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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