2.1.3 Distance function As another example, we consider the distance function d(a, an) to the boundary an of an open, bounded set Q R". In some ways the HJ equation for this function is simpler than that of the optimal control problem described above, and we can more easily explain viscosity solutions and issues of uniqueness, etc, in this context The distance function is defined by d(x,092)=inf|r- Note that the infimum here is always attained, not necessarily uniquely, since an is compact and y H -yl is continuous; denote by r(ar)c an the set of minimizing y We write V(x)=d(x,09) for simplicity, and consider V(a)as a function on the closed set Q2. It can be verified that V(r)is a non-negative Lipschitz continuous function. In fact, we shall see that V is the unique continuous viscosity solution of VV|-1=0in9 (20) satisfying the boundary condition V=0 on aQ Equations(20)and(21)constitute a Dirichlet problem Example 2.1 Q=(1, 1)CR. Here, aQ=f-1, 1 and Q2=[-1, 1.Then v(a) 1+xif-1<x<0 1-xif0<x<1 which is Lipschitz continuous, and differentiable except at x =0. At each point +0 V solves the HJ equation(20), and V satisfies the boundary condition(21)(V(-1 v(1)=0), see figure2. Note that T(x)=-1for-1≤x<0,丌(x)=1for0<x≤1, and T(0 1, 1. The Lipschitz function Vi(a) 1 also satisfies(20)ae. and (21); there are many other such functions Dynamic programming. The distance function satisfies a simple version of the dynamic programming principle: for any r>0 we have V(x)=,inf{x-2|+V(z)} We will use this later to show that V is a viscosity solution of (20), but for now we discuss d derive(222.1.3 Distance Function As another example, we consider the distance function d(x, ∂Ω) to the boundary ∂Ω of an open, bounded set Ω ⊂ Rn . In some ways the HJ equation for this function is simpler than that of the optimal control problem described above, and we can more easily explain viscosity solutions and issues of uniqueness, etc, in this context. The distance function is defined by d(x, ∂Ω) = inf y∈∂Ω |x − y|. (18) Note that the infimum here is always attained, not necessarily uniquely, since ∂Ω is compact and y 7→ |x − y| is continuous; denote by π(x) ⊂ ∂Ω the set of minimizing y. We write V (x) = d(x, ∂Ω) (19) for simplicity, and consider V (x) as a function on the closed set Ω. It can be verified that V (x) is a non-negative Lipschitz continuous function. In fact, we shall see that V is the unique continuous viscosity solution of |∇V | − 1 = 0 in Ω (20) satisfying the boundary condition V = 0 on ∂Ω. (21) Equations (20) and (21) constitute a Dirichlet problem. Example 2.1 Ω = (−1, 1) ⊂ R1 . Here, ∂Ω = {−1, 1} and Ω = [−1, 1]. Then V (x) =  1 + x if − 1 ≤ x ≤ 0 1 − x if 0 ≤ x ≤ 1 which is Lipschitz continuous, and differentiable except at x = 0. At each point x 6= 0 V solves the HJ equation (20), and V satisfies the boundary condition (21) (V (−1) = v(1) = 0), see Figure 2. Note that π(x) = −1 for −1 ≤ x < 0, π(x) = 1 for 0 < x ≤ 1, and π(0) = {−1, 1}. The Lipschitz function V1(x) = |x| − 1 also satisfies (20) a.e. and (21); there are many other such functions. Dynamic programming. The distance function satisfies a simple version of the dynamic programming principle: for any r > 0 we have V (x) = inf |x−z|<r {|x − z| + V (z)}. (22) We will use this later to show that V is a viscosity solution of (20), but for now we discuss and derive (22). 9