The Green function for elliptic systems in two dimensions

TL;DR

This paper constructs the Green function for elliptic systems in two dimensions with measurable coefficients, considering mixed boundary conditions in Lipschitz domains, and introduces a boundary-adapted BMO space for solutions.

Contribution

It develops a method to construct Green functions for elliptic systems with mixed boundary conditions in 2D, using a novel boundary-adapted BMO space.

Findings

Constructed Green functions for elliptic systems in 2D.

Extended the theory to mixed boundary conditions in Lipschitz domains.

Provided a boundary-adapted BMO space for solutions.

Abstract

We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. We consider the elliptic system in a Lipschitz domain with mixed boundary conditions. Thus we specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We require a corkscrew or non-tangential accessibility condition on the set where we specify Dirichlet boundary conditions. Our proof proceeds by defining a variant of the space $BMO(\partial \Omega)$ that is adapted to the boundary conditions and showing that the solution exists in this space. We also give a construction of the Green function with Neumann boundary conditions and the fundamental solution in the plane.