Green's function estimates for time measurable parabolic operators on polyhedrons and polyhedral cones

TL;DR

The article derives sharp Gaussian-type bounds for Green's functions of time-measurable parabolic operators on non-smooth 3D domains, specifically polyhedral cones and polyhedra. By introducing mixed weights that track singular behavior near vertices, edges, and the boundary, and by identifying admissible exponent ranges via geometric and spectral data, it provides weighted Green's function estimates that generalize classical Gaussian bounds to domains with corners and edges. These results enable weighted Sobolev and SPDE regularity theories on non-smooth domains and rely on reductions to wedge and half-space models together with cone-analytic techniques. The framework sets the stage for further regularity results for (stochastic) parabolic equations on polyhedral geometries.

Abstract

We provide Green's function estimates for parabolic operators on polyhedrons and polyhedral cones in $\mathbb{R}^3$. These estimates incorporate mixed weights, which include appropriate powers of the distances to the vertices, the edges, and the boundary of the domains. The allowable ranges for the weight parameters are explicitly determined by the geometry of the domains.

Figures (4)

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Theorems & Definitions (50)

• Remark 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• proof
• Remark 2.6
• Proposition 2.7
• proof
• Theorem 2.8
• Remark 2.9