Problems to Chapter 7


### Problems to Chapter 7

Problem 1.

1. Find the solutions that depend only on $r$ of the equation \begin{equation*} \Delta u:=u_{xx}+u_{yy}=0. \end{equation*}
2. Find the solutions that depend only on $\rho$ of the equation \begin{equation*} \Delta u:=u_{xx}+u_{yy}+u_{zz}=0. \end{equation*}
3. (bonus) In $n$-dimensional case prove that if $u=u(r)$ with $r=(x_1^2+x_2^2+\ldots+x_n^2)^{\frac{1}{2}}$ then $$\Delta u = u_{rr}+ \frac{n-1}{r}u_r=0.$$
4. (bonus) In $n$-dimensional case prove ($n\ne 2$) that $u=u(r)$ satisfies Laplace equation as $x\ne 0$ iff $u=Ar^{2-n}+B$.

Problem 2. Suppose that $u$ is a harmonic function on the open unit disc $\mathbb{D}=\{(x,y)\colon x^2+y^2 < 1\}\subset\mathbb{R}^2$ which is continuous on the closed unit disc $\overline{\mathbb{D}}={(x,y)\colon x^2+y^2 \leq 1}$ and is equal to $g(\theta)$ on the boundary unit circle $S^1={(x,y)\,|\, x^2+y^2 = 1}$.

1. Determine the maximum value that $u$ takes on the closed disc $\overline{\mathbb{D}}$.
2. Determine $u(0)$.

\begin{align} &g(\theta)=\theta (2\pi -\theta)\qquad\qquad0\le \theta<2\pi;\\[6pt] &g(\theta) = \left\{\begin{aligned} &\cos(\theta), && |\theta|\le \tfrac{\pi}{4}, \\ &\tfrac{1}{\sqrt{2}},&& \theta\in\bigl[-\pi,-\tfrac{\pi}{4}\bigr)\cup\bigl(\tfrac{\pi}{4},\pi\bigr]; \end{aligned} \right.\\[6pt] &g(\theta)=\sin (\tfrac{\theta}{2})\qquad\qquad 0\le \theta<2\pi; \\[6pt] &g(\theta) = \theta^4, \qquad\qquad -\pi\le \theta\le \pi;\\[4pt] &g(\theta) = \left\{\begin{aligned} &\sec^2(\theta), && |\theta|\le \tfrac{\pi}{4}, \\ &2, && \theta\in\bigl[-\pi,-\tfrac{\pi}{4}\bigr)\cup\bigl(\tfrac{\pi}{4},\pi\bigr]; \end{aligned}\right.\\[6pt] &g(\theta)=\frac{\theta}{1+\theta^2}\qquad 0-\pi\le \theta \le \pi;\\[6pt] &g(\theta) = \theta(2\pi-\theta), \qquad 0\le \theta \le 2\pi; \\[6pt] &g(\theta) = \left\{\begin{aligned} &-|\sin(\theta)|, && |\theta|\le \tfrac{\pi}{2}, \\ &-1, && \theta\in\bigl[-\pi,-\tfrac{\pi}{2}\bigr)\cup\bigl(\tfrac{\pi}{2},\pi\bigr]; \end{aligned}\right.\\[6pt]& g(\theta) = \theta^2 - \theta^4, \qquad -\pi\le \theta \le \pi;\\[6pt] &g(\theta)=\frac{\sin(\theta)}{2+\cos (\theta)}\qquad -\pi\le \theta <\pi;\\[6pt] &g(\theta)=|\sin (\theta)| \qquad\qquad 0\le \theta <2\pi.
\end{align}

Problem 3. Using the proof of mean value theorem (see Subsection 7.2.4) prove that if $\Delta u\ge 0$ in $B(y,r)$ then

1. $u(y)$ does not exceed the mean value of $u$ over the sphere $S(y,r)$ bounding this ball: $$u(y)\le \frac{1}{\sigma_n r^{n-1}}\int_{S(y,r)} u\,dS.$$
2. $u(y)$ does not exceed the mean value of $u$ over this ball $B(y,r)$: $$u(y)\le \frac{1}{\omega_n r^n}\int_{B(y,r)} u\,dV.$$

3. Formulate similar statements for functions satisfying $\Delta u\le 0$ (in the next problem we refer to them as (a)' and (b)').

Definition 1.

1. Functions having property (a) (or (b) does not matter) of the previous problem are called subharmonic.

2. Functions having property (a)' (or (b)' does not matter) are called superharmonic.

Problem 4.

1. Using the proof of maximum principle prove the maximum principle for subharmonic functions and minimum principle for superharmonic functions.

2. Show that minimum principle for subharmonic functions and maximum principle for superharmonic functions do not hold (Hint: construct counterexamples with $f=f(r)$).

3. Prove that if $u,v,w$ are respectively harmonic, subharmonic and superharmonic functions in the bounded domain $\Omega$, coinciding on its boundary ($u|_\Sigma=v|_\Sigma=w|_\Sigma$) then in $w\ge u \ge v$ in $\Omega$.

Problem 5. Using Newton shell theorem (see Subsection 7.3.1), prove that if Earth was a homogeneous solid ball, then the gravity pull inside of it would be proportional to the distance to the center.

Problem 6. Prove that if $u(\mathbf{x})$ is harmonic, so is $v(\mathbf{x})=r^{2-n}u(\mathbf{x}/r^2)$, $r:=|\mathbf{x}|$.

Hint. Use that $\Delta u =(r ^{n-1}u_r)_r +r^{-2}\Lambda u$, where $\Lambda$ is an operator with respect to angular variables $\mathbf{x}/r$, which are not affected by inversion $\mathbf{x}\mapsto \mathbf{x}/|\mathbf{x}|^2$, while radial variable $r:=|\mathbf{x}|\mapsto \rho=r^{-1}$.

If $n=2$, then $\Delta =\partial_{\bar{z}}\partial_z$ with $\partial_{\bar{z}}=\frac{1}{2}(\partial_x+i\partial_y)$ $\partial_{z}=\frac{1}{2}(\partial_x-i\partial_y)$. Then $v(z)=u(1/\bar{z})$.

Problem 7. Using method of reflection (studied earlier for different equations) construct Green function for

1. Dirichlet problem
2. Neumann problem

for Laplace equation in

1. half-plane
2. half-space

since we know that in the whole plane and space they are just potentials \begin{gather} \frac{1}{2\pi}\log \bigl((x_1-y_1)^2+(x_2-y_2)^2\bigr)^{\frac{1}{2}},\label{eq-H7.P.4}\\ -\frac{1}{4\pi} \bigl((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2\bigr)^{-\frac{1}{2}} \end{gather} respectively.

Problem 8. Apply method of descent described in Subsection 9.1.4 but to Laplace equation in $\mathbb{R}^2$ and starting from Coulomb potential in $3D$ $$U_3(x,y,z)=-\frac{1}{4\pi} \bigl(x^2+y^2+z^2\bigr)^{-\frac{1}{2}},$$ derive logarithmic potential in $2D$ $$U_2(x,y,z)=\frac{1}{2\pi}\log \bigl(x^2+y^2\bigr)^{\frac{1}{2}}, \label{eq-7.P.7}$$ Hint. You will need to calculate diverging integral $\int_0^\infty U_3 (x,y,z)$. Instead consider $\int_0^N U_3 (x,y,z)$, subtract constant (f.e. $\int_0^N U_3 (1,0,z)$) and then tend $N\to \infty$.

Problem 9. Apply method of descent, described in Subsection 9.1.4 but now looking for stationary solution of $-\Delta u=f(x_1,x_2,x_3)$ instead of non-stationary solution of \begin{align} & u_{tt}-\Delta u=f(x_1,x_2,x_3),\\ & u|_{t=0}=g(x_1,x_2,x_3),\\ & u_t|_{t=0}=h(x_1,x_2,x_3) \end{align} start from Kirchhoff formula (9.1.12) and derive for $n=3$ (7.2.10) with $G(x,y)$ equal to (\ref{eq-7.P.7}) here.

Problem 10. Apply method of descent but to Laplace equation in $\mathbb{R}^2$ and starting from Coulomb potential in $3D$ $$U_3(x,y,z)=-\frac{1}{4\pi} \bigl(x^2+y^2+z^2\bigr)^{-\frac{1}{2}},$$ derive logarithmic potential in $2D$ $$U_2(x,y,z)=\frac{1}{2\pi}\log \bigl(x^2+y^2\bigr)^{\frac{1}{2}},$$ Hint. You will need to calculate diverging at $z=\infty$ integral $\int_0^\infty U_3 (x,y,z)\,dz=\int_0^\infty U_3 (r,0,z)\,dz$ with $r=(x^2+y^2)^{1/2}$. To do this we need to regularize it. The easiest way is to to this is to consider integral $\int_0^\infty \partial_r U_3 (r,0,z)\,dz$, converging at $z=\infty$, and find its primitive by $r$.

Alternatively, one can consider $\int_0^N U_3 (x,y,z)\,dz$, subtract constant (f.e. $\int_0^N U_3 (1,0,z)\,dz$), and then tend $N\to \infty$.