Boundary value problems for elliptic systems with block structure

Abstract: I’ll consider a very simple elliptic PDE in the upper half-space: divergence form, transversally independent coefficients and no mixed transversal-tangential derivatives. In this case, the Dirichlet problem can formally be solved via a Poisson semigroup, but there might not be a heat semigroup. The construction is rigorous for L2 data. For other data classes X (Lebesgue, Hardy, Sobolev, Besov,…) the question, whether the corresponding Dirichlet problem is well-posed, is inseparably tied to the question, whether there is a compatible Poisson semigroup on X.

On a "semigroup space" the infinitesimal generator has (almost?) every operator theoretic property that one can dream of and these can be used to prove well-posedness. But it turns out that there are genuinely more "well-posedness spaces" than "semigroup spaces". For example, up to boundary dimension n=4 there is a well-posed BMO-Dirichlet problem, whose unique solution has no reason to keep its tangential regularity in the interior of the domain.

I’ll give an introduction to the general theme and discuss some new results, all based on a recent monograph jointly written with Pascal Auscher.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.