The purpose of these lectures is to give a quick and elementary, yet rigorous, presentation of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are nonsmooth and standard approaches in order to define a ¿weak solution¿ do not apply because the standard idea of using ¿integration-by-parts¿ in order to pass derivatives to smooth test functions by duality, is not available. After developing some main theory, we will focus on applications in the modern field of Calculus of Variations in L¿.
Part I General Theory
- History, Examples, Motivation and First Definitions
- Second Definitions and Basic Analytic Properties of the Notions
- Stability Properties of the Notions and Existence via Approximation
- Mollification of Viscosity Solutions and Semiconvexity
- Existence of Solution to the Dirichlet Problem via Perron's Method
-Comparison results and Uniqueness of Solution to the Dirichlet Problem
Part II Applications
- Minimisers of Convex Functionals and Viscosity Solutions of the Euler-Lagrange PDE
Existence of Viscosity Solutions to the Dirichlet Problem for the ¿Laplacian
The course will be held by Prof. Dr. Nicos Katzourakis, Department of Mathematics and Statistics, University of Reading
