The Green Function for Elliptic Systems in Two Dimensions

J. L. Taylor, S. Kim, R. M. Brown

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


We construct the fundamental solution or Green function for a divergence form elliptic system in two dimensions with bounded and measurable coefficients. Our main goal is construct the Green function for the operator with mixed boundary conditions in a Lipschitz domain. 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(Ω) 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.

Original languageEnglish
Pages (from-to)1574-1600
Number of pages27
JournalCommunications in Partial Differential Equations
Issue number9
StatePublished - Sep 2013

Bibliographical note

Funding Information:
Seick Kim is supported by NRF Grant No. 2010-0008224 and R31-10049 (WCU program). This work was partially supported by a grant from the Simons Foundation (#195075 to Russell Brown).


  • BMO
  • Green function
  • Mixed boundary conditions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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