Korevaar-Schoen $p$-energies and their $\Gamma$-limits on Cheeger spaces

Abstract

This paper studies properties of $\Gamma$-limits of Korevaar-Schoen $p$-energies on a Cheeger space. When $p>1$, this kind of limit provides a natural $p$-energy form that can be used to define a $p$-Laplacian, and whose domain is the Newtonian Sobolev space $N^{1,p}$. When $p=1$, the limit can be interpreted as a total variation functional whose domain is the space of BV functions. When the underlying space is compact, the $\Gamma$-convergence of the $p$-energies is improved to Mosco convergence for every $p \ge 1$.