A priori estimates for general elliptic and parabolic boundary value problems over irregular domains

TL;DR

This paper develops a unified framework for establishing a priori estimates and regularity results for a wide class of elliptic and parabolic boundary value problems on irregular domains, applicable to many real-world scientific fields.

Contribution

It introduces a comprehensive approach that handles both local and nonlocal boundary conditions simultaneously, extending solvability and regularity results to complex irregular regions.

Findings

Established solvability for general boundary value problems.

Derived global regularity results for heat equations with complex boundary conditions.

Provided a priori estimates applicable to various scientific applications.

Abstract

We investigate the realization of a myriad of general local and nonlocal inhomogeneous elliptic and parabolic boundary value problems over classes of irregular regions. We present a unified approach in which either local or nonlocal Neumann, Robin, and Wentzell boundary value problems are treated simultaneously. We establish solvability and global regularity results for both the stationary and time-dependent heat equations governed by general differential operators with unbounded measurable coefficients and various boundary conditions at once, first on a general framework, and then by presenting concrete important examples of irregular domains, Wentzell-type boundary conditions, and nonlocal maps. As a consequence, we develop a priori estimates for multiple differential equations under various situations, which are tied to a large number of applications performed over real world regions, such heat transfer, electrical conductivity, stable-like processes (probability theory), diffusion of medical sprays in the bronchial trees, and oceanography (among many others).