Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schr\"odinger equations with exterior conditions

We consider Dirichlet exterior value problems related to a class of non-local Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain existence and uniqueness of weak solutions. Next we prove a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier, valid on compact subsets of the domain, and a full anti-maximum principle by restricting to fractional Schr\"odinger operators. Furthermore, we show a maximum principle for narrow domains, and a refined elliptic ABP-type estimate. Finally, we obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.Comment: 35 pages, Liouville-type theorems for semi-linear equations adde

Dirichlet Form Theory and its Applications

Theory of Dirichlet forms is one of the main achievements in modern probability theory. It provides a powerful connection between probabilistic and analytic potential theory. It is also an effective machinery for studying various stochastic models, especially those with non-smooth data, on fractal-like spaces or spaces of infinite dimensions. The Dirichlet form theory has numerous interactions with other areas of mathematics and sciences. This workshop brought together top experts in Dirichlet form theory and related fields as well as promising young researchers, with the common theme of developing new foundational methods and their applications to specific areas of probability. It provided a unique opportunity for the interaction between the established scholars and young researchers