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 \(\mathcal{E}^p\).