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Fall, Mouhamed Moustapha

Regularity results for nonlocal equations and applications. (English) Zbl 1450.35093

Calc. Var. Partial Differ. Equ. 59, No. 5, Paper No. 181, 52 p. (2020).
Summary: We introduce the concept of \(C^{m,\alpha}\)-nonlocal operators, extending the notion of second order elliptic operator in divergence form with \(C^{m,\alpha}\)-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 \(C^{0,\alpha}\)-coefficients and the classical Schauder estimates for \(C^{m+1,\alpha}\)-coefficients. We further apply the regularity results for \(C^{m,\alpha}\)-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.

Cited in 16 Documents

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J15 Second-order elliptic equations
35R11 Fractional partial differential equations
47G20 Integro-differential operators

Keywords:

Hölder continuity of the gradient; Schauder estimates; optimal higher order regularity estimates
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Full Text: DOI arXiv

References:

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