Regularity versus singularity for elliptic problems in two dimensions

created by beck on 11 Nov 2009
modified on 16 Feb 2015

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Inserted: 11 nov 2009
Last Updated: 16 feb 2015

In two dimensions every weak solution to a nonlinear elliptic system $\rm{div } a(x,u,Du)=0$ has Hölder 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.