Regularity results for nonlocal equations and applications
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Calculus of Variations
Regularity results for nonlocal equations and applications Mouhamed Moustapha Fall1 Received: 13 February 2020 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce the concept of C m,α -nonlocal operators, extending the notion of second order elliptic operator in divergence form with C m,α -coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Hölder continuity of the gradient of the solutions in the case of C 0,α -coefficients and the classical Schauder estimates for C m+1,α -coefficients. We further apply the regularity results for C m,α -nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided. Mathematics Subject Classification 35B65 · 47G20
1 Introduction We are concerned with a class of (not necessarily translation invariant) elliptic equations driven by nonlocal operators of fractional order. We extend in the nonlocal setting some key existing results for elliptic equations in divergence form with C m,α -coefficients. For a better description of how far the results in this paper extend to the fractional setting those available in the classical case, we start by recalling some main results of the classical local theory. We consider a weak solution u ∈ H 1 () to the equation N
∂i (ai j (x)∂ j u) = f
in ,
(1.1)
i, j=1
Communicated by A. Malchiodi. The author’s work is supported by the Alexander von Humboldt foundation. He thanks Joaquim Serra, for his interest in this work and with whom he had stimulating discussions that help to improve the first section of this paper. He also thanks the anonymous referee for useful comments.
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Mouhamed Moustapha Fall [email protected]; [email protected] African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal, B.P. 14 18, Mbour, Senegal 0123456789().: V,-vol
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where, is a bounded open subset of R N , f ∈ L loc (), p > N /2, and the matrix coefficients ai j are measurable functions and satisfy, for every x ∈ , the following properties: (i) ai j (x) = a ji (x)
for all i, j = 1, . . . , N ,
(1.2) 1 for all i, j = 1, . . . , N . δi j κ In the regularity theory for elliptic equations in divergence form with measurable coefficients, the De Giorgi–Nash–Moser theory provides a priori C 0,α0 () estimates for weak solutions to (1.1), for some α0 = α0 (N , p, κ), see e.g. [33]. The range or value of the largest Hölder exponent α0 is known in general once the coefficients are sufficiently regular. For instance, 0,β if ai j ∈ C() then u ∈ Cloc () for all β < min(2 − N / p, 1). Now Hölder continuous coefficients ai j yield Hölder continuity of the gradient of u. Namely, if ai j ∈ C 0,α (), for (ii) κδi j ≤ ai j (x) ≤
1,min(1− N ,α)
p (), provided 2− N / p > 1. Moreover the Schauder some α ∈
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