The mean value property: old and new.

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.