Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
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We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation 1/2 in R n. Our energy estimates hold for every nonlinearity 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 , 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 in R n.
CitationCabré, X.; Cinti, E. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. "Discrete and continuous dynamical systems. Series A", Novembre 2010, vol. 28, núm. 3, p. 1179-1206.