Poster in PDF format

final report in PDF format

Overview

The p-Laplacian operator holds a central place in nonlinear analysis, and it arises naturally in many biological, physical, and engineering models that involvethe p-energy, a non-linear analogue of the classical Dirichlet energy.

The solutions of problems related to the p-Laplacian have properties that change drastically as the parameter p varies: when p=1, the problems are highly singular and directly related to mean curvature and the theory of minimal surfaces; as p goes from 1 to 2, one returns to the well-understood linear Laplacian; as p increases from 2 to ∞, the problems become increasingly degenerate and more closely connected to variational problems in L^∞.

Problems related to the limiting cases p=1 and p=∞ still pose many difficult challenges, but recent developments have provided promising new methods for attacking them. In particular, much work has recently been done on the statistical and game-theoretic aspects of operators related to the p-Laplacian, leading to new approximations of p-harmonic functions and to new insights into their properties.

Connections between the p-Laplacian and stochastic games, for example, have led to a surprisingly direct proof of the Harnack inequality for p-harmonic functions in the degenerate case, when p > 2. When 1 ≤ p < 2, on the other hand, understanding the local statistics of the 1-Laplacian has made problems in the singular regime far more accessible.

These new approaches have been successful in recovering and simplifying known results.

We also mention here that there are some long-standing open problems related to the p-Laplacian (unique continuation, the strong comparison principle, and complete spectral analysis).

Another key feature is the role of p-Laplacian in subelliptic structures.

This workshop is an opportunity for experts in this area to meet in a stimulating environment where they can share their ideas and focus on these problems.

Objectives

 The workshop will bring together a diverse group of mathematicians investigating connections between nonlinear partial differential equations and probability, broadly interpreted. Talks will enable participants to update one another on the latest developments in the field, and some time will be set for participants to collaborate and discuss in person on problems of common interest.