Ross Ogilvie      28th October, 2024
______________________________________________________________________________________________________________________________

27.

To be or not to be

Consider the Dirichlet problem for the Laplace equation $\mathrm{\Delta }u=0$ on $\mathrm{\Omega }$ with $u=g$ on $\partial \mathrm{\Omega }$, where $\mathrm{\Omega }\subset {ℝ}^n$ is an open and bounded subset and $g$ is a continuous function. We know from the weak maximum principle that there is at most one solution. In this question we see that for some domains, existence is not guaranteed.

(a)

Consider $\mathrm{\Omega }=B(0,1)\setminus \{0\}$, so that the boundary $\partial \mathrm{\Omega }=\mathit{\partial 𝐵}(0,1)\cup \{0\}$ consists of two components. We write $g(x)=g_1(x)$ for $x\in \mathit{\partial 𝐵}(0,1)$ and $g(0)=g_2$. Show that there does not exist a solution for $g_1(x)=0$ and $g_2=1$.

Hint. Use Lemma 3.23, even if we haven’t reached it in lectures yet. (3 points)

(b)

Generalise this: What are the necessary and sufficient conditions on $g$ for the Dirichlet problem to have a solution on this domain? (3 points)

(c)

Generalise again: What can you say about the Dirichlet problem for bounded domains whose boundaries have isolated points? (1 point)

28.

Do nothing by halves

Let $H_1^{+}=\{x=(x_1,\dots ,x_n)\in {ℝ}^n\mid x_1>0\}$ be the upper half-space and $H_1^0=\{x=(x_1,\dots ,x_n)\in {ℝ}^n\mid x_1=0\}$ the dividing hyperplane. We call $R_1(x)=(-x_1,x_2,\dots ,x_n)$ reflection in the plane $H^0$.

(a)

A reflection principle for harmonic functions Let $u\in C^2(\overline{H_1^{+}})$ be a harmonic function that vanishes on $H_1^0$. Show that the function $v:{ℝ}^n\to ℝ$ defined through reflection

$$v(x)=\{\begin{array}{cc}u(x)\hfill & \text{ for }{x}_1\ge 0\hfill \\ -u(R_1(x))\hfill & \text{ for }{x}_1<0\hfill \end{array}$$

is harmonic. (3 points)

(b)

Green’s function for the half-space Show that Green’s function for $H_1^{+}$ is

$$G(x,y)=\mathrm{\Phi }(x-y)-\mathrm{\Phi }(R_1(x)-y).$$

(2 points)

(c)

Green’s function for the half-ball Compute the Green’s function for $B^{+}$. (3 points)
Hint. Make use of both the Green’s function for the ball and part (b).

29.

Teach a man to fish

(a)

Using the Green’s function of $H_1^{+}$ from the previous question, derive the following formal integral representation for a solution of the Dirichlet problem $\mathrm{\Delta }u=0\text{ in }{H}_1^{+},u{|}_{H_1^0}=g$

$$u(x)=\frac{2x_1}{n{\omega }_n}{\int \; <!--\; nolimits\; -->}_{H_1^0}\frac{g(z)}{|x-z{|}^n}d\sigma (z)$$

Here, ‘formal’ means that you do not need to prove that the integrals are finite/well-defined.
(3 points)

(b)

Show that if $g$ is periodic (that is, there is some vector $L\in {ℝ}^{n-1}$ with $g(x+L)=g(x)$ for all $x\in {ℝ}^{n-1}$) then so is the solution. (2 points)

(c)

Now consider the plane $n=2$ with $g$ function with compact support. Approximate the value of $u(x)$ for large $|x|$. Feel free to modify this question as you see fit, what interesting things can you say about the growth of $u$? (Bonus Points as deserved)