Case(i). Now suppose there exists a subsequence e,;→0asi→∞ such that e;∈Og orye;∈0.Ifxs;∈9, V(xa1)-V2(ve:)≤V(x1)-V(ve)→0 (89) asi→∞, or if y;∈Og, V(x;)-V2(ye;)≤(xe;)-V(ve;)→0 asi→o; hence(78) This completes the proof The distance function is the unique viscosity solution of(20),(21). At first sight Theorem 2.3 does not apply to(20),(21). This is because equation(20) does not have the additive V-term that(69)has, and this term was used in an essential way in the proof of Theorem 2.3. In fact, in general viscosity solutions to equations of the form I(, VV)=0 may not be unique! For instance, in the context of the Bounded Real Lemma both the available storage and required supply are viscosity solutions of equations of the type( 91) It turns out that comparison/uniqueness for HJ equation(20) for the distance function can be proved, either directly using additional hypothesis(such as convexity 3, Theorem II.5.9), or via a transformation as we now show We use the Kruskoy transformation. a useful trick. Define W=重(V)=1-e where V is the distance function(19). Then W is a viscosity solution of W(x)+|VV(x)-1=0in9, W=0 on aQ by the general properties of viscosity solutions mentioned above. By Theorem 2.3, we see that w is the unique viscosity solution of (93), and hence V=()-1(W)=-log(1-W) is the unique viscosity solution of (20),(21). Comparison also follows in the same way 2.3.2 Cauchy Problem In this section we simply state without proof an example of a comparison/uniqueness result, 3, Theorem III.3.15]. There are many results like this available, with variot kinds of structural assumptions(e.g(95),(96)which must be checked in order to applyCase (ii). Now suppose there exists a subsequence εi → 0 as i → ∞ such that xεi ∈ ∂Ω or yεi ∈ ∂Ω. If xεi ∈ ∂Ω, V1(xεi ) − V2(yεi ) ≤ V2(xεi ) − V2(yεi ) → 0 (89) as i → ∞, or if yεi ∈ ∂Ω, V1(xεi ) − V2(yεi ) ≤ V1(xεi ) − V1(yεi ) → 0 (90) as i → ∞; hence (78). This completes the proof. The distance function is the unique viscosity solution of (20), (21). At first sight Theorem 2.3 does not apply to (20), (21). This is because equation (20) does not have the additive V -term that (69) has, and this term was used in an essential way in the proof of Theorem 2.3. In fact, in general viscosity solutions to equations of the form H(x, ∇V ) = 0 (91) may not be unique! For instance, in the context of the Bounded Real Lemma both the available storage and required supply are viscosity solutions of equations of the type (91). It turns out that comparison/uniqueness for HJ equation (20) for the distance function can be proved, either directly using additional hypothesis (such as convexity [3, Theorem II.5.9]), or via a transformation as we now show. We use the Kruskov transformation, a useful trick. Define W = Φ(V ) 4 = 1 − e −V , (92) where V is the distance function (19). Then W is a viscosity solution of W(x) + |∇V (x)| − 1 = 0 in Ω, W = 0 on ∂Ω, (93) by the general properties of viscosity solutions mentioned above. By Theorem 2.3, we see that W is the unique viscosity solution of (93), and hence V = Ψ(W) 4 = Φ−1 (W) = − log(1 − W) (94) is the unique viscosity solution of (20), (21). Comparison also follows in the same way. 2.3.2 Cauchy Problem In this section we simply state without proof an example of a comparison/uniqueness result, [3, Theorem III.3.15]. There are many results like this available, with various kinds of structural assumptions (e.g. (95), (96)) which must be checked in order to apply them. 20