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

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