On the integrability of ?finite energy? solutions for p-harmonic equations

Abstract

We prove a higher integrability result for the gradient of solutions to some degenerate elliptic PDEs, whose model arises in the study of mappings with finite distortion. The nonnegative function $$\mathcal{K}(x)$$ which measures the degree of degeneracy of ellipticity bounds lies in the exponential class, i.e. $$\exp (\lambda \mathcal{K}(x))$$ is integrable for some λ > 0. Our result states that if λ is sufficiently large, then the gradient of a “finite energy” solution actually belongs to the Zygmund space L p logαL,α ≥ 1.