Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions

Abstract

(1.2) div Φσ(u,Du) = Φξ(u,Du). Using compactness methods based on the translation invariance of the equation (1.2), and a priori estimates in C1 norm, we prove various properties of bounded entire solutions of (1.2), such as a sharp inequality for the gradient, energy monotonicity and optimal growth, Liouville type results, and one-dimensional symmetry. An important role in this program is played by the function