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arXiv · 2308.06075

Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

Carlo Alberto Antonini·Matteo Cozzi
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Abstract

We address some regularity issues for mixed local-nonlocal quasilinear operators modeled upon the sum of a $p$-Laplacian and of a fractional $(s, q)$-Laplacian. Under suitable assumptions on the right-hand sides and the outer data, we show that weak solutions of the Dirichlet problem are $C^{1, \theta}$-regular up to the boundary. In addition, we establish a Hopf type lemma for positive supersolutions. Both results hold assuming the boundary of the reference domain to be merely of class $C^{1, \alpha}$, while for the regularity result we also require that $p > s q$.

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Carlo Alberto Antonini, Matteo Cozzi. 2023-08-11. Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type. https://arxiv.org/abs/2308.06075

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