Uniform Hölder regularity with small exponent in competition-fractional diffusion systems

Abstract

For a class of competition-diffusion nonlinear systems involving the s-power of the Laplacian, s ∈ (0, 1), of the formwe prove that L ∞ boundedness implies C 0,α boundedness for α > 0 sufficiently small, uniformly as β → +∞. This extends to the case s = 1/2 part of the results obtained by the authors in the previous paper [arXiv:1211.6087v1].2010 Mathematics Subject Classification. Primary: 35J65; secondary: 35B40 35B44 35R11 81Q05 82B10.

Variational Problems with Long-Range Interaction

“…belong to the same functional class [13,25], and hence in particular satisfy the same free-boundary condition, that is |∂ ν u i (x 0 )| = |∂ ν u j (x 0 )| on the regular part of the free boundary. A similar difference has been observed in [27,28,29] in the case of fractional operators, that is when the non-locality is in the differential operator.…”

On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition

“…Competition-diffusion nonlinear systems with k-components involving the fractional Laplacian have been the object of a recent literature, starting with [19,20], where the authors provided asymptotic estimates for solutions to systems of the form…”

On s-harmonic functions on cones

“…For the case of standard diffusion, both the asymptotic analysis and the properties of the segregated limiting profiles are fairly well understood, we refer to [13,15,16,24,28] and references therein. Instead, when the diffusion is nonlocal and modeled by the fractional Laplacian, the only known results are contained in [29,30,31,32]. As shown in [29,30], estimates in Hölder spaces can be obtained by the use of fractional versions of the Alt-Caffarelli-Friedman (ACF) and Almgren monotonicity formulae.…”

“…For the statement, proof and applications of the original ACF monotonicity formula we refer to the book by Caffarelli and Salsa [10] on free boundary problems. Let us state here the fractional version of the spectral problem beyond the ACF formula used in [29,30]: consider the set of 2-partitions of S n−1 as…”

“…A classical result by Friedland and Hayman, [21], yields ν ACF = 1 (case s = 1), and the minimal value is achieved for two half spheres; this equality is the core of the proof of the classical Alt-Caffarelli-Friedman monotonicity formula. It was proved in [29] that ν ACF s is linked to the threshold for uniform bounds in Hölder norms for competition-diffusion systems, as the interspecific competition rate diverges to infinity, as well as the exponent of the optimal Hölder regularity for their limiting profiles. It was also conjectured that ν ACF s = s for every s ∈ (0, 1).…”