Properties. Some properties of viscosity solutions 1.(Consistency. If V E C(@)is a viscosity solution of (1), then for any point a E Q at which v is differentiable we have F(a,v(a), VV(a))=0 2. If V is locally Lipschitz continuous in Q, then F(a,V(), VV(r))=0 a.e. in Q 3. (Stability ) Let VEC(S)(n>0 be viscosity solutions of FN(,V(a), Vv(a))=0 in Q and assume VN- V locally uniformly in 9, and FN- F locally uniformly in 2×R×Rn,asN→∞. Then v∈C(9) is a viscosity solution of(1) 4.(Monotonic change of variable. Let V E C(Q)be a viscosity solution of (1)and 业∈C(R) be such thatΦ(t)>0. Then w=重(V) is a viscosity solution of F(a,yW()),y(W(a)vW(a))=0 (31) where=更-1 2.2 Value Functions are Viscosity Solutions 2.2.1 The Distance Function is a Viscosity Solution We showed in Example 2.2 that in the specific case at hand the distance function is a viscosity solution. Let's now consider the general case. We use the dynamic programming principle(22) to illustrate a general methodology Subsolution property. Let EC(Q2)and suppose that V-o attains a local maximum at o E Q; so there exists r>0 such that the ball B(o, r)CQ and (x)-(x)≤V(xo)-(xo)x∈B(xo,r) We want to show that Vo(xo)|-1≤ Let hE R, and set a =To+th. Then for t>0 sufficiently small E B(ao, r),and from (32) -(o(o+ th)-(ao))<-((o+th) ) V(o Now from the dynamic programming principle(22)we ve (xo)≤th+V(xo+th)Properties. Some properties of viscosity solutions: 1. (Consistency.) If V ∈ C(Ω) is a viscosity solution of (1), then for any point x ∈ Ω at which V is differentiable we have F(x, V (x), ∇V (x)) = 0. 2. If V is locally Lipschitz continuous in Ω, then F(x, V (x), ∇V (x)) = 0 a.e. in Ω. 3. (Stability.) Let V N ∈ C(Ω) (N ≥ 0) be viscosity solutions of F N (x, V N (x), ∇V N (x)) = 0 in Ω, and assume V N → V locally uniformly in Ω, and F N → F locally uniformly in Ω × R × Rn , as N → ∞. Then V ∈ C(Ω) is a viscosity solution of (1). 4. (Monotonic change of variable.) Let V ∈ C(Ω) be a viscosity solution of (1) and Ψ ∈ C 1 (R) be such that Φ0 (t) > 0. Then W = Φ(V ) is a viscosity solution of F(x, Ψ(W(x)), Ψ 0 (W(x))∇W(x)) = 0 (31) where Ψ = Φ−1 . 2.2 Value Functions are Viscosity Solutions 2.2.1 The Distance Function is a Viscosity Solution We showed in Example 2.2 that in the specific case at hand the distance function is a viscosity solution. Let’s now consider the general case. We use the dynamic programming principle (22) to illustrate a general methodology. Subsolution property. Let φ ∈ C 1 (Ω) and suppose that V −φ attains a local maximum at x0 ∈ Ω; so there exists r > 0 such that the ball B(x0, r) ⊂ Ω and V (x) − φ(x) ≤ V (x0) − φ(x0) ∀ x ∈ B(x0, r). (32) We want to show that |∇φ(x0)| − 1 ≤ 0. (33) Let h ∈ Rn , and set x = x0 + th. Then for t > 0 sufficiently small x ∈ B(x0, r), and so from (32), −(φ(x0 + th) − φ(x0)) ≤ −(V (x0 + th) − V (x0)) (34) Now from the dynamic programming principle (22) we have V (x0) ≤ t|h| + V (x0 + th) (35) 12