Math 6410 • ODE

\begin{enumerate}
\item[(a)] Consider a string clamped at the end points \(a,b\) with
\(u(a,t)=u_a\), \(u(b,t)=u_b\), where \(u(x,t)\) is the string's deviation from the
horizontal rest position. The kinetic energy of the string is
\[
  E_k=\frac{1}{2}\int_a^b \rho u_t^2\dd s,\qquad
  \dd s=\sqrt{1+u_x^2}\dd x,
\]
where \(\rho(x,t)\) is the mass density. The potential energy due to stretching of
the string is
\[
  E_p=\int_a^b k\left(\sqrt{1+u_x^2}-1\right)\dd x,
\]
where \(k(x,t)\) is the elastic coefficient.
\begin{enumerate}
\item[i.] Find the first variation of the action
\[
  A=\int_0^T(E_k-E_p)\dd t.
\]
\item[ii.] Integrating by parts, imposing the boundary conditions, and equating the
first variation to zero, derive the PDE
\[
  \left(\rho u_t\sqrt{1+u_x^2}\right)_t
  =
  \left(k(1+u_x^2)^{-1/2}u_x\right)_x.
\]
Assuming \(\rho,k\) are constants and using the approximation \(\abs{u_x}\ll1\) derive
the classical one-dimensional wave equation.
\end{enumerate}
\item[(b)] Consider the functional
\[
  Y[u]=\int_D\left(\frac{1}{2}\abs{\nabla u}^2+\frac{1}{2}u^2+fu\right)\dd x,
\]
where \(D\) is a bounded domain in \(\R^n\) and \(f\) is a bounded, continuous function
in \(D\). Suppose that there exists a minimizer \(u^*\) of \(Y[u]\) in the Sobolev space
\(H_1(D)\). Use this to prove existence and uniqueness of the weak formulation of
the PDE (for which \(u\in C^1(D)\))
\[
  -\nabla^2u+u=-f,\qquad x\in D\subset\R^n,\qquad
  \partial_nu=0,\qquad x\in\partial D.
\]
[Hint: show that the weak formulation can be expressed as
\((u,\psi)_{H_1(D)}=0\) for all \(\psi\in H_1(D)\).]
\end{enumerate}