Viscosity Solutions

Lu = aij(x)Diju for u ∈ C(Ω). Suppose u ∈ C(Ω) is a supersolution in Ω, i.e. Lu ≤ 0. Then if φ ∈ C(Ω) satisfies Lφ > 0, we get L(u− φ) < 0 in Ω, hence by the maximum principle, u− φ does not have interior local minima in Ω. Put differently, if φ ∈ C(Ω) is such that u− φ has a local minimum at x0 ∈ Ω, then necessarily Lφ(x0) ≤ 0. Geometrically, u− φ having a local minimum at x0 means that the graph of φ touches the graph of u from below at x0, if we shift φ by a constant. Definition 1.1. Let f ∈ C(Ω). We say that u ∈ C(Ω) is a viscosity supersolution (resp. subsolution) of Lu = f in Ω if for all x0 ∈ Ω and all functions φ ∈ C(Ω) such that u− φ has a local minimum (resp. maximum) at x0, the inequality Lφ(x0) ≤ f(x0) (resp. ≥ f(x0))