17:00
The role of small space dimensions in the regularity theory of elliptic problems
Abstract
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.