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An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions
Cabré Vilagut, Xavier; Serra Montolí, Joaquim
Universitat Politècnica de Catalunya. Departament de Matemàtiques; Universitat Politècnica de Catalunya. EDP - Equacions en Derivades Parcials i Aplicacions
We study nonlinear elliptic equations for operators corresponding to non-stable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) Lévy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen-Cahn type equations with a non-stable Lévy diffusion. These operators may still be realized as local operators using a system of PDEs - in the spirit of the extension problem of Caffarelli and Silvestre.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística
Differential equations, Elliptic
Conjecture of De Giorgi
One-dimensional symmetry
Sums of fractional Laplacians
Equacions diferencials el·líptiques
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
info:eu-repo/semantics/submittedVersion
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