We will describe the Signorini, or lower-dimensional obstacle problem, for a uniformly elliptic, divergence form operator L = div(A(x)nabla) with Lipschitz continuous coefficients. We will give an overview of this problem and discuss some recent developments, including the optimal regularity of the solution and the $C^{1,alpha}$ regularity of th...
A body of literature has developed concerning "cloaking by anomalous localized resonance". The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, ˆ‡ · (a(x)ˆ‡u(x)) = f(x). The complex-valued coefficient has a matrix-shell-core geometry, with real part equal to 1 in the matrix and the core,...
Creator:
Lu, Jianfeng (Duke University)
Created:
2012-09-14
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
L. Carleson introduced the measures which bear his name to solve an interpolationproblem for analytic functions (Ann. of Math.,1962), establishing their relationship withthe existence of nontangential limits at the boundary. These measures were subsequentlyunderstood within the larger context of duality of tent spaces. Carleson measures haveplay...
Creator:
Pipher, Jill (Brown University)
Created:
2012-05-30
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic input data. Besides stochastic loadings, via perturbation theory, our approach covers also elliptic problems on stochastic domains or with stochastic coefficients. The solution's two-point correlation satisfies a determini...
Creator:
Harbrecht, Helmut (Universität Stuttgart)
Created:
2010-10-21
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.
Consider the Dirichlet problem in a Lipschitz domain in the plane.Suppose that the boundary data is in BMO. I will show that, if thecoefficients have small imaginary part and are independent of one ofthe coordinates, then solutions to the Dirichlet problem satisfy aCarleson-measure condition.
Created:
2012-06-02
Contributed By:
University of Minnesota, Institute for Mathematics and its Applications.