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

15 November 2010
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.
  • Partial Differential Equations Seminar