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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.

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain Ω ⊂ R n , n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N , D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p 0 > 1 depending on the domain Ω such that for p in the interval (1, p 0 ), the mixed problem with Neumann data in the space L p (N ) and Dirichlet data in the Sobolev space W 1,p (D) has a unique solution with the non-tangential maximal function of the gradient of the solution in L p (∂Ω). We also obtain results for p = 1 when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces. 1 2 J.L. TAYLOR, K.A. OTT, AND R.M. BROWN data [9]. The mixed boundary value problem in Lipschitz domains appears as an open problem in Kenig's CBMS lecture notes [16, Problem 3.2.15]. There is a large literature on boundary value problems in polyhedral domains and we do not attempt to summarize this work here. See the work of Bȃcuţȃet. al. [1] for recent results for the mixed problem in polyhedral domains and additional references.Under mild restrictions on the boundary data we can use energy estimates to show that there exists a solution of the mixed problem with ∇u in L 2 of the domain. Our goal in this paper is to obtain more regularity of the solution and, in particular, to show that ∇u lies in L p (∂Ω). Brown [2] showed that the solution satisfies ∇u ∈ L 2 (∂Ω) when the data f N is in L 2 (N ) and f D is in the Sobolev space W 1,2 (D) for a certain class of Lipschitz domains. Roughly speaking, his results hold when the Dirichlet and Neumann portions of the boundary meet at an angle strictly less than π. In this same class of domains, Sykes and Brown [31] obtain L p results for 1 < p < 2 and I. Mitrea and M. Mitrea [24] establish well-posedness in an essentially optimal range of function spaces. Lanzani, Capogna and Brown [20] establish L p results in two dimensional graph domains when the data comes from weighted L 2 -spaces and the Lipschitz constant is less than one. The aforementioned results rely on a variant of the Rellich identity. The Rellich identity cannot be used in the same way in general Lipschitz domains because it produces estimates in L 2 , and even in smooth domains simple examples show that we cannot expect to have solutions with gradient in L 2 (∂Ω).Ott and Brown [26] establish conditions on Ω, N , and D which ensure uniqueness of solutions of the L p -mixed problem and they also establish conditions on Ω, N , D and f N and f D which ensure that solutions to the L p -mixed problem exist. All of this work is done under an additional geometric assumption on the boundary of D. More specifically, the authors address solvability of the mixed problem for the Laplacian in bounded Lipschitz domains under the assumption that the boundary between D and N (relative to ∂Ω) is l...

We consider the eigenvalues of an elliptic operatorwhere u = (u 1 , ..., u m ) t is a vector valued function and a αβ (x) are (n × n) matrices whose elements a αβ ij (x) are at least uniformly bounded measurable real-valued functions such thatfor any combination of α, β, i, and j. We assume we have two non-empty, open, disjoint, and bounded sets, Ω and Ω, in R n , and add a set Tε of small measure to form the domain Ωε. Then we show that as ε → 0 + , the Dirichlet eigenvalues corresponding to the family of domains {Ωε} ε>0 converge to the Dirichlet eigenvalues corresponding to Ω 0 = Ω ∪ Ω. Moreover, our rate of convergence is independent of the eigenvalues.In this paper, we consider the Lamé system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre-Hadamard ellipticity condition. Mathematics Subject Classification Numbers: 35, 43Keywords: eigenvalues, elliptic systems, perturbed domainswhere ε 0 (J) depends on the multiplicity of σ 0 J . Moreover, the rate a > 0 is independent of any eigenvalue and C only depends on the eigenvalue σ 0 J and the distance from σ 0 J to nearby eigenvalues.