Sharp energy estimates for nonlinear fractional diffusion equations

By X. Cabré, Eleonora Cinti. 2012

112 citations recorded by the source corpus.

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 $$s$$.

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

View Paper: https://www.semanticscholar.org/paper/2e2fe9f2b141db48dccfb12841a32e3f1b847d45

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The Pohozaev Identity for the Fractional Laplacian (2012, background): “For such extension, some Pohozaev type identities are proved in [4–6].”
Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions (2011, background): “12 below) and when n = 3 and s ≥ 1/2 (as a consequence of a result from [8]).”
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]).”