JournalsaihpcVol. 36, No. 2pp. 523–543

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

  • Mostafa Fazly

    Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
  • Yannick Sire

    Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA
Symmetry properties for solutions of nonlocal equations involving nonlinear operators cover
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Abstract

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.

Cite this article

Mostafa Fazly, Yannick Sire, Symmetry properties for solutions of nonlocal equations involving nonlinear operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, pp. 523–543

DOI 10.1016/J.ANIHPC.2018.07.004