Marco Biroli (Politecnico di Milano)
Markov functionals and nonlinear Dirichlet forms
Abstract.
We introduce the notion of p-homogeneous Markov functionals (p>1) and of the nonlinear Dirichlet form (related to a p-homogeneous Markov functionals (p>1). We investigate in particular the Riemannian case. In this case we prove an Harnack inequality for the positive harmonics relative to a nonlinear Dirichlet form and the Holder continuity of the harmonics. The boundary behaviour is also investigated. For quasi-minima related to a p-homogeneous Markov functionals (p>1) (in the Riemannian case) we prove the local Holder continuity and an Harnack type inequality for positive quasi-minima. In this case we have to modify the techniques of Giaquinta-Giusti and Di Benedetto-Trudinger for the Euclidean case, due to the fact that only a scaled p-Poincaré inequality is assumed (then a scaled q-Poincaré inequality with q>p>0 does not hold in general).

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