Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 76, pp. 1-11.

Title: Heat kernel estimates for fourth-order non-uniformly elliptic operators with
non-strongly convex symbols
       
Authors: Gerassimos Barbatis (National and Kapodistrian Univ. of Athens, Greece)
         Panagiotis Branikas (National and Kapodistrian Univ. of Athens, Greece)
         
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 induced
by 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. 

Submitted November 4, 2021. Published November 18, 2022.
Math Subject Classifications: 35K40, 47D06, 35K65, 35K67.
Key Words: Heat kernel estimates; higher order operators; singular-degenerate coefficients.