A function V E C(@) is a viscosity subsolution of (1) if, for any o E C(@)and any al maximum to EQ of V-o we have F(x0,V(xo),Vo(xo)≤0 A function V E C( )is a viscosity supersolution of(1)if, for any oE C(Q2)and any local minimum To E Q2 of V-o we have F(x0,V(xo),Vo(x0)≥ A function V E C(Q2)is a viscosity solution of(1) if it is both a subsolution and a This definition may at first sight appear strange, though in practice it is often easy to use. Note that derivatives in(24 ) and(25) appear only on the smooth function There re a number of equivalent formulations, and the key point is that the definitions relate sub-or superdifferentials(of functions which need not be differentiable) to inequalities associated with the HJ equation The superdiferential of a function V E C(Q2)is defined by +v(x)={A∈R: lim sup y y→x,y∈ The subdifferential of a function V E C(S)is defined by Dv(x)={A∈R”: liminf V()-(a)-(y-) If V E C(S) then D+v(r)= D-v(a)=fvv(a)). In general, A E D+v(a)iff there exists E C(Q)such that Vo(r)=A and V-o has a local maximum at and A E D-V() iff there exists o E C(@2)such that Vo(a)=A and V-o has a loc minimum at ?r Therefore the viscosity definition is equivalently characterized by F(x,V(x),)≤0VA∈Dv(x) F(x,V(x),A)≥0VA∈Dv( Example 2.2 Continuing with Example 2. 1, we see that {1}if-1<x<0 D+V(r)= d-v(ar) {-1}if0<x<1 V(0)=[-1,1] Consequently V is ity solution of(20) However, the function Vi is not is viscosity solution, since OE D-Vi(0)=[1,1],and 0-120A function V ∈ C(Ω) is a viscosity subsolution of (1) if, for any φ ∈ C 1 (Ω) and any local maximum x0 ∈ Ω of V − φ we have F(x0, V (x0), ∇φ(x0)) ≤ 0 (24) A function V ∈ C(Ω) is a viscosity supersolution of (1) if, for any φ ∈ C 1 (Ω) and any local minimum x0 ∈ Ω of V − φ we have F(x0, V (x0), ∇φ(x0)) ≥ 0 (25) A function V ∈ C(Ω) is a viscosity solution of (1) if it is both a subsolution and a supersolution. This definition may at first sight appear strange, though in practice it is often easy to use. Note that derivatives in (24) and (25) appear only on the smooth function φ. There are a number of equivalent formulations, and the key point is that the definitions relate sub- or superdifferentials (of functions which need not be differentiable) to inequalities associated with the HJ equation. The superdifferential of a function V ∈ C(Ω) is defined by D +V (x) = {λ ∈ Rn : lim sup y→x, y∈Ω V (y) − V (x) − λ(y − x) |x − y| ≤ 0} (26) The subdifferential of a function V ∈ C(Ω) is defined by D −V (x) = {λ ∈ Rn : lim inf y→x, y∈Ω V (y) − V (x) − λ(y − x) |x − y| ≥ 0} (27) If V ∈ C 1 (Ω) then D+V (x) = D−V (x) = {∇V (x)}. In general, λ ∈ D+V (x) iff there exists φ ∈ C 1 (Ω) such that ∇φ(x) = λ and V − φ has a local maximum at x; and λ ∈ D−V (x) iff there exists φ ∈ C 1 (Ω) such that ∇φ(x) = λ and V − φ has a local minimum at x. Therefore the viscosity definition is equivalently characterized by F(x, V (x), λ) ≤ 0 ∀ λ ∈ D +V (x) (28) and F(x, V (x), λ) ≥ 0 ∀ λ ∈ D −V (x) (29) Example 2.2 Continuing with Example 2.1, we see that D+V (x) = D−V (x) =  {1} if − 1 < x < 0 {−1} if 0 < x < 1 D+V (0) = [−1, 1], D−V (0) = ∅. (30) Consequently V is a viscosity solution of (20). However, the function V1 is not is viscosity solution, since 0 ∈ D−V1(0) = [−1, 1], and |0| − 1 6≥ 0. 11