Variational problems with free boundaries for the fractional Laplacian

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Abstract

We discuss properties (optimal regularity, nondegeneracy, smoothness of the free boundary etc.) of a variational interface problem involving the fractional Laplacian; due to the nonlocality of the Dirichlet problem, the task is nontrivial. This difficulty is bypassed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a codimension 2 (degenerate) free boundary. © European Mathematical Society 2010.

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Caffarelli, L. A., Roquejoffre, J. M., & Sire, Y. (2010). Variational problems with free boundaries for the fractional Laplacian. Journal of the European Mathematical Society, 12(5), 1151–1179. https://doi.org/10.4171/JEMS/226

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