Regularity versus singularities for elliptic problems in two dimensions

Abstract

In two dimensions every weak solution to a nonlinear elliptic system $\mathrm{div}\mathrm{a}(\mathrm{x},\mathrm{u},\mathrm{Du})=0$ has Hölder continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent $p\ge 2$. We give an example showing that this result cannot be extended to the subquadratic case, i.e. that weak solutions are not necessarily continuous if $1<p<2$. Furthermore, we discuss related results for variational integrals.

arXiv comment11 pages, revised and slightly extended version