Viscosity solutions

Abstract

In this section we want to present the notion of viscosity solution for equations having the general form

$$E(x,u(x),\nabla u(x),{\nabla ^2}u(x)) = 0.$$

((5.1))

The idea behind this approach is a second-order comparison principle, which makes it suitable for dealing with both elliptic and parabolic problems. Consistently with this goal, we shall assume u to be defined on some locally compact domain A ⊂ ℝⁿ, so that we require every point in the domain A to have a compact neighborhood. This topological assumption is actually very useful, as it allows to deal at the same time with open and closed domains, as well as with domains of the form ℝⁿ⁻¹ × [0, ∞), which typically occur in the study of parabolic problems.