Abstract
We study the nonlinear fractional equation \((-\Delta )^su=f(u)\) in \(\mathbb R ^n,\) for all fractions \(0<s<1\) and all nonlinearities \(f\). For every fractional power \(s\in (0,1)\), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension \(n=3\) whenever \(1/2\le s<1\). This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation \(-\Delta u=f(u)\) in \(\mathbb R ^n\). It remains open for \(n=3\) and \(s<1/2\), and also for \(n\ge 4\) and all \(s\).
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Both authors were supported by grants MINECO MTM2011-27739-C04-01 (Spain) and GENCAT 2009SGR345 (Catalunya). The second author was partially supported by University of Bologna (Italy), funds for selected research topics, and by the ERC Starting Grant “AnOptSetCon” n. 258685.
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Communicated by O.Savin.
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Cabré, X., Cinti, E. Sharp energy estimates for nonlinear fractional diffusion equations. Calc. Var. 49, 233–269 (2014). https://doi.org/10.1007/s00526-012-0580-6
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DOI: https://doi.org/10.1007/s00526-012-0580-6