Nonlinear Dirichlet Forms Associated with Quasiregular Mappings

If \((\mathcal{E}, \mathcal{D})\) $(\mathcal{E},\mathcal{D})$ is a symmetric, regular, strongly local Dirichlet form on \(L^2 (X,m)\) $L^2(X,m)$ , admitting a carré du champ operator \(\Gamma \) $\mathrm{\Gamma }$ , and \(p>1\) $p>1$ is a real number, then one can define a nonlinear form \(\mathcal{E}^p\) ${\mathcal{E}}^p$ by the formula \( \mathcal{E}^p(u,v) = \int _{X} \Gamma (u)^\frac{p-2}{2} \Gamma (u,v)dm , \) $${\mathcal{E}}^p(u,v)={\int }_X\mathrm{\Gamma }{(u)}^{\frac{p-2}{2}}\mathrm{\Gamma }(u,v)dm,$$ where u, v belong to an appropriate subspace of the domain \(\mathcal{D}\) $\mathcal{D}$ . We show that \(\mathcal{E}^p\) ${\mathcal{E}}^p$ 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 \(W_0^{1,p}\) $W_0^{1,p}$ . Using the above procedure, for each n-dimensional quasiregular mapping f we construct a nonlinear Dirichlet form \(\mathcal{E}^n\) ${\mathcal{E}}^n$ ( \(p=n\) $p=n$ ) such that the components of f become harmonic functions with respect to \(\mathcal{E}^n\) ${\mathcal{E}}^n$ . Finally, we obtain Caccioppoli type inequalities in the intrinsic metric induced by \(\mathcal{E}\) $\mathcal{E}$ , for harmonic functions with respect to the form \(\mathcal{E}^p\) ${\mathcal{E}}^p$ .