Well-posedness in $L^p$ for elliptic boundary value problems

Lundi 19 mars 2012 14:00-15:00 - Svitlana Mayboroda - University of Minnesota

Résumé : 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, $\mathbbR^n+1_+$. 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 $A$ independent on the transversal direction to the boundary. In combination with our earlier perturbation theorems, this result shows that the Dirichlet boundary value problem is well-posed in some $L^p$, $p<\infty$, whenever (roughly speaking) $|A(x,t)-A(x,0)|^2 dxdt/t$ is a small Carleson measure. Such a result was only known in the setting of real symmetric matrices [D. Jerison, C. Kenig, 1981]. The non-symmetric case was open since then, and ultimately had to be approached by completely different techniques. In 2000 Kenig, Koch, Pipher, and Toro established the well-posedness for non-symmetric matrices in dimension $2$. The present work pertains to all dimensions $n\geq 2$.
This is joint work with S. Hofmann, C. Kenig, and J. Pipher.

Lieu : bât. 425 - 113-115

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