaries of minimizers $ u \ge 0 $ of the Alt-Phillips functional involving negative power potentials
$ \int_\Omega \left(|\nabla u|^2 + u^{-\gamma} \chi_{\{u>0\}}\right) \, dx, \quad \quad \gamma \in (0, 2). $
These estimates remain uniform as the parameter $ \gamma \to 2 $. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $ \gamma \to 2 $. The results are based on the $ \Gamma $-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional
$ \int_{\Omega} |\nabla u|^2 dx + Per_{\Omega}(\{ u = 0\}), $
considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.
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Daniela De Silva, Ovidiu Savin. Uniform density estimates and $ \Gamma $-convergence for the Alt-Phillips functional of negative powers[J]. Mathematics in Engineering, 2023, 5(5): 1-27. doi: 10.3934/mine.2023086
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