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July  2010, 28(3): 1179-1206. doi: 10.3934/dcds.2010.28.1179

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, Spain

Received  April 2010 Published  April 2010

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.
Keywords: entire solutions., Half-Laplacian, energy estimates, symmetry properties.
Mathematics Subject Classification: Primary: 35J60, 35R10; Secondary: 35B05, 35J20, 60J7.
Citation: Xavier Cabré, Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1179-1206. doi: 10.3934/dcds.2010.28.1179
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