Non-uniqueness for the $p$-harmonic flow

http://dx.doi.org/10.4153/CMB-1997-021-9
Canad. Math. Bull. 40(1997), 174-182

  Published:1997-06-01
 Printed: Jun 1997

If $f_0\colon\Omega\subset \R^m\to S^n$ is a weakly $p$-harmonic map from a bounded smooth domain $\Omega$ in $\R^m$ (with $2<p<m$) into a sphere and if $f_0$ is not stationary $p$-harmonic, then there exist infinitely many weak solutions of the $p$-harmonic flow with initial and boundary data $f_0$, {\it i.e.,} there are infinitely many global weak solutions $f\colon\Omega\times \R_+\to S^n$ of \begin{gather*} \partial_tf-\rmdiv(|\nabla f|^{p-2}\nabla f)=| f = f_0\quad \mbox{on the parabolic boundary of $\Omega\times \R_+$.} \end{gather*} We also show that there exist non-stationary weakly $(m-1)$-harmonic maps $f_0\colon B^m\to S^{m-1}$.