The regularity problem for the Laplace equation and boundary Poincaré inequalities in rough domains

Xavier Tolsa (ICREA - Universitat Autònoma de Barcelona - CRM)

05-Jul-2022, 16:00-17:00 (21 months ago)

Abstract: Given a bounded domain $\Omega \subset \mathbb R^n$, one says that the $L^p$-regularity problem is solvable for the Laplace equation in $\Omega$ if, given any continuous function $f$ defined in $\partial \Omega$ and the harmonic extension $u$ of $f$ to $\Omega$, the non-tangential maximal function of the gradient of $u$ can be controlled in $L^p$ norm by the tangential derivative of $f$ in $\partial\Omega$. Up to quite recently this was only known to hold for Lipschitz domains (in some range of $p$'s). In my talk I will explain a recent result with Mihalis Mourgoglou where we show that the $L^p$-regularity is also solvable in more general domains, such as 2-sided chord-arc domains. In the solution of this problem, the Poincaré inequality in the boundary of the domain plays an important role. I will also discuss this issue and a related joint result with Olli Tapiola where we show that the boundaries of 2-sided chord-arc domains support 1-Poincaré inequalities.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control

Audience: researchers in the topic


Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

Organizers: Simon Blatt*, Philipp Reiter*, Armin Schikorra*, Guofang Wang
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