Math 641-600 — Fall 2021

Assignment 12 - These are not to be handed in.

1. Section 4.3: 5, 6

2. Suppose that $Lu= u''+\lambda u$, with Dom$(L)=\{u\in L^2[0,\infty):Lu\in L^2[0,\infty)\ \text{and}\ u(0)=0\}$, where $\lambda\in \mathbb C \setminus [0,\infty)$. In addition, choose $\text{Im}\sqrt{\lambda}>0$. Show that the Green's function for $L$ is given by \[ g(x,y, \lambda)=\begin{cases} \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} xe^{i\sqrt{\lambda} y} & 0\le x\le y<\infty\\ \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} y)e^{i\sqrt{\lambda}x} & 0\le y\le x<\infty \end{cases}\] Hint: follow the procedure used in Keener, pg. 150, to solve a similar problem.

3. Consider a Sturm-Liuville operator $Lu=-\frac{d}{dx}p(x)\frac{du}{dx}+q(x)u$, where $p\in C^1[0,1]$ and satisfies $p(x)>0$. In addition, we suppose that $q\in C[0,1]$ and is real valued. If Dom$(L)=\{u\in L^2[0,1]:Lu\in L^2[0,1] \ \text{and }u(0)=0, \ u'(1)+u(1)=0\}$. Show that a Green's function exists if and only if there is no homogeneos solution $u$ to $Lu=0$ that is also in Dom$(L)$.

4. Let $Lu=-\frac{d}{dx}x\frac{du}{dx}$, with Dom$(L)=\{u\in L^2[0,1]: Lu\in L^2[0,1] \ \text{and}\ \lim_{x\downarrow 0}xu'(x)=0, \ u(1)=0\}$. Find the Green's function for $L$.

5. Use the Courant-Fischer Theorem to show that the eigevalues of the Sturm-Liousville operator $L$ defined in problem 3 above, subject to $u(0)=0$ and the three boundary conditions $\{u'(1)=0\}$ (Neumann), $\{u'(1)+\sigma u(1)=0,\ \sigma>0\}$ (Mixed) and $\{u(1)=0\}$ (Dirichlet), are ordered so that $\lambda_n^N \le \lambda^\sigma_n\le \lambda^D_n$.