Elliptic equations

Saturday, May 30, 2015 - 3:30pm - 4:00pm
Mariana Smit Vega Garcia (Universität Duisburg-Essen)
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 the regular part of the free boundary.
Saturday, June 2, 2012 - 2:00pm - 2:50pm
Jill Pipher (Brown University)
Saturday, June 2, 2012 - 11:30am - 12:00pm
Consider the Dirichlet problem in a Lipschitz domain in the plane.
Suppose that the boundary data is in BMO. I will show that, if the
coefficients have small imaginary part and are independent of one of
the coordinates, then solutions to the Dirichlet problem satisfy a
Carleson-measure condition.
Friday, June 1, 2012 - 10:00am - 10:50am
Jill Pipher (Brown University)
Wednesday, May 30, 2012 - 2:00pm - 2:50pm
Jill Pipher (Brown University)
L. Carleson introduced the measures which bear his name to solve an interpolation
problem for analytic functions (Ann. of Math.,1962), establishing their relationship with
the existence of nontangential limits at the boundary. These measures were subsequently
understood within the larger context of duality of tent spaces. Carleson measures have
played a fundamental role in the theory of elliptic boundary value problems, especially
in determining solvability of boundary value problems in the context of non-smooth real
Friday, September 14, 2012 - 10:50am - 11:20am
Jianfeng Lu (Duke University)
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, ∇ · (a(x)∇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, and -1 in the shell; one is interested in understanding the resonant behavior of the solution as the imaginary part of a(x) decreases to zero (so that ellipticity is lost).
Thursday, May 31, 2012 - 11:00am - 11:50am
Jill Pipher (Brown University)
Thursday, October 21, 2010 - 2:00pm - 3:00pm
Helmut Harbrecht (Universität Stuttgart)

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 deterministic boundary value problem with the two-fold tensor product operator on the two-fold tensor tensor product domain.

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