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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.09930 |
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Table of Contents:
- This article is concerned with ``up to $C^{2, α}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities. We then prove the interior $C^{1,α}$-regularity and the $C^{1,α}$-regularity up to the boundary of weak solutions, which extends previous results by the authors [X. Su, E. Valdinoci, Y. Wei and J. Zhang, Math. Z. (2022)], where the nonlinearities considered were of subcritical type. In addition, we establish the interior $C^{2,α}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,α}$-regularity up to the boundary for all $s\in(0,\frac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.