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^plog^{αL,α ≥ 1}.