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

TL;DR

This work develops a unified variational framework to analyze a broad class of local and nonlocal elliptic and parabolic boundary value problems on irregular domains. It introduces generalized operators $\mathcal{A}$ and $\mathcal{B}$, nonlocal boundary data $\mathcal{J}_{\Omega}$ and $\Theta_{\Gamma}$, and a Wentzell-type boundary form $\Lambda_{\Gamma}$ with boundary measure $\mu$, yielding a coercive, continuous bilinear form $\mathcal{E}_{\mu}$. The authors prove solvability, global regularity, and $L^\infty$-type a priori estimates for weak solutions, along with inverse-positivity principles, for both elliptic and parabolic problems across a wide spectrum of irregular domains (e.g., $(\epsilon,\delta)$-domains, Reifenberg flat domains, ramified fractals) and boundary geometries (including Koch snowflakes). They also construct and analyze nonlocal operators (Besov-type maps and Dirichlet-to-Neumann) within this setting, enabling fractional-type interior and boundary data and broad applications to diffusion, heat transfer, and transport on fractal interfaces. The results provide robust tools for modeling physical processes on rough surfaces and irregular regions, with implications for engineering, physics, and probability (stable-like processes).

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).

Figures (2)

• Figure 1: Koch 4-Crystal and 5-Crystal, First and Second Iterations
• Figure 2: Figure above: the domain $\Omega_1$ generated by the maps $F_i$ (taken from ACH-SAB06-01); Figure below: the domain $\Omega_2$ generated by the maps $G_i$ when $\tau=\tau^{\ast}$ (taken from ACH08).

Theorems & Definitions (61)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Theorem 1
• Remark 2
• Lemma 1
• Proposition 1
• proof
• Definition 4