open access

https://doi.org/10.14943/99357

https://hdl.handle.net/2115/82542

Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by p-Laplacian

英語

2021-08-30

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1137

1-29

We consider a one-Laplace equation perturbed by p-Laplacian with 1 < p < ∞. We prove that a weak solution is continuously differentiable (C1) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show C1-regularity of the solution at the boundary of a facet where the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limit is a constant function by establishing a Liouville-type result, which is proved by showing a strong maximum principle. Our argument is rather elementary since we assume that the solution is convex. A few generalization is also discussed.

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