Seminar Analysis: Lisa Beck (Bonn)

An introduction to (partial) regularity theory for elliptic problems

In this seminar we will give a survey on some aspects of the classical regularity theory for W^1,p-solutions to elliptic problems (convex variational integral or elliptic systems), restricting ourselves to simple model cases and explaining the challenges behind proving such results. For scalar valued solutions full regularity (continuous or even better) can be established under very mild assumptions, which is nowadays known as the De Giorgi-Nash-Moser theory. In the vectorial case instead, the various component functions and their partial derivative can interact in such a way that the system or variational integral under consideration allows discontinuous or even unbounded solutions, and in fact various counterexamples to full regularity have been constructed. As a consequence, only partial regularity can be expected, in the sense that the solution (or its gradient) is locally continuous outside of a negligible set (the singular set). We will give some heuristics on the general
approach to partial regularity results and then we briefly discuss how in some particular situations (small space dimensions, special structure conditions) an upper bound on the Hausdorff dimension of the singular set can be obtained.

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