HW4 (MTH847 Fall 2017)

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.


  1. 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 \( u\equiv 0\).
  2. 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.)
  3. 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.)
  4. 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 \[ v(0,x) = h(x), \qquad \partial_t v(0,x) = 0.\]
  5. 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.)
  6. 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.)
  7. 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.)
  8. 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.)
  9. 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.
  10. 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 \[ \sum_{k,\ell = 0}^d \partial_{x_k} (m_{k\ell} Q_{\ell j}) = \Box \phi \cdot \partial_{x_j} \phi .\]