Energy Estimates and 1-d Symmetry for Nonlinear Equations Involving the Half-laplacian
نویسندگان
چکیده
We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation (−∆)u = f(u) in R. 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 −∆u = f(u) in R.
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