Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations

Abstract

We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, C:={x:Du(x)=0}. Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, s?(12,1). Due to inapplicability of comparison principle for the equations under consideration, one can not use the classical Perron's method for the existence of a solution. However, using the Lipschitz estimates established in theorem and vanishing viscosity method, we get the existence of solution. We further prove interior C1,? regularity of the viscosity solutions using an improvement of the flatness technique when s is close enough to 1.