### 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*.

### Problems

- 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 \(\triangle v = g\).
- 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 \(\mathbb{R}^d \setminus \{0\}\). - 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.)* - 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 \(r\).)* - 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.) - Verify that away from the origin \(\{z = 0\}\), the fundamental solution \(\Phi(z)\) verifies \(\triangle \Phi = 0\).

*(Hint: do it in polar coordinates.)* - 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.)* - 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, \[ \max_{\overline{\Omega}} u = \max_{\partial\Omega} u.\]

- 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.
- Let \(v\) be a harmonic function. Show that \(u:= |\nabla v|^2\) is subharmonic.