### General Info

Remember: for every question you **must** indicate all the people with whom you collaborated. If you didn't collaborate with anyone else on a given problem, write "Collaborators: none."

Please write legibly, or type-up your solutions in LaTeX. The grader will greatly appreciate it.

This problem set will be due *in class, Monday, October 30*.

### Problems

- Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^d\). Suppose \(u\in C^2([0,T]\times \overline{\Omega})\) solves the initial-boundary value problem \[ \partial_t u - \triangle u + u^3 = 0 \] with \[ u(0,x) = 0 ~~ (x\in \Omega), \qquad u(t,y) = 0 ~~ (y\in \partial\Omega).\] Prove, using the energy method, that \( u\equiv 0\).
- Prove that the general solution to the partial differential equation
\[ \partial_x \partial_y u(x,y) = 0 \]
is
\[ u(x,y) = F(x) + G(y).\]
(
*Hint: the equation implies*\( \partial_y u\)*is independent of*\(x\).*Apply fundamental theorem of calculus twice on a rectangle in the*\(x,y\)*coordinates.*) - Making use of the previous question, show that, in general, the initial value problem for \( \partial^2_{xy} u = 0\) with data
\[ u(0,y) = f(y), \qquad \partial_x u(0,y) = g(y) \]
has either no solutions, or infinitely many solutions.

(*Remark: compare this to the analogous discussion when we talked about the theorem of Cauchy-Kovalevskaya.*) - Prove that if \(u\in C^\infty(\mathbb{R} \times \mathbb{R}^d)\) solves \(\Box u = 0\) with the initial conditions \[ u(0,x) = 0, \qquad \partial_t u(0,x) = h(x),\] then the function \( v := \partial_t u\) solves \(\Box v = 0\) with the initial conditions \[ v(0,x) = h(x), \qquad \partial_t v(0,x) = 0.\]
- Let \(\vec{E}, \vec{B} \in C^\infty(\mathbb{R}\times \mathbb{R}^3; \mathbb{R}^3)\) solve
*Maxwell's equations*\begin{gather} \partial_t \vec{E} = \mathrm{curl}~ \vec{B} \newline \partial_t \vec{B} = - \mathrm{curl}~ \vec{E} \newline \mathrm{div} \vec{E} = \mathrm{div} \vec{B} = 0 \end{gather} (where \(\mathrm{div} \vec{E} = \sum_{i = 1}^3 \partial_{x_i} E_i\) and \(\mathrm{curl}\vec{E} = (\partial_{x_2} E_3 - \partial_{x_3} E_2, \partial_{x_3} E_1 - \partial_{x_1} E_3, \partial_{x_1} E_2 - \partial_{x_2} E_1) \)). Prove that the components of the vectors \(\vec{E}, \vec{B}\) satisfy \[ \Box E_i = 0, \qquad \Box B_i = 0.\] (*Remark: Maxwell's equations describe the dynamics of a free electromagnetic field. The above shows that the electromagnetic field strengths obey wave equations.*) - Let \(\vec{u} \in C^\infty(\mathbb{R}\times \mathbb{R}^3; \mathbb{R}^3)\) solve the
*linear isotropic elastic equations*\[ - \partial^2_{tt} \vec{u} + \mu \triangle \vec{u} + (\lambda + \mu) \vec{\nabla}( \mathrm{div} \vec{u}) = 0\] (here \(\mu > 0\) is the shear modulus of the material and \(\lambda\) [which can be negative] is Lamé's first parameter for the material). Show that the quantities \(\mathrm{div}~\vec{u}\) and \(\mathrm{curl}~\vec{u}\) each solve a wave equation, though with (in general) different wave speeds.

(*Remark: the wave corresponding to*\(\mathrm{div}~\vec{u}\)*is the "P-wave" in seismology, and the wave corresponding to*\(\mathrm{curl}~\vec{u}\)*is the "S-wave". Your results should show that for materials in which*\(\lambda \geq 0\),*the P-wave propagates faster than the S-wave.*) - Let \(u\in C^2(\mathbb{R}\times\mathbb{R})\) solve the initial value problem for the wave equation:
\begin{gather}
\Box u = - \partial^2_{tt} u + \partial^2_{xx} u = 0\newline
u(0,x) = g(x) \newline
\partial_t u(0,x) = h(x)
\end{gather}
Assume \(g,h\) both have compact support. Define the
*kinetic energy*of \(u\) to be \[ k(t) = \frac12 \int_{\mathbb{R}} [\partial_t u(t,x)]^2 ~\mathrm{d}x \] and the*potential energy*to be \[ p(t) = \frac12 \int_{\mathbb{R}} [\partial_x u(t,x)]^2 ~\mathrm{d}x.\] Prove, with the help of the fundamental solution of the wave equation in one dimension, that the total energy is conserved (i.e. \(k(t) + p(t)\) is constant in \(t\)). (*Remark: this statement is also true in higher dimensions, as we see in the "energy method" section of the course for wave equations.*) - Continuing from the previous question, using again the fundamental solution, prove that there exists a time \(T\) depending on \(g,h\) such that \(k(t) - p(t) = 0\) for all \(t > T\).

(*Remark: this second statement is known as the "equipartition of energy". In higher dimensions, in general, it is only true asymptotically [in the sense that*\( \lim_{t\to +\infty} k(t) - p(t) = 0\)*]; this result is most easily derived through Fourier analysis.*) - Let \(u\) solve the initial value problem for the wave equation in 3 dimensions:
\begin{gather}
\Box u = 0 \newline
u(0,x) = g(x) \newline
\partial_t u(0,x) = h(x)
\end{gather}
where we assume \( g, h\in C^\infty_c(\mathbb{R}^3)\). Prove, using the Kirchhoff parametrix, that
- for every \(x_0\in \mathbb{R}^3\), the function \(t \mapsto u(t,x_0)\) has compact support.
- for every \(t_0 \in \mathbb{R}\), the function \(x \mapsto u(t_0,x)\) has compact support.

- Let \( \phi\in C^2(\mathbb{R}\times\mathbb{R}^d)\), identify \(\partial_{x_0}\) with \(\partial_t\). Denote by \(m\) the \((d+1)\times(d+1)\) diagonal matrix indexed by \(i,j\in \{0, \ldots, d\}\) such that \(m_{00} = -1\) and \(m_{ii} = 1\) for every \(i > 0\). (This \(m\) is known as the "Minkowski metric".) Define the (symmetric) matrix-valued function \(Q\) whose components are given by
\[ Q_{ij} = \partial_{x_i} \phi \partial_{x_j}\phi - \frac12 m_{ij} ( \sum_{k = 1}^d [\partial_{x_k} \phi]^2 - [\partial_{x_0} \phi]^2) \]
where \(i,j\in \{0, 1, 2, \ldots, d\}\).

Verify that \[ \sum_{k,\ell = 0}^d \partial_{x_k} (m_{k\ell} Q_{\ell j}) = \Box \phi \cdot \partial_{x_j} \phi .\]