The Green function for elliptic systems in two dimensions

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.

References in corpus (4)

• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• The mixed problem in Lipschitz domains with general decompositions of the boundary
• Neumann functions for second order elliptic systems with measurable coefficients
• The mixed problem for the Lamé system in two dimensions

Cited by in corpus (9)

• Neumann functions for second order elliptic systems with measurable coefficients
• The Green function for the mixed problem for the linear Stokes system in domains in the plane
• Green's function for second order parabolic systems with Neumann boundary condition
• A Heterogeneous Stochastic FEM Framework for Elliptic PDEs
• Green's function for nondivergence elliptic operators in two dimensions
• The mixed problem for the Lamé system in two dimensions
• Heat kernel for the elliptic system of linear elasticity with boundary conditions
• Homogenization Theory of Elliptic System with Lower Order Terms for Dimension Two
• Quantitative Homogenization with Relatively Soft Inclusions and Interior Estimates