Let us prove(78). Now the fact that Φ(x,x)≤Φ2(xe,ye) te -yel ≤V(x)-V2(v)≤C for suitable C>0(recall V2 is bounded on Q2), and so x-y|≤(Ce) Therefore xe-y→0as and by continuity, V2(e)-v2(ye)-0 as E-0; hence(79) gives lre-ye12 We now need to consider where the points e, ye lie Case(i). Suppose ag, ye E Q(both interior points), for all sufficiently small e >0. Let o()=(2)--,o2()=12()+-uP Now o1, 2 E C( Q2), ae is a local maximum for V1-2, and ye is a local minimum for V2-o1. Alse 2(xe) The viscosity sub- and supersolution definition implies V1(x)+H(xe,)≤0 V2(v)+H(ye,)≥0. Subtracting we have (x)-V2(y)+H( <0 and using the assumption on H V(x2)-V(v)≤u1(|x-/1+2-ml This implies 重(xe,y)≤u1(xe-yl(1+ and hence(78) follows using( 81)and( 82) 19Let us prove (78). Now the fact that Φε(xε, xε) ≤ Φε(xε, yε) implies |xε − yε| 2 2ε ≤ V2(xε) − V2(yε) ≤ C (79) for suitable C > 0 (recall V2 is bounded on Ω), and so |xε − yε| ≤ (Cε) 1/2 . (80) Therefore |xε − yε| → 0 as ε → 0, (81) and by continuity, V2(xε) − V2(yε) → 0 as ε → 0; hence (79) gives |xε − yε| 2 2ε → 0 as ε → 0. (82) We now need to consider where the points xε, yε lie. Case (i). Suppose xε, yε ∈ Ω (both interior points), for all sufficiently small ε > 0. Let φ1(y) = V1(xε) − |xε − y| 2 2ε , φ2(x) = V2(yε) + |x − yε| 2 2ε , (83) Now φ1, φ2 ∈ C 1 (Ω), xε is a local maximum for V1 − φ2, and yε is a local minimum for V2 − φ1. Also, ∇φ1(yε) = xε − yε ε = ∇φ2(xε). (84) The viscosity sub- and supersolution definition implies V1(xε) + H(xε, xε−yε ε ) ≤ 0, V2(yε) + H(yε, xε−yε ε ) ≥ 0. (85) Subtracting we have V1(xε) − V2(yε) + H(xε, xε − yε ε ) − H(yε, xε − yε ε ) ≤ 0 (86) and using the assumption on H V1(xε) − V2(yε) ≤ ω1(|xε − yε|(1 + |xε − yε| ε )). (87) This implies Φε(xε, yε) ≤ ω1(|xε − yε|(1 + |xε − yε| ε )), (88) and hence (78) follows using (81) and (82). 19