Heat kernel estimates for fourth-order non-uniformly elliptic operators with non-strongly convex symbols

Abstract

We obtain heat-kernel estimates for fourth-order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This entails certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are inducedby the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequality and an interpolation inequality, which are related to the singularity or degeneracy of the coefficients.