Nonlocal self-improving properties

Abstract

International audienceSolutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by integral(Rn)integral(Rn)[u(x) - u(y)][eta(x) - eta(y)]K(x, y) dx dy = integral(Rn) f eta dx for all eta is an element of C-c(infinity) (R-n), where the kernel K( . ) is a measurable function and satisfies the bounds 1/Lambda vertical bar x - y vertical bar(n+2 alpha) <= K(x, y) <= Lambda/vertical bar x - y vertical bar(n+2 alpha) with 0 < alpha < 1, Lambda > 1, while f is an element of L-loc(q)(R-n) for some q > 2n/(n + 2 alpha). The main result states that there exists a positive, universal exponent delta equivalent to delta(n, alpha, Lambda, q) such that for every weak solution u the self-improving property u is an element of W-alpha,W-2 (R-n) double right arrow u is an element of W-loc(alpha+delta,2+delta) (R-n) holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Holder-type inequalities in R-2n and which is implied by delicate covering and exit-time arguments. In turn, such reverse Holder inequalities are based on the concept of dual pairs, that is, pairs (mu, U) of measures and functions in R-2n which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order alpha

Ground state solutions of scalar field fractional Schrödinger equations

“…and for which condition (30) holds. Since the functional J λ,μ is coercive, we deduce by (32) that the sequence {u j } j∈N ⊂ E n s (V ) is bounded.…”

“…See also [42] where the open problem given in [3] was solved. We also mention here, for completeness, some very interesting regularity results for fractional problems proved recently in [29,30]. Motivated by this large interest in the current literature, under suitable conditions on the potential V and exploiting variational methods, we are concerned in the present paper with the study of multiple solutions for the following fractional parametric problem…”

Nonlinear commutators for the fractional p-Laplacian and applications

“…The fractional p-Laplacian has recently received quite some interest, for example we refer to [2,9,10,21,18,16,13,17,23]. Higher regularity is one interesting and very challenging question where only very partial results are known, e.g.…”

A Potential Space Estimate for Solutions of Systems of Nonlocal Equations in Peridynamics

“…Some of these approaches use reverse Hölder inequalities and nonlocal Gehring-type lemmas as in [13], or prove the result via a commutator estimate as in [22] and in [3] using a functional analytic approach. The approaches in [13,22] are local in nature and application of appropriate embedding estimates in their work show improved local differentiability will lead to improved local integrability. The approach in [3] is rather robust and implies local and global regularity results.…”

“…One notices that in the course of the proof of Theorem 3.1 we actually proved that weak solutions u of the coupled nonlocal system satisfy both higher integrability and higher differentiability on a Sobolev scale. This self-improvement of solutions is of independent interest that has been studied in [3,13] for nonlocal elliptic scalar equations. We restrict our attention to regularity of solutions in the scale of the Bessel potential spaces.…”