Ebook: Regularity Estimates for Nonlinear Elliptic and Parabolic Problems: Cetraro, Italy 2009 <P>Editors: Ugo Gianazza, John Lewis</P>

The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. In particular considerable strides have been made in regularity estimates for degenerate and singular elliptic and parabolic equations over the last several years, and in many unexpected and challenging directions. Because of all these recent results, it seemed high time to create an overview that would highlight emerging trends and issues in this fascinating research topic in a proper and effective way. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields.

---

The purpose of these notes is to explore some simple relations between Markovian path and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants, and Gaussian Markov fields such as the fre field. These relations are first studied in complete generality for the finite discrete setting, then partly generalized to specific examples in infinite and continuous spaces Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics.- Regularity of Supersolutions.- Introduction to random Tug-of-War games and PDEs.- The Problems of the Obstacle in Lower Dimension and for the Fractional Laplacian