Paweł Strzelecki | Some problems on harmonic functions

Here are some problems on harmonic functions. If you wish to present a solution of one of them on Tuesday Oct 31 or Tuesday Nov 7 (and collect a few informal brownie points in front of your classmates), please feel free to do so. You might send me an email a few days before.

Problem 1. Let $u$ be a harmonic function on $\mathbb{R}^n$ . Fix a nonnegative function $\varphi\in C_0^\infty(\mathbb{R}^n)$ which depends only on $r=|x|=(x_1^2+\ldots+x_n^2)^{1/2}$ and satisfies $\int \varphi(x)\, dx=1$ .

For $\varepsilon>0$ set $\varphi{_\varepsilon} (x) =\varepsilon^{-n}\varphi(x/\varepsilon)$ and consider the convolution $u_{\varepsilon}$ of the functions $u$ and $\varphi_{\varepsilon}$ , given by

$u_\varepsilon(x)\equiv u\ast\varphi_\varepsilon(x) = \int_{\mathbb{R}^n} \varphi_\varepsilon(x-y) u(y)\, dy\, .$

Do the following:

prove that $u_{\varepsilon} \equiv u$ ;
deduce that harmonic functions are smooth.

Problem 2. Let $u$ be a harmonic function on $\mathbb{R}^n$ . Assume that for some constant $C>0$ we have

$|u(x)| \le C(1+ |x|)^m, \qquad x\in \mathbb{R}^n.$

Prove that $u$ is a polynomial of degree at most $m$ .

Hint: use estimates for derivates that follow from the mean value formula.

Problem 3. Check that the maximum principle does not hold in unbounded domains (hint: work in dimension 2).

Problem 4 (Liouville theorem for harmonic functions). Prove that a harmonic function $u$ on $\mathbb{R}^n$ such that $\sup u = M < +\infty$ must be constant.

Hint: use Harnack’s inequality for balls.

Problem 5. Let $u$ be a harmonic function on $\mathbb{R}^n$ . Assume that $a\in \mathbb{R}^n$ and the radii $0< r_1 < r_2 < r_3$ form a geometric sequence: $r_2^2=r_1\, r_3$ . Let $\sigma$ denote the surface measure on the unit sphere $\mathbb{S}^{n-1}$ . Prove that

$\int_{\mathbb{S}^{n-1}}\ u(a+r_{1} x)\, u(a+r_{3} x)\, d\sigma(x) =\int_{\mathbb{S}^{n-1}}\ u^2(a+r_2x)\, d\sigma(x)\, .$

Problem 6. Assume that $u$ is harmonic on $B(0,2)\setminus \{0\} \subset \mathbb{R}^n$ and

$\lim_{x\to 0}\frac{u(x)}{\Gamma(x)} = 0\, ,$

where $\Gamma$ stands for the fundamental solution of the Laplace operator. Prove that $u$ has a finite limit at $0$ and can be extended to a harmonic function on $B(0,2)$ .