Ir al contenido principal

## Due May 13

### Problem 1

Let $X$ be a closed subspace of the Hilbert space $\mathscr H$.
1. $X^\perp = \{x\in\mathscr H: x\perp X \}$ is a closed subspace of $\mathscr H$.
2. $\mathscr H \cong X\oplus X^\perp$

### Problem 2

If $g$ is the weak derivative of $f\in L^2(\mathbb R^d)$ with respect to $x_j$, then
$\hat g(\xi) = 2\pi i \xi_j \hat f(\xi).$

### Problem 3

Let $\mathscr H^1(\Omega)$ be the set of equivalence classes in $H^1(\Omega)$ under the relation $f\sim g$ if and only if $f-g$ is a constant.
1. $\mathscr H^1(\Omega)$ is a vector space.
2. The bilinear form $\mathcal E$ is an inner product on $\mathscr H^1(\Omega)$.
3. $\mathscr H^1(\Omega)$ is a Hilbert space with respect to $\mathcal E$.
4. $H_0^1(\Omega)$ is a closed subspace of $\mathscr H^1(\Omega)$.

### Problem 4

Let $\Omega$ be a bounded $C^1$-domain in $\mathbb R^d$. Then there exists a constant $A>0$ such that, for every $f\in H^1(\Omega)$,
$\displaystyle\int_{\partial\Omega} |f|^2 d\sigma \le A ||f||^2_{H^1},$
where $||\cdot||^2_{H^1}$ is the norm in $H^1(\Omega)$ induced by the inner product
$\displaystyle \langle f, g \rangle_1 = \int_\Omega f \cdot \bar g + \nabla f \cdot \overline{\nabla g}$.
Hence, the restriction $f\mapsto f|_{\partial\Omega}$ is a bounded operator from $H^1(\Omega)$ into $L^2(\partial\Omega,d\sigma)$. (Hint: Extend the normal field $\nu$ from $\partial\Omega$ into $\Omega$ --$\Omega$ is a $C^1$-domain-- and apply the divergence theorem to the field $F = |f|^2 \nu$.)

### Problem 5

Consider, in the disk $\mathbb D$, the functions $u_k(r,\theta) = r \cos k\theta$, $k\in\mathbb N$, in polar coordinates.
1. The $u_k$ are orthogonal in $H^1(\mathbb D)$, with respect to $\langle \cdot, \cdot \rangle_1$ of the previous problem. (Hint: Use integration in polar coordinates.)
2. $||u_k||_{H^1} \sim k$, and thus $f = \sum_k a_k u_k \in H^1(\mathbb D)$ if and only if $\sum_k |k a_k|^2 < \infty$.
3. Construct a function $f\in L^2(\mathbb S^1)$ that is not the restriction of an $H^1(\mathbb D)$ function to the circle $\mathbb S^1.$
Thus, the restriction operator $f\mapsto f|_{\partial\Omega}$ is not onto $L^2(\partial\Omega,d\sigma)$.