Regularity of solutions to a class of variable-exponent fully nonlinear elliptic equations

Abstract

In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the p(x)-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class C 1,κ for a universal constant 0 < κ < 1. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.

A fully nonlinear degenerate free transmission problem

“…Next, we apply the matrix inequality (15) to special vectors as to recover information about the eigenvalues of X − Y . First, apply (15) to vectors of the form (z, z) ∈ R 2d to get…”

“…We then conclude that all the eigenvalues of (X − Y ) are less than or equal to 4L 2 + 2ι. Now, by applying (15) to…”

“…The case of variable exponents θ = θ(x) is the topic of [15]. In that paper the authors examine the regularity of viscosity solutions under the assumption that θ : B 1 → R is a continuous function.…”