## Introduction

This course will cover the theory of viscosity solutions for nonlinear first and second order partial differential equations (PDEs). The notion of viscosity solution is based on a comparison principle (or maximum principle), and is the natural notion of weak solution for many fully nonlinear PDEs of first and second order. Applications of viscosity solutions include problems in optimal control, differential games, mathematical finance, curvature motion of curves and surfaces, calculus of variations in Linfinity, among many other problems.

Topics include the comparison principle for viscosity solutions for first and second order PDEs, the method of vanishing viscosity, the Perron method, homogenization, a thorough study of monotone finite difference approximations for viscosity solutions, including convergence rates, and possibly other topics.

Grades will be determined based on a selection of homework problems and a final presentation of a paper from the field. The prerequisites are a graduate course in real analysis and at least one semester of a graduate course in PDEs. Please contact the instructor (jcalder at umn dot edu) if you are interested in the course and do not have the prerequisites.

The animation above is a numerical simulation of the mean curvature motion PDE, which moves a curve in the plane in the direction of its inward normal vector with a speed equal to its curvature. Mean curvature motion arises in physical systems that involve surface tension, such as soap film/bubbles and biological cell membranes, and many other fields of pure and applied mathematics.

For a closed curve in the plane, curvature motion can be studied in the classical setting; that is, the solution exists globally in time and is smooth. For surfaces in 3D and higher dimensions, singularities can develop in the surface, such as topological changes (e.g., the number of connected components can change), and we are unable to interpret solutions classically beyond such singularities. Viscosity solutions offer the correct notion of generalized mean curvature motion, and describe how the surface evolves after formation of singularities. This is one example of a topic we will study in the course.