Mathematical Research Letters

Volume 7 (2000)

Number 5

Regularity of weak solutions to critical exponent variational equations

Pages: 651 – 656

DOI: http://dx.doi.org/10.4310/MRL.2000.v7.n5.a11

Authors

Karen K. Uhlenbeck (The University of Texas)

Jeff A. Viaclovsky (Massachusetts Institute of Technology)

Abstract

We present a general method for proving regularity of weak solutions to variational equations with critical exponent nonlinearities. We will focus primarily on the $C^{\infty}$ regularity of $L^2_2$ solutions to a nonlinear fourth order variational \mbox{equation} in $4$ dimensions. This equation was considered by Chang, Gursky, and Yang in \cite{CGY}, where regularity was obtained only for minimizers using techniques from Morrey~\cite{Morrey} and Schoen-Uhlenbeck~\cite{SU}. The \mbox{methods} in this paper apply to a more general class of critical exponent variational equations in $n$ dimensions with leading term a power of the Laplacian.

Full Text (PDF format)