14. Sobolev spaces

For \(k \in {\mathbb N}_0\) and \(1 \leq p < \infty\), we define the Sobolev norms

\[ \| u \|_{W_p^k(\Omega)} := \left( \sum_{|\alpha| \leq k} \| D^\alpha u \|_{L_p}^p \right)^{1/p}, \]

for \(k \in {\mathbb N}_0\) we set

\[ \| u \|_{W_\infty^k(\Omega)} := \max_{|\alpha| \leq k} \| D^\alpha u \|_{L_\infty}. \]

In both cases, we define the Sobolev spaces via

\[ W_p^k(\Omega) = \{ u \in L_1^{loc} : \| u \|_{W_p^k} < \infty \} \]

In the previous chapter we have seen the importance of complete spaces. This is the case for Sobolev spaces:

Theorem: The Sobolev space \(W_p^k(\Omega)\) is a Banach space.

Proof: Let \(v_j\) be a Cauchy sequence with respect to \(\| \cdot \|_{W_p^k}\). This implies that \(D^\alpha v_j\) is a Cauchy sequence in \(L_p\), and thus converges to some \(v^\alpha\) in \(\|. \|_{L_p}\).

We verify that \(D^\alpha v_j \rightarrow v^\alpha\) implies \(\int_\Omega D^\alpha v_j \varphi \, dx \rightarrow \int_\Omega v^\alpha \varphi \, dx\) for all \(\varphi \in {\cal D}\). Let \(K\) be the compact support of \(\varphi\). There holds

\[\begin{eqnarray*} \int_\Omega (D^\alpha v_j - v^\alpha) \varphi \, dx & = & \int_{K} (D^\alpha v_j - v^\alpha) \varphi \, dx \\ & \leq & \| D^\alpha v_j - v^\alpha \|_{L_1(K)} \| \varphi \|_{L_\infty} \\ & \leq & \| D^\alpha v_j - v^\alpha \|_{L_p(K)} \| \varphi \|_{L_\infty} \rightarrow 0 \end{eqnarray*}\]

Finally, we have to check that \(v^\alpha\) is the weak derivative of \(v\):

\[\begin{eqnarray*} \int v^\alpha \varphi \, dx & = & \lim_{j\rightarrow \infty} \int_\Omega D^\alpha v_j \varphi \, dx \\ & = & \lim_{j\rightarrow \infty} (-1)^{|\alpha|} \int_\Omega v_j D^\alpha \varphi \, dx = \\ & = & (-1)^\alpha \int_\Omega v D^\alpha \varphi \, dx. \end{eqnarray*}\]

\(\Box\)

An alternative definition of Sobolev spaces were to take the closure of smooth functions in the domain, i.e.,

\[ \widetilde W_p^k := \overline{ \{ C^\infty (\Omega) : \|.\|_{W_p^k} \leq \infty \} }^{\|.\|_{W_p^k}}. \]

A third one is to take continuously differentiable functions up to the boundary

\[ \widehat W_p^k := \overline{ C^\infty (\overline{\Omega}) }^{\|.\|_{W_p^k}}. \]

Under moderate restrictions, these definitions lead to the same spaces:

Theorem: Let \(1 \leq p < \infty\). Then \(\widetilde W_p^k = W_p^k\).

Definition: A domain is called a Lipschitz domain, if its boundary \(\partial \Omega\) can be locally represented by a Lipschitz-continuous function.

Are these domains Lipschitz domains?

A precise definition which applies also to unbounded domains is the following:

Definition: The domain \(\Omega\) has a Lipschitz boundary, \(\partial \Omega\), if there exists a collection of open sets \(O_i\), a positive parameter \(\varepsilon\), an integer \(N\) and a finite number \(L\), such that for all \(x \in \partial \Omega\) the ball of radius \(\varepsilon\) centered at \(x\) is contained in some \(O_i\), no more than \(N\) of the sets \(O_i\) intersect non-trivially, and each part of the boundary \(O_i \cap \Omega\) is a graph of a Lipschitz function \(\varphi_i : {\mathbb R}^{d-1} \rightarrow {\mathbb R}\) with Lipschitz norm bounded by \(L\).

Theorem: Assume that \(\Omega\) has a Lipschitz boundary, and let \(1 \leq p < \infty\). Then \(\widehat W_p^k = W_p^k\).

The case \(W_2^k\) is special, it is a Hilbert space. We denote it by

\[ H^k(\Omega) := W_2^k(\Omega). \]

The inner product is

\[ (u,v)_{H^k} := \sum_{|\alpha| \leq k} (D^\alpha u, D^\alpha v)_{L_2} \]

In the following, we will prove most theorems for the Hilbert spaces \(H^k\), and state the general results for \(W_p^k\).

We define semi-norms

\[ | u |_{H^k}^2 = \left( \sum_{|\alpha| = k} \| D^\alpha u \|_{L_2}^2 \right)^{1/2} \]

The null-space of these semi-norms consists of polynomials up to order \(k-1\).