$Variational problems with free boundaries for the fractional Laplacian$

Variational problems with free boundaries for the fractional Laplacian

We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discov...

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Veröffentlicht in:	Journal of the European Mathematical Society : JEMS 2010-01, Vol.12 (5), p.1151-1179
Hauptverfasser:	Caffarelli, Luis, Roquejoffre, Jean-Michel, Sire, Yannick
Format:	Artikel
Sprache:	eng
Schlagworte:	Partial differential equations
Online-Zugang:	Volltext
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Beschreibung
Zusammenfassung:	We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.
ISSN:	1435-9855 1435-9863
DOI:	10.4171/JEMS/226