Elliptic regularity estimates with optimized constants and applications

TL;DR

This paper refines classical elliptic regularity estimates by optimizing constants in key inequalities, providing sharper tools for elliptic PDE analysis and applications to conjectures and spectral theory.

Contribution

It introduces optimized constants in fundamental elliptic estimates, demonstrating their optimality and applying these results to the Landis conjecture and spectral bounds.

Findings

Optimized constants in maximum principle and H"older estimates

Counterexamples confirming optimality of estimates

Applications to Landis conjecture and spectral estimates

Abstract

We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\"older continuous gradients, and prove versions of the generalized maximum principle, the $C^{1,\alpha}$-estimate, the Hopf-Oleinik lemma, the boundary weak Harnack inequality and the differential Harnack inequality, in which the constant is optimized with respect to the norms of the coefficients of the operator and the size of the domain. Our estimates are complemented by counterexamples which show their optimality. We also give applications to the Landis conjecture and spectral estimates.