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 BM O(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.

The Green function for the mixed problem for the linear Stokes system in domains in the plane

“…The argument can be made rigorous by approximating the δ-functions by normalized characteristic functions of balls centered at x and y and letting the radii tend to zero. See the work of Taylor, Kim, and Brown [30] for a similar argument.…”

A heterogeneous stochastic FEM framework for elliptic PDEs

“…For d ≤ 3, based on the estimates of Green's function in [16,34], we have that for fixed x i and ω ∈ Ω, (2.5) where C i depends on x i , λ min , λ max and the domain D, but does not depend on the dimension of ω. Integrating (2.5) in Ω, (2.6) where ψ j i (y) and η j i (ω), j = 1, .…”

Heat kernel for the elliptic system of linear elasticity with boundary conditions

“…We also prove that if the gradient of weak solutions of the system Lu = 0 satisfy the growth condition called Dirichlet property (see H3 in Section 2.5), then the heat kernel has a Gaussian upper bound; see Theorem 3.10. The Dirichlet property is known to hold in the case when Ω is a Lipschitz domain in R 2 and D satisfies the corkscrew condition (see [25]) or when the coefficients and domains are sufficiently smooth and D ∩N = ∅ (see [9]). As an application, we construct Green's function for the elliptic system from the heat kernel and in the presence of Dirichlet property, we show that the Green's function has the usual bound of C|x − y| 2−n (or logarithmic bound if n = 2); see Theorem 4.3.…”

“…We should mention that our paper, though technically more involved, is an extension of their method. The elliptic Green function for (MP) in two dimensional domains is constructed in Taylor et al [25] by a different method not involving the heat kernel.…”

“…Therefore, the conclusions of Theorem 3.1 are valid. In fact, if Ω is a Lipschitz domain and D is a (possibly empty) set satisfying the corkscrew condition, i.e., for each x ∈ ∂D (where the boundary is taken with respect to ∂Ω) and r ∈ (0, r 0 ), we may find x r ∈ D so that |x − x r | ≤ r and dist(x r , ∂Ω \ D) ≥ M −1 r, where r 0 > 0 and M > 0 are constants, then it is known that H3 holds; see [25]. Therefore, the conclusions of Theorem 3.10 are valid in this case.…”