On some regularity properties of mixed local and nonlocal elliptic equations

TL;DR

This paper investigates the regularity of solutions to mixed local and nonlocal elliptic equations, establishing new estimates and regularity results up to the boundary, including for critical nonlinearities and fractional Laplacian operators.

Contribution

It extends existing regularity results for mixed local-nonlocal elliptic equations to include critical nonlinearities and boundary regularity, with sharp exponents for all fractional orders.

Findings

Established $L^ abla$ bounds for solutions with critical nonlinearities.

Proved interior and boundary $C^{1,eta}$ regularity for solutions.

Proved interior and boundary $C^{2,eta}$ regularity for all fractional orders.

Abstract

This article is concerned with ``up to $C^{2, \alpha}$-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 $C^{1,\alpha}$-regularity and the $C^{1,\alpha}$-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 $C^{2,\alpha}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,\alpha}$-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.