The Hamilton-Jacobi equation u_t + H(∇u) = 0 with convex H has the Hopf-Lax solution formula:
Au(x,t) = min_y {u₀(y) + t·L((x-y)/t)}, where L is the Legendre transform of H
Bu(x,t) = u₀(x - H'(0)t)
Cu(x,t) = ∫H(ξ)û₀(ξ)e^{iξx}dξ
Du(x,t) = max_y {u₀(y) - |x-y|²/(2t)}
The Hopf-Lax formula gives the viscosity solution as a minimum over all 'paths' from initial data to the point (x,t). L = H* is the Legendre-Fenchel transform (convex conjugate) of H, playing the role of the Lagrangian in the variational principle. For H(p) = |p|²/2, this gives u(x,t) = min_y{u₀(y) + |x-y|²/(2t)}.
Question 2 True / False
Hamilton-Jacobi equations are closely connected to optimal control theory.
TTrue
FFalse
Answer: True
The value function V(x,t) = inf over controls of {cost + terminal value} satisfies the Hamilton-Jacobi-Bellman equation V_t + H(x, ∇V) = 0, where H(x,p) = min_a{f(x,a)·p + L(x,a)}. This connects PDE theory to dynamic programming: solving the HJ equation is equivalent to solving the optimization problem.
Question 3 Short Answer
Why do classical solutions of Hamilton-Jacobi equations break down?
Think about your answer, then reveal below.
Model answer: Characteristics carrying different initial slopes cross in finite time, causing the gradient ∇u to become multi-valued
For H(p) = |p|²/2, characteristics are straight lines with slopes depending on ∇u₀. When the initial data has varying slope, faster characteristics overtake slower ones, and the classical solution ceases to exist. The viscosity solution develops corners (kinks) where ∇u is discontinuous.
Question 4 Multiple Choice
The eikonal equation |∇u| = 1 is a stationary Hamilton-Jacobi equation whose viscosity solution is:
AThe distance function to the boundary
BA harmonic function
CThe Green's function
DA constant
The viscosity solution of |∇u| = 1 in Ω with u = 0 on ∂Ω is u(x) = dist(x, ∂Ω), the distance to the boundary. The gradient of the distance function has magnitude 1 wherever it is differentiable, and it satisfies the equation in the viscosity sense everywhere.