# 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."

This problem set will be due in class, Friday September 29.

### Problems

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.
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