Sobolev spaces - embedding - exercise

Question

I have to show that $H^{2}(\Omega) \subset \subset H^1(\Omega)$. I think the Arzela - Ascoli theorem can help.. .I dont know how to start this exercise . I am beginner in Sobolev spaces.. someone can give me a hint ?

Thank you

Answer 1

The Rellich-Kondrachov theorem implies that for bounded Lipschitz domains $\Omega$ the embedding $H^1(\Omega)\subset L^2(\Omega)$ is compact. To use this fact, observe that the $H^2$ norm of a function is comparable to the sum of $H^1$ norms of its partial derivatives. Thus, if $(u_k)$ is a bounded sequence in $H^2$, then the partial derivatives $\partial u_k/\partial x_1$ form a bounded sequence in $H^1$. By Rellich-Kondrachov the sequence of derivatives has a strongly convergent subsequence in $L^2$. Same works for derivatives with respect to other variables. Thus, $\nabla u_k$ converge strongly in $L^2$. You can get the convergence of $u_k$ themselves in $L^2$ from the Poincaré inequality after choosing a further subsequence for which the averages $u_\Omega$ converge. (The latter is possible because the averages are bounded.)

For the future: the geometric properties of $\Omega$ and of its boundary are important when Sobolev spaces are involved. They should be stated in the question.