Abstract
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 language | English |
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Pages (from-to) | 1574-1600 |
Number of pages | 27 |
Journal | Communications in Partial Differential Equations |
Volume | 38 |
Issue number | 9 |
DOIs | |
State | Published - 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).
Keywords
- BMO
- Green function
- Mixed boundary conditions
ASJC Scopus subject areas
- Analysis
- Applied Mathematics