Energy estimates and 1-D symmetryfor nonlinear equations involving the half-Laplacian

Xavier Cabré; Eleonora Cinti

doi:10.3934/dcds.2010.28.1179

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2010, Volume 28, Issue 3: 1179-1206. Doi: 10.3934/dcds.2010.28.1179

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Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

Xavier Cabré^1, and
Eleonora Cinti^2,

1.
ICREA and Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada 1, Diagonal 647, 08028 Barcelona
2.
Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada 1, Diagonal 647, 08028 Barcelona

Received: March 31, 2010

Published: August 2010

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Abstract

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)$^1/2 $u=f(u)$ in R ⁿ. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable.
As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in R ⁿ.

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Mathematics Subject Classification: Primary: 35J60, 35R10; Secondary: 35B05, 35J20, 60J75.

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