Math 6410 • ODE

Let \(a(x)\in C^\infty(\overline{U})\), where \(U\) is a smooth bounded connected domain,
\(U\subset\R^n\). Assume \(a(x)\) is uniformly bounded away from zero on \(U\). Consider the
initial-boundary value problem
\[
(*)\quad
\begin{cases}
  u_t=\operatorname{div}(aDu), & \text{in }U\times(0,\infty),\\
  u=0, & \text{on }\partial U\times(0,\infty),\\
  u=g, & \text{on }U\times\{0\}.
\end{cases}
\]
\begin{enumerate}
\item[(a)] If \(g\) minimizes
\(\lambda=\int_U a\lvert Dg\rvert^2\dd x\) over all \(g\in H_0^1(U)\) satisfying
\(\int_U g^2\dd x=1\), characterize \(\int_U u(x,1)^2\dd x\) in terms of \(\lambda\).
\item[(b)] Let \(J(g)=\int_U u(x,1)^2\dd x\), where \(u\) solves \((*)\) with initial
condition \(g\), and denote
\[
  B=\{g\in L^2(U):\norm{g}_2=1\}.
\]
Denote by \(g_1\) the minimizer of \(\lambda\) from part (a). Prove
\[
  \sup_{g\in B} J(g)
\]
is attained by \(g_1\). Does \(J(g)\) have a minimizer over \(B\)? What is
\(\inf_{g\in B}J(g)\)?
\end{enumerate}