The role of small space dimensions in the regularity theory of elliptic problems

The role of small space dimensions in the regularity theory of elliptic problems

Mon, 15/11/2010
17:00 		Lisa Beck (Scuola Normale Superiore di Pisa) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
The role of small space dimensions in the regularity theory of elliptic problems
Let $ u \in W^{1,p}(\Omega,\R^N) $, $ \Omega $ a bounded domain in $ \R^n $, be a minimizer of a convex variational integral or a weak solution to an elliptic system in divergence form. In the vectorial case, various counterexamples to full regularity have been constructed in dimensions $ n \geq 3 $, and it is well known that only a partial regularity result can be expected, in the sense that the solution (or its gradient) is locally continuous outside of a negligible set. In this talk, we shall investigate the role of the space dimension $ n $ on regularity: In arbitrary dimensions, the best known result is partial regularity of the gradient $ Du $ (and hence for $ u $) outside of a set of Lebesgue measure zero. Restricting ourselves to the partial regularity of $ u $ and to dimensions $ n \leq p+2 $, we explain why the Hausdorff dimension of the singular set cannot exceed $ n-p $. Finally, we address the possible existence of singularities in two dimensions.