Mixed Boundary Value Problems for non-Divergence Type Elliptic Equations in Unbounded Domain

TL;DR

This paper studies the behavior of solutions to non-divergence elliptic equations with mixed boundary conditions in unbounded domains, establishing growth and decay principles based on boundary structure and capacity.

Contribution

It introduces a Phragmén-Lindelöf type principle for non-divergence elliptic equations with mixed boundaries in unbounded domains, linking solution behavior to boundary capacity and admissibility conditions.

Findings

Established growth/decay estimates depending on boundary capacity.

Linked solution behavior to boundary structure and admissibility conditions.

Provided a framework for qualitative analysis of solutions in unbounded domains.

Abstract

We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragm\'en-Lindel\"of type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers converging to infinity.