期刊
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
卷 49, 期 1-2, 页码 233-269出版社
SPRINGER HEIDELBERG
DOI: 10.1007/s00526-012-0580-6
关键词
Fractional laplacian; Energy estimates; Symmetry properties
资金
- MINECO (Spain) [MTM2011-27739-C04-01]
- GENCAT (Catalunya) [2009SGR345]
- University of Bologna (Italy)
- ERC [258685]
- ICREA Funding Source: Custom
- European Research Council (ERC) [258685] Funding Source: European Research Council (ERC)
We study the nonlinear fractional equation (-Delta)(s) u = f (u) in R-n, for all fractions 0 < s < 1 and all nonlinearities f. For every fractional power s is an element of (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 <= 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 R-n. It remains open for n = 3 and s < 1/2, and also for n >= 4 and all s.
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