Regularity versus singularities for elliptic problems in two dimensions

Abstract

In two dimensions every weak solution to a nonlinear elliptic system $\rm{div} a(x,u,Du)=0$ has H\"older continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent $p \geq 2$. 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.