Mixed Boundary Value Problems on Cylindrical Domains

TL;DR

This paper investigates second-order divergence-form systems on cylindrical domains with mixed boundary conditions, establishing well-posedness and characterizing solutions under certain coefficient assumptions using advanced analytical methods.

Contribution

It introduces new a priori estimates and well-posedness results for mixed boundary value problems on cylindrical domains with coefficients close to structured models.

Findings

Established a priori estimates for solutions.

Proved well-posedness under specific coefficient conditions.

Characterized solutions with bounded non-tangential maximal functions.

Abstract

We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients A are close to coefficients A\_0 that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if A\_0 has a special structure. We obtain a complete characterization of weak solutions whose gradient either has an L^2-bounded non-tangential maximal function or satisfies a Lusin area bound. Our method relies on the first-order formalism of Axelsson, McIntosh, and the first author and the recent solution of Kato's conjecture for mixed boundary conditions due to Haller-Dintelmann, Tolksdorf, and the second author.