Regularity versus singularities for elliptic problems in two dimensions

TL;DR

This paper investigates the regularity of solutions to nonlinear elliptic systems in two dimensions, showing solutions are Hölder continuous for growth exponent p ≥ 2, but not necessarily for 1 < p < 2, with implications for variational integrals.

Contribution

It establishes the regularity threshold at p ≥ 2 and provides a counterexample for subquadratic growth, clarifying the limits of solution regularity in elliptic problems.

Findings

Solutions are Hölder continuous for p ≥ 2.

Counterexample shows solutions need not be continuous for 1 < p < 2.

Results impact understanding of regularity in variational problems.

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.