Nonlinear Dirichlet forms associated with quasiregular mappings

Abstract

If $(\mathcal{E},\mathcal{D})$ is a symmetric, regular, strongly local Dirichlet form on $L^2(X,m)$, admitting a carré du champ operator $\Gamma$, and $p>1$ is a real number, then one can define a nonlinear form ${\mathcal{E}}^p$ by the formula $${\mathcal{E}}^p(u,v)={\int }_X\Gamma (u{)}^{\frac{p-2}{2}}\Gamma (u,v)dm,$$ where $u$, $v$ belong to an appropriate subspace of the domain $\mathcal{D}$. We show that ${\mathcal{E}}^p$ 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 ${\mathcal{E}}^n$ ($p=n$) such that the components of $f$ become harmonic functions with respect to ${\mathcal{E}}^n$. Finally, we obtain Caccioppoli type inequalities in the intrinsic metric induced by $\mathcal{E}$, for harmonic functions with respect to the form ${\mathcal{E}}^p$.