A priori estimates for nonlinear elliptic complexes

Abstract

We give a priori estimates for nonlinear partial differential equations in which the ellipticity bounds degenerate. The so-called distortion function which measures the degree of degeneracy of ellipticity is exponentially integrable, thus not necessarily bounded. The right spaces of the gradient of the solutions to such equations turn out to be the Orlicz-Zygmund spaces $L^p\log^\alpha L(\Omega)$. It is the first time the regularity results for genuine nonisotropic PDEs have been successfully treated. Recent advances in harmonic analysis, especially the ${\mathcal H }^{1}-BMO$ duality theory, play the key role in our arguments. Applications to the geometric function theory are already in place [16].