HW2 (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, Friday September 29.


  1. Prove that the Laplacian is rotationally invariant. More precisely, prove that if \(\triangle u = f\) where \(u,f\) are real-valued functions on \(\mathbb{R}^d\), then for any orthogonal \(d\times d\) matrix \(O\), the functions \( v(x) = u(Ox)\) and \(g(x) = f(Ox)\) satisfy \(\triangle v = g\).
  2. Let \(u:\mathbb{R}^d \to\mathbb{R}\) be a harmonic function. Show that its Kelvin transform \[ Ku(x) := \frac{1}{|x|^{d-2}} u(x / |x|^2) \] is a harmonic function on \(\mathbb{R}^d \setminus \{0\}\).
  3. Let \(\Omega = \mathbb{R}^d \setminus \overline{B(0,1)}\). Suppose \(u \in C^2(\Omega; \mathbb{R})\) is a harmonic function. Prove that, for every \(x\in \mathbb{R}^d\), the function \(\phi: (|x| + 1, \infty) \to\mathbb{R}\) given by \[ \phi(r) = \frac{1}{|\partial B(x,r)|} \int_{\partial B(x,r)} u(y) ~\mathrm{d}S(y) \] can be expressed in the form \[ \phi(r) = A \int_{|x|+1}^r s^{1-d} ~\mathrm{d}s + B\] for some constants \(A,B\) depending on \(x\) but independent of \(r\).
    (Hint: emulate the proof of the mean value theorem.)
  4. Fix \(k\in \mathbb{N}\). Let \(u:\mathbb{R}^2 \to \mathbb{R}\) be given by \(u(x,y) = \Re (x + iy)^{k}\) (where \(\Re\) stands for the "real part"). Let \(v:\mathbb{R}^2 \to \mathbb{R}\) be given by \( v(x,y) = \Re \exp[ (x + iy)^{k} ]\). Prove that both \(u\) and \(v\) are harmonic functions.
    (Comment: in fact, the components of any meromorphic function are harmonic when regarded as mappings on \(\mathbb{R}^2\) with the poles removed. This shows that in spite of the integral bound from Question 3, there is no bound on the growth rate of \( \max_{\partial B(0,r)} |u| \) as a function of \(r\).)
  5. In class we discovered the fundamental solution for the Poisson's equation \[ - \triangle u = f \] which allows us to write \[ u(x) = \int_{\mathbb{R}^d} \Phi(x-y) f(y) ~\mathrm{d}y \] for some function $\Phi$; our results covered the cases \(d = 2\) and \(d \geq 3\). Complete the coverage by finding, analogously, the fundamental solution in the case $d = 1$. (Justify your answer.)
  6. Verify that away from the origin \(\{z = 0\}\), the fundamental solution \(\Phi(z)\) verifies \(\triangle \Phi = 0\).
    (Hint: do it in polar coordinates.)
  7. Prove that if \(f\in C^2_c(\mathbb{R}^2)\) is such that \[ \int_{\mathbb{R}^2} f(y) ~\mathrm{d}y = 0 \] then any bounded solution \(u\) of \(-\triangle u = f\) must take the form \[ u(x) = C + \int_{\mathbb{R}^2} \Phi(x-y) f(y) ~\mathrm{d}y \] for some constant \(C\).
    (Hint: notice that \(\int \Phi(x) f(y) ~\mathrm{d}y = 0\). First obtain a quantitative estimate for the difference \(\Phi(x-y) - \Phi(x)\) as \(|x| \to \infty\) when \(y\in \mathbb{R}^2\) is held fixed; use this to argue that the function \(x \mapsto \int \Phi(x-y) f(y) ~\mathrm{d}y\) is bounded.)
  8. Let \(\Omega \subset \mathbb{R}^d\) be bounded and open. Given a function \(u\in C^2(\overline{\Omega}; \mathbb{R})\), we say that it is subharmonic if \[ - \triangle u \leq 0 \quad \text{in }\Omega.\] Prove that:
    • If \(u\) is subharmonic, then \[ u(x) \leq \frac{1}{| B(x,r) |} \int_{B(x,r)} u(y) ~\mathrm{d}y, \quad \text{for all } B(x,r) \subseteq \Omega.\]
    • The maximum (but not minimum) principle holds for subharmonic \(u\): that is, \[ \max_{\overline{\Omega}} u = \max_{\partial\Omega} u.\]
  9. Let \(\phi: \mathbb{R}\to\mathbb{R}\) be smooth and convex. Let \(v\) be a harmonic function. Show that \(u := \phi\circ v\) is subharmonic.
  10. Let \(v\) be a harmonic function. Show that \(u:= |\nabla v|^2\) is subharmonic.