Gaussian estimates vs. elliptic regularity on open sets

TL;DR

This paper explores the relationship between Gaussian estimates, elliptic regularity, and harmonic analysis for elliptic operators with mixed boundary conditions on complex open sets, extending existing theorems to broader contexts.

Contribution

It generalizes an equivalence theorem to mixed boundary conditions and non-Lipschitz domains, unifying properties of heat kernels, harmonic functions, and energy growth.

Findings

Established equivalence between Gaussian bounds and elliptic regularity for complex boundary conditions.

Extended the theorem to open sets beyond Lipschitz domains.

Provided kernel bounds enabling harmonic analysis on rough open sets.

Abstract

Given an elliptic operator $L= - \mathrm{div} (A \nabla \cdot)$ subject to mixed boundary conditions on an open subset of $\mathbb{R}^d$, we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, H\"older continuity of $L$-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet we prove consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.