Variational problems with free boundaries for the fractional Laplacian

Luis Caffarelli; Jean-Michel Roquejoffre; Yannick Sire

Journal of the European Mathematical Society (2010)

  • Volume: 012, Issue: 5, page 1151-1179
  • ISSN: 1435-9855

Abstract

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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.

How to cite

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Caffarelli, Luis, Roquejoffre, Jean-Michel, and Sire, Yannick. "Variational problems with free boundaries for the fractional Laplacian." Journal of the European Mathematical Society 012.5 (2010): 1151-1179. <http://eudml.org/doc/277768>.

@article{Caffarelli2010,
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.},
author = {Caffarelli, Luis, Roquejoffre, Jean-Michel, Sire, Yannick},
journal = {Journal of the European Mathematical Society},
keywords = {fractional Laplacian; free boundary; fractional Laplacian; extension formula; free boundary problem; regularity; regularity of the free boundary; positive density},
language = {eng},
number = {5},
pages = {1151-1179},
publisher = {European Mathematical Society Publishing House},
title = {Variational problems with free boundaries for the fractional Laplacian},
url = {http://eudml.org/doc/277768},
volume = {012},
year = {2010},
}

TY - JOUR
AU - Caffarelli, Luis
AU - Roquejoffre, Jean-Michel
AU - Sire, Yannick
TI - Variational problems with free boundaries for the fractional Laplacian
JO - Journal of the European Mathematical Society
PY - 2010
PB - European Mathematical Society Publishing House
VL - 012
IS - 5
SP - 1151
EP - 1179
AB - 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.
LA - eng
KW - fractional Laplacian; free boundary; fractional Laplacian; extension formula; free boundary problem; regularity; regularity of the free boundary; positive density
UR - http://eudml.org/doc/277768
ER -

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