Special Issue dedicated to Numerical Methods for Viscosity Solutions and Applications

摘要

The theory of viscosity solutions for nonlinear PDEs was born in the 1980s and is still an active and important field of research in mathematics. It has provided a sound framework for a number of differential models of great importance in applications. Such problems are characterized by common mathematical features including: 1.Solutions with low regularity. Viscosity solutions are often continuous, but not differentiable, and singularities may encode information important to applications; for example, switching curves in control problems. Moreover, in some applications even continuity may not be present. 2.Strong nonlinearities and possible degeneracies of differential operators. 3.Solutions defined over high dimensional spaces; for example, the Hamilton-Jacobi-Bellman equations for optimal flight trajectories evolve on an underlying space of at least six dimensions, while stochastic control problems from mathematical finance may have twenty dimensions.