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Title of Journal: Calc Var

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Abbravation: Calculus of Variations and Partial Differential Equations

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Springer Berlin Heidelberg

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DOI

10.1002/asna.18901231605

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1432-0835

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Sharp energy estimates for nonlinear fractional di

Authors: Xavier Cabré Eleonora Cinti
Publish Date: 2012/12/30
Volume: 49, Issue: 1-2, Pages: 233-269
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Abstract

We study the nonlinear fractional equation Delta su=fu in mathbb R n for all fractions 0s1 and all nonlinearities f For every fractional power sin 01 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 onedimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension n=3 whenever 1/2le s1 This result is the analogue of a conjecture of De Giorgi on onedimensional symmetry for the classical equation Delta u=fu in mathbb R n It remains open for n=3 and s1/2 and also for nge 4 and all sBoth authors were supported by grants MINECO MTM201127739C0401 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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  2. Fractional Poincaré inequality with finite total Q -curvature
  3. Solitary waves for a class of quasilinear Schrödinger equations in dimension two
  4. Thick clusters for the radially symmetric nonlinear Schrödinger equation
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  7. Duality for rectified cost functions
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  17. Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations
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