Recommended exercises, second part

The list below will be filled out gradually during the course (as we cover the corresponding subjects).

Recommended exercises, Sobolev spaces:

1. Prove that if $LaTeX: u\in L^1_{\mathrm{loc}}(U)$ $u\in L^1_{\mathrm{loc}}(U)$ and

$LaTeX: \int_U u\phi dx=0\ \forall\phi\in C^\infty_c (U)$ $\int_U u\phi dx=0\ \forall\phi\in C^\infty_c (U)$

then LaTeX: u=0 $u=0$ a.e.

2. Prove that there is a partition of unity as stated in the proof of Theorem 5.3.2 (Global approximation by smooth functions).

3. Why is the boundary not straightened out in the proof of Theorem 5.3.3 (Global approximation by functions smooth up to the boundary) (as opposed to Theorem 5.4.1 (Extension Theorem))?

5. The following exercises from the book: 5.10.2 (starting "Assume $LaTeX: 0<\beta<\gamma\leq1$ $0<\beta<\gamma\leq1$ "), 5.10.4 (starting "Assume LaTeX: n=1 $n=1$ and $LaTeX: u\in W^{1,p}(0,1)$ $u\in W^{1,p}(0,1)$ "), 5.10.7 (starting "Assume that $U$ is bounded and there exists a smooth vector field"), 5.10.8 (starting Let $U$ be bounded, with a LaTeX: C^1 $C^1$ boundary. Show that"), 5.10.10 (starting "(a) Integrate by parts to prove"), 5.10.14 (starting "Verify that if LaTeX: n>1 $n>1$ , the unbounded function"), 5.10.17 (Chain rule).

Recommended exercises, second-order elliptic equations:

1. The following exercises from the book: 6.6.1 (starting "Consider Laplace's equation with potential function c."), 6.6.2 (starting "Let $LaTeX: Lu=-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}+cu$ $Lu=-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}+cu$ "), 6.6.4 (starting "Assume LaTeX: U $U$ is connected."), 6.6.8 (starting "Let $u$ be a smooth solution of the uniformly elliptic equation"), 6.6.10 (starting "Assume $U$ is connected. Use (a) energy methods"), 6.6.12 (starting "We say that the uniformly elliptic operator"), 6.6.14 (starting "Let $LaTeX: \lambda_1$ $\lambda_1$ be the principal eigenvalue of the uniformly elliptic, nonsymmetric operator").

Recommended exercises, parabolic equations:

The following exercises from the book 7.5.1 (starting "Prove that there is at most one smooth solution of this initial/boundary-value problem for the heat equation with Neumann boundary conditions", 7.5.2 (starting "Assume LaTeX: u $u$ is a smooth solution of") and 7.5.7 (starting "Suppose $u$ is a smooth solution of") (some or all of these exercises may be solved in the exercise session).