Math 6410 • ODE

Consider the scalar problem
\[
\begin{aligned}
  u_t+H(Du) &= 0, && \text{in } \R^n\times(0,\infty),\\
  u &= x_1, && \text{on } \R^n\times\{t=0\}.
\end{aligned}
\]
Assume the Hamiltonian
\[
  H(p)=\frac{1}{4}\sum_{j=1}^n\frac{1}{a_j}(p_j-b_j)^2,
\]
where the coefficients \(a_j,b_j\) are fixed real numbers, with \(a_j>0\).
\begin{enumerate}
\item[(a)] Use the Legendre transform
\(H^*(q)=\sup_p\{p\cdot q-H(p)\}\) to find the Lagrangian.

\item[(b)] Setting the coefficients \(a_j=1\), \(b_j=0\) for all \(j\), use the Hopf-Lax
formula
\[
  u(x,t)=\min_y\left\{tL\left(\frac{x-y}{t}\right)+g(y)\right\}
\]
to find an explicit weak solution to the problem above.
\end{enumerate}