ENERGY ESTIMATES AND 1-D SYMMETRY FOR NONLINEAR EQUATIONS INVOLVING THE HALF-LAPLACIAN

By X. Cabré, Eleonora Cinti. 2010

106 citations recorded by the source corpus.

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation 1/2 in R n. Our energy estimates hold for every nonlinearity and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension , we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation in R n.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5d321c0c2314e2816660399668d266bed52821a1

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Citations in the corpus

Hitchhiker's guide to the fractional Sobolev spaces (2011, unknown): “, [18, 88, 13, 17, 78], etc.”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “See [13, 60, 10, 11, 12] for progresses in this direction and [32, 3, 56, 25, 30, 31] for the state-of-the-art results on the conjecture in the classical case, formally obtained by taking s = 1.”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, background): “In particular, it is shown that a multidimensional standing wave solution must be one dimensional (without counting the dimension of extension) for certain low dimensions and s (see also [16,17]).”
Nonlocal Diffusion and Applications (2015, background): “NONLOCAL DIFFUSION AND APPLICATIONS 51 For n “ 3, the proof can be found in [13] for s P ””
Sharp energy estimates for nonlinear fractional diffusion equations (2012, methodology): “1 in [4], we consider a C∞ function K(x), defined on R with compact support in B1 and such that ∫ Rn K(x)dx = 1.”