José González Llorente, Universitat Autónoma de Barcelona
The classical mean value property of harmonic functions plays a remarkable
role in Geometric Function Theory and it is the fundamental key of the interplay
between Potential Theory, Probability and Brownian motion.
Things are more delicate when the differential operator is not linear. A relevant
exemple is the p-laplacian for 1 < p < ∞. The p-laplacian
generalizes the usual laplacian (recovered when p = 2) and arises in a natural way
when minimizing the p-norm of the gradient in a Dirichlet problem. It is connected
to many other questions in Function Theory, PDE’s, elasticity theory, etc... In the
last years there has been an increasing interest in searching probabilistic models
for the p- laplacian and the starting point is to identify which is the natural mean
value property associated to it.
In the course we will first review some striking results about the classical mean
value property and its connection to harmonicity. We will then focus on the (nonlinear)
mean value property associated to the p-laplacian, including simple probabilistic
interpretations. Finally we will describe some new results, generalizations
and open questions.