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.

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