Sharp energy estimates for nonlinear fractional diffusion equations

  • Xavier CabréEmail author
  • Eleonora Cinti


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 \(s\).


Fractional laplacian Energy estimates Symmetry properties 

Mathematics Subject Classification

35R11 35B06 35B08 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.ICREA and Universitat Politècnica de CatalunyaDepartament de Matemàtica Aplicada 1BarcelonaSpain
  2. 2.Dipartimento di MatematicaUniversità degli Studi di Bologna BolognaItaly

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