Title: The Dirichlet Problem and Boundary Regularity: How To Attain the Boundary Values in a Good Way

Abstract: In the classical Dirichlet boundary value problem one seeks a solution to some differential equation so that it takes prescribed boundary values. I will first focus on the Dirichlet problem for harmonic functions, and explain why it is not always solvable so that the boundary data are taken pointwise. A natural question is then how to relax the problem so that it always has a reasonable solution. This will lead to a discussion of so-called regular and irregular boundary points, where the boundary data are or are not attained as limits. I will then turn to the corresponding nonlinear theory for p-harmonic functions on Rⁿ and metric spaces.

Profile: Anders Björn is a Professor of Mathematics at Linköping University, Sweden, where he also received his PhD in 1995. He was a postdoc at University of Michigan, Ann Arbor, and has later spent longer research periods at Charles University in Prague, University of Cincinnati and the Mittag-Leffler Institute in Stockholm. His research is in analysis on metric spaces, mainly in collaboration with Jana Björn. In particular, he is interested in p-harmonic functions, partial differential equations and various minimization problems. He studies various solving methods and properties of the solutions, such as their interior and boundary regularity and growth estimates. This is research in pure mathematics and its aim is to provide rigorous fundamentals for and a better understanding of some problems, which could come from natural sciences and other fields. Analysis on metric spaces makes it possible to study such questions simultaneously in many different settings, for example on very rough sets and for highly nonhomogeneous media. It also brings new insight into which properties and assumptions are really essential for the theory and which are the main obstructions. A popular-scientific description of Anders's research is at this website.

Language: English, no interpretation.

Target audience: General audience / everyone at OIST and beyond.
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