Nonlinear Dirichlet forms associated with quasiregular mappings

Abstract

If $({\cal E}, {\cal D})$ is a symmetric, regular, strongly local Dirichlet form on $L^2 (X,m)$, admitting a carr\'{e} du champ operator $\Gamma$, and $p>1$ is a real number, then one can define a nonlinear form ${\cal E}^p$ by the formula $$ {\cal E}^p(u,v) = \int_{X} \Gamma(u)^\frac{p-2}{2} \Gamma(u,v)dm , $$ where $u$, $v$ belong to an appropriate subspace of the domain ${\cal D}$. We show that ${\cal E}^p$ is a nonlinear Dirichlet form in the sense introduced by P. van Beusekom. We then construct the associated Choquet capacity. As a particular case we obtain the nonlinear form associated with the $p$-Laplace operator on $W_0^{1,p}$. Using the above procedure, for each $n$-dimensional quasiregular mapping $f$ we construct a nonlinear Dirichlet form ${\cal E}^n$ ($p=n$) such that the components of $f$ become harmonic functions with respect to ${\cal E}^n$. Finally, we obtain Caccioppoli type inequalities in the intrinsic metric induced by ${\cal E}$, for harmonic functions with respect to the form ${\cal E}^p$.