Boundary regularity of solutions to variable-exponent degenerate mixed elliptic equations

We investigate the boundary regularity of solutions to a class of variable-exponent gradient degenerate mixed fully nonlinear local-nonlocal elliptic Dirichlet problems. A crucial feature of the operators under consideration is that they degenerate on the set of critical points, C : = { x ∈ Ω : D u ( x ) = 0 } . First, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, s ∈ ( 1 / 2 , 1 ) (Theorem 1.5), under general conditions. Due to inapplicability of the comparison principle for the equations under consideration, the classical Perron’s method for the existence of a solution cannot be employed. However, utilizing the Lipschitz estimates established in Theorem 1.5 and “vanishing viscosity” method, we prove the existence of a solution. Subsequently, we establish the interior C 1 , δ -regularity of viscosity solutions using an improvement of the flatness technique when s is close enough to 1 (Theorem 1.6). Furthermore, under suitable assumptions, we establish the Hölder regularity of solutions up to the boundary (Theorem 1.15), a result that is new even for analogous nonlocal Dirichlet problems.