A Priori Estimates for General Elliptic and Parabolic Boundary Value Problems Over Irregular Domains

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).