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### Problem set 11, PDE

In all problems, $\Omega\subset\R^n$ is a $C^1$ bounded domain, and derivatives are understood as weak derivatives.

## Problem 1

The space $H^1(\Omega)$ is a Hilbert space with respect to the inner product

$\displaystyle \langle u, v \rangle_{H^1} = \int_\Omega \big( u\cdot v + \nabla u \cdot \nabla v \big).$

## Problem 2

The $H_0^1$ and $H^1$ norms are equivalent in the space $H_0^1(\Omega)$.

## Problem 3

The restriction $u\mapsto u|_{\mathbb \partial\Omega}$ is bounded from $H^1(\Omega)$ into $L^2(\partial\Omega)$. (Hint: Extend the normal field $\nu$ from $\partial\Omega$ to $\bar\Omega$ and use the divergence theorem, together with the Cauchy inequality.)

## Problem 4

Consider, in the disk $\mathbb D$, the functions $u_k(r,\theta) = r \cos k\theta$, $k\in\N.$

1. The $u_k$ are orthogonal in $H^1(\mathbb D)$, with respect to the $H^1$ product. (Hint: Use integration in polar coordinates; note that you also need to calculate the gradient $\nabla$ of a function in $\mathbb D$ 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 $\mathbb S^1$.