A brief review on function spaces

In this chapter we collect some results on various function space we will use throughout the book. One essential property of many function space we will consider is that they are complete, i.e. they are either Banach or Hilbert space, see Section Relevant concepts from functional analysis.

Note

This section barely scratches at the surface of the topic, we will only summarize (and not even prove) the most essential results we need later one in this course.

Also, this chapter will be a work in progress during the entire course, as we will add relevant results here whenever we need them elsewhere.

Measure and integration theory, Lebesque spaces

Lebesque integration theory provides a powerful generalization of the Riemann integral which makes sure that the set of so-called Lebesque-integrable functions turns into a Banach space when endowed with a suitable norm. Nowdays, in most standard text books, Lebesque integration theory is presented as part of the curriculum on Measure and Integration theory, see Chapter 9-10 in [Browder, 2012] for a quick introduction. To this end,

Definition 1 (Lebesque spaces)

Let \(\Omega \subset \RR^n\) be a open domain.

Then the Lebesque spaces \(L^p(\Omega)\) are defined by

(6)\[L^p(\Omega) = \{ f : \Omega \to \RR \text{ is measurable and } \|f\|_{L^p(\Omega)} < \infty \}.\]

Here, the \(L^p\)-norm \(\|\cdot\|_{L^p(\Omega)}\) is defined by

(7)\[\begin{split}\|f\|_{L^p(\Omega)} := \begin{cases} \Bigl( \int_{\Omega} |f(x)|^p \dx \Bigr)^{1/p} &\quad 1 \leqslant p < \infty \\ \mrm{ess}\sup_{\Omega} |u| &\quad p=\infty \end{cases}\end{split}\]

Sometimes we write \(\|f\|_{p,\Omega}\) instead of \(\|f\|_{L^p(\Omega)}\). A function \(f \in L^p(\Omega)\) is often called \(L^p\)-integrable.

We also introduce the space of locally \(L^p\)-integrable functions on \(\Omega\); that is, functions that are \(L^p\) integrable on every compact subset \(K \Subset \Omega\),

(8)\[L^p_{\mrm{loc}}(\Omega) = \{ f: \Omega \to \RR | f \in L^p(K) \; \forall K \Subset \Omega \}.\]

TODO

Introduce inner product on \(L^2\).

Lemma 1 (Determining uniqueness through testing)

Let \(u_1, u_2 \in L^1_{\mrm{loc}}(\Omega)\) and assume that

\[\int_{\Omega} (u_1 - u_2) \phi \dx = 0 \quad \forall \phi \in C^{\infty}_c(\Omega).\]

Then \(u_1 = u_2\) almost every in \(\Omega\); that is, up to set of measure \(0\).

Remark 2

In this setting, \(\phi\) is typically called a test function. When determining whether two functions are equal, the previous lemma roughly states that you can do this by comparing their “actions” on suitable test functions \(\phi\) instead of comparing their values at (almost) every point.

Here, the “action” is simply the resulting number computed from multiplying the functions in question with the test function \(\phi\) and integrating over \(\Omega\).

Sobolev spaces

Weak derivatives

Let start with a motivating example. Let \(u \in C^k(\Omega)\) and \(\phi \in C^{\infty}_c(\Omega)\). Using Green’s theorem and taking into account that \(\phi = 0\) on a open neighborhood of the boundary of \(\Omega\), we see that

(9)\[\int_{\Omega} \partial_{x_i} u \phi \dx = - \int_{\Omega} u \partial_{x_i} \phi \dx,\]

and iterating this formula, we observe that for any multiindex \(\alpha \in \NN^n\) with \(\alpha \leqslant k\), it holds that

(10)\[\int_{\Omega} \partial^{\alpha} u \phi \dx = (-1)^{|\alpha|} \int_{\Omega} u \partial^{\alpha} \phi \dx.\]

Note that the integral expression on the right-hand side of (10) makes perfectly sense even for \(u\in L^1_{\mrm{loc}}\) and not only \(u\in C^k(\Omega)\). This leads to a possibility to generalize or weakened the classical definition of derivatives.

Definition 2 (Weak derivative)

Let \(\alpha \in \NN^n\) be a multiindex and \(u, u_{\alpha} \in L^1_{\mrm{loc}}(\Omega)\). We say that \(u_{\alpha}\) is \(\alpha\)-th weak derivative of \(u\) if

\[\int_{\Omega} u_{\alpha} \phi \dx = (-1)^{|\alpha|} \int_{\Omega} u \partial^{\alpha} \phi \dx\]

holds for all \(\phi \in C^{\infty}_c(\Omega)\).

Lemma 2 (Uniqueness of weak derivatives)

If \(u \in L^1_{\mrm{loc}}(\Omega)\) possesses an \(\alpha\)-th weak derivative, it is uniquely defined in \(L^1_{\mrm{loc}}(\Omega)\).

Proof. For two weak derivatives \(u_{\alpha}\) and \(\tilde{u}_{\alpha}\) we have that

\begin{align} \int_{\Omega} u_{\alpha} \phi &= (-1)^{|\alpha|} \int_{\Omega} u \partial^{\alpha} \phi dx \\ \int_{\Omega} \tilde{u}_{\alpha} \phi &= (-1)^{|\alpha|} \int_{\Omega} u \partial^{\alpha} \phi dx \end{align}

and by substracting the second from the first inequality, we obtain that

\[\int_{\Omega} (u_{\alpha} - \tilde{u}_{\alpha} ) \phi dx = 0 \quad \forall \phi \in C_c^{\infty}(\Omega),\]

and thus \(u_{\alpha} = \tilde{u}_{\alpha}\) almost everywhere by Lemma 1.

Exercise 2 (Relation between the modulus function and the Heaviside function)

Let \(\Omega = (-1,1)\) and set

\[\begin{split}u(x) &= x \\ H(x) &= \begin{cases} -1 &\quad x \in (-1,0) \\ 1 &\quad x \in [0, 1) \end{cases}\end{split}\]

By simply using the definition of the weak derivative, show that \(H(x)\) is the weak derivative of \(u\).

Definition 3 (Sobolev spaces)

• \(W^{k,p}(\Omega) := \{ u \in L^p(\Omega) |\, \partial^{\alpha}u \text{ exists and belongs to } L^p(\Omega) \, \forall \alpha \text{ with } |\alpha| \leqslant k \} \)

• For \(p=2\), we usually write

  \[H^k(\Omega) := W^{k,2}(\Omega)\]

  Note that the \(\| \cdot \|_{H^k(\Omega)}\) is induced by the inner product

  \[(v,w)_{H^k(\Omega)} := \sum_{|\alpha| \leqslant k} (\partial^{\alpha} v, \partial^{\alpha} w)_{L^2(\Omega)}\]

• For \(u \in W^{k,p}(\Omega)\), we set

  \[\begin{split}\| u \|_{W^{k,p}({\Omega})} := \|u\|_{k,p,\Omega} := \begin{cases} \Bigl( \sum_{|\alpha| \leqslant k} \| \partial^{\alpha} u \|_{L^p(\Omega)}^p \Bigr)^{1/p} & 1\leqslant p < \infty, \\ \sum_{|\alpha| \leqslant k} \| \partial^{\alpha} u \|_{L^{\infty}(\Omega)} & p = \infty. \end{cases}\end{split}\]

• We set

  \[W_0^{k,p}(\Omega) := \overline{C_c^{\infty}(\Omega)}^{\|\cdot\|_{k,p,\Omega}},\]

  that is, the topological closure of \(C_c^{\infty}(\Omega)\) in \(W^{k,p}(\Omega)\).

Remark 3

\(W_0^{k,p}(\Omega)\) can be understood as the closed subspace consisting of those function \(\phi\) in \(W^{k,p}(\Omega)\) which are limits of sequences \(\{\phi_n\}_{n=1}^\infty \subset C_c^{\infty}(\Omega)\).

Later we will need the following important result known as Poincaré inequality.

Theorem 3 (Poincaré inequality)

Let \(\Omega\) be an open and bounded subset of \(\RR^n\) and suppose then there is a constant \(C_P = C_P(p,n,\Omega)\) such that

\[\|u \|_{L^p(\Omega)} \leqslant C_P \|\nabla u \|_{L^p(\Omega)}.\]

for any \(u \in W^{1,p}_0(\Omega)\).

Proof. For a proof we refer to [Evans, 2010] (p. 279).

Corollary 2

On \(W^{1,p}_0(\Omega)\), the \(\| \cdot\|_{W^{1,p}(\Omega)}\) is equivalent to the norm

\[\| u \|_{W^{1,p}_0(\Omega)} := \| \nabla u \|_{L^{p}(\Omega)} \]

Corollary 2

On \(W^{1,p}_0(\Omega)\), the \(\| \cdot\|_{W^{1,p}(\Omega)}\) is equivalent to the norm

\[\| u \|_{W^{1,p}_0(\Omega)} := \| \nabla u \|_{L^{p}(\Omega)} \]

Proof. A simple application of the Poincaré application yields

\[\|\nabla u\|_{\Omega}^p \leqslant \| u \|_{\Omega}^p + \|\nabla u\|_{\Omega}^p \leqslant (1+C_P) \|\nabla u\|_{\Omega}^p.\]

TODO

Introduce dual spaces of \(H^1\) and \(H^1_0\).

Approximation results

Poincaré inequalties

Trace operators