Ross Ogilvie      21st October, 2024
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23.

Subharmonic Functions

Let $\mathrm{\Omega }\subset {ℝ}^n$ be an open and connected region. A continuous function $v:\overline{\mathrm{\Omega }}\to ℝ$ is called subharmonic if for all $x\in \mathrm{\Omega }$ and $r>0$ with $B(x,r)\subset \mathrm{\Omega }$ it lies below its spherical mean: $v(x)\le 𝒮[v](x,r)$.

(a)

Let $v_1,v_2$ be two subharmonic functions. Show that $v=\max <!--\; FUNCTION\; APPLICATION\; -->(v_1,v_2)$ is subharmonic.
(1 point)

(b)

Suppose that $v$ is twice continuous differentiable. Show that $v$ is subharmonic if and only if $-\mathrm{\Delta }v\le 0$ in $\mathrm{\Omega }$. (3 points)

(c)

Prove that every subharmonic function obeys the maximum principle: If the maximum of $v$ can be found inside $\mathrm{\Omega }$ then $v$ is constant. (2 points)

24.

Liouville’s Theorem

Liouville’s theorem (3.10 in the script) says that if $u$ is bounded and harmonic, then $u$ is constant. In this question we give a geometric proof in ${ℝ}^2$ using ball means defined when $\overline{B(x,r)}\subset \mathrm{\Omega }$ through

$$M[v](x,r)=\frac{1}{{\omega }_n{r}^n}{\int \; <!--\; nolimits\; -->}_{B(x,r)}v(y)dy.$$

Theorem 3.5 states that if $u$ is harmonic then it obeys $u(x)=M[u](x,r)$. This beautiful proof comes from the article Nelson, 1961.

(a)

Consider two points $a,b$ in the plane which are distance $2d$ apart. Now consider two balls, both with radius $r>d$, centred on the two points respectively. Show that the area of the intersection is (2 bonus points)

$$\mathrm{area}B(a,r)\cap B(b,r)=2r^2\mathrm{acos}(d{r}^{-1})-2d\sqrt{r^2-d^2}$$

(b)

Suppose that $v$ is a bounded function on the plane: $-C\le v(x)\le C$ for all $x$ and some constant $C$. Show that (2 points)

$$|M[v](a,r)-M[v](b,r)|\le \frac{2C}{{\omega }_2}\left(\pi -2\mathrm{acos}(d{r}^{-1})-\frac{2d}{r}\sqrt{1-d^2{r}^{-2}}\right)$$

(c)

Let $u\in C^2({ℝ}^2)$ be a bounded harmonic function. Complete the proof that $u$ is constant.
(2 points)

25.

Weak Tea

As in the script, for any test function $\psi \in 𝒟((0,R))$, define a test function ${\tilde{\psi }}_a\in B(a,R)$ by

$${\tilde{\psi }}_a(x)=\frac{\psi (|x-a|)}{n{\omega }_n|x-a{|}^{n-1}}.$$

(a)

Describe the support of $\tilde{\psi }$ in terms of the support of $\psi$. (1 point)

(b)

We have defined the spherical mean of a distribution using the formula ${𝒮}_a[F](\psi )=F({\tilde{\psi }}_a)$ for $\psi \in 𝒟((0,R))$. Compute ${𝒮}_a[\delta ]$ for $a\ne 0$ and $a=0$. Does it have weak mean value property? (3 points)

(c)

Take a function $u\in C({ℝ}^n)$ and fix a radius $r$. Consider the function $x\mapsto 𝒮[u](x,r)$. Prove

$$F_{𝒮[u](x,r)}(\phi )=F_u(x\mapsto 𝒮[\phi ](x,r)).$$

(2 points)

(d)

The spherical mean for distributions, as we have defined it, has the center point fixed and takes a test function on $(0,\infty )$ instead of a radius. The formula in the previous part defines a different sort of spherical mean of a distribution ${\widetilde{𝒮}}_r[F]\in {𝒟}^{\prime }({ℝ}^n)$: $$\begin{array}{lll}\hfill {\widetilde{𝒮}}_r[F](\phi )=F(x\mapsto 𝒮[\phi ](x,r))& & \hfill \end{array}$$

What properties would you expect of this spherical mean applied to a harmonic distribution? What are the advantages and disadvantages of this spherical mean compared to the one in the lecture script? What interesting observations can you make about this?
(Bonus points as deserved.)

26.

Back in the saddle

Suppose that $u\in C^2({ℝ}^2)$ is a harmonic function with a critical point at $x_0$. Assume that the Hessian of $u$ has non-zero determinant. Show that $x_0$ is a saddle point. Explain the connection to the maximum principle. (4 points)