โ Scribed by Dynkin E.B.
No coin nor oath required. For personal study only.
This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that will be of interest to anyone who works on applications of probabilistic methods to mathematical analysis. The book is suitable for graduate students and research mathematicians interested in probability theory and its applications to differential equations. Also of interest by this author is Diffusions, Superdiffusions and Partial Differential Equations in the AMS series, Colloquium Publications.
๐ SIMILAR VOLUMES
Suppose L is a second-order elliptic differential operator in R d and D is a bounded, smooth domain in R d . Let 1 < ฮฑ โค 2 and let ฮ be a closed subset of โD. It is known [13] that the following three properties are equivalent: (ฮฑ) ฮ is โ-polar; that is, ฮ is not hit by the range of the correspondi
This Letter presents some special features of a class of integrable PDEs admitting billiard-type solutions, which set them apart from equations whose solutions are smooth, such as the KdV equation. These billiard solutions are weak solutions that are piecewise smooth and have first derivative discon