Applied Math/PDE Seminar: Svitlana Mayboroda (U. of Minnesota) Boundary value problems for elliptic operators with real non-symmetric coefficients

4607B South Hall

One of the simplest and the most important results in elliptic theory is the maximum principle. It provides sharp estimates for the solutions to elliptic PDEs in $L^infty$ in terms of the corresponding norm of the boundary data. It holds on arbitrary domains for all (real) second order divergence form elliptic operators $- div A nabla$. The well-posedness of boundary problems in $L^p$, $p<infty$, is a far more intricate and challenging question, even in a half-space. In particular, it is known that some smoothness of $A$ in $t$, the transversal direction to the boundary, is needed.

In the present talk we shall discuss the well-posedness in $L^p$ for elliptic PDEs associated to matrices $A$ of real (possibly non-symmetric) coefficients independent on the transversal direction to the boundary. In combination with our earlier perturbation theorems, this result shows that the Dirichlet and Regularity boundary value problems are well-posed in some $L^p$, $1<p<infty$, whenever (roughly speaking) $|A(x,t)-A(x,0)|^2 dxdt/t$ is a small Carleson
measure.


This is joint work with S. Hofmann, C. Kenig, and J. Pipher.

 

Friday, 18 April, 2014

Contact:

Carlos Garcia-Cervera

Phone: 18055638873

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