23-34 Xifeng Su, Enrico Valdinoci Yuanhong Wei, and Jiwen Zhang

On some regularity properties of mixed local and nonlocal elliptic equations (174K, LaTeX) May 18, 23

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Abstract. This article is concerned with ``up to $C^{2, lpha}$-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 $C^{1,lpha}$-regularity and the $C^{1,lpha}$-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 $C^{2,lpha}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,lpha}$-regularity up to the boundary for all $s\in(0,rac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.

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