Regularity versus singularities for elliptic problems in two dimensions

2009/10/30 by Beck, Lisa
#35J47 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.0910.5876

Abstract

In two dimensions every weak solution to a nonlinear elliptic system \rmdiv a(x,u,Du)=0 has Hölder continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent p ≥ 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.

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