Search papers, labs, and topics across Lattice.
This paper establishes uniform logarithmic Hessian estimates for solutions of non-uniformly elliptic equations in non-divergence form, a significant improvement over previous results. The proof leverages a novel measure estimate from convex analysis, circumventing complex localization or covering arguments. A constructed counterexample demonstrates the necessity of the logarithmic exponent, ruling out stronger power-type estimates in the planar case.
Logarithmic Hessian estimates, not power-law, are the sharpest possible for non-uniformly elliptic equations in non-divergence form.
We establish uniform logarithmic Hessian estimates for solutions of non-uniformly elliptic equations in non-divergence form. The proof is based on a measure estimate from convex analysis without localization or covering arguments. Moreover, we construct a counterexample that rules out uniform power-type Hessian estimates and shows the sharpness of the logarithmic exponent in the plane. Finally, we extend our estimates to strong solutions and apply them to obtain regularity results for Grushin-type equations.
