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Problem set 11, PDE

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

Problem 1

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

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

Problem 2

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

Problem 3

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

Problem 4

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

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

  2. $latex ||u_k||_{H^1} \sim k$, and thus $latex f = \sum_k a_k u_k \in H^1(\mathbb D)$ if and only if $latex \sum_k |k a_k|^2 < \infty$.

  3. Construct a function $latex f\in L^2(\mathbb S^1)$ that is not the restriction of an $latex H^1(\mathbb D)$ function to the circle $latex \mathbb S^1$.