Functional analysis tools

1. Basic definitions

1.1. Norms and scalar products

Let \(E\) be a vector space.

Definition: Norm

\(\|.\|\) : \(E \rightarrow \RR\) is a norm on \(E\) if it verifies :

(N1): \( \left( \| x \| = 0 \right) \Longrightarrow (x=0) \)
(N2): \(\forall\, \lambda\in\RR,\; \forall x\in E, \quad \| \lambda x \| = |\lambda| \; \| x \| \)
(N3): \(\forall\, x,y \in E, \quad \| x+ y \| \le \|x \| + \|y\|\) (triangular inequality)

For \(E=\RR^n\) and \(x=(x_1,\ldots,x_n) \in\RR^n\), we define the norms

\[\| x \|_1 = \sum_{i=1}^n |x_i| \qquad \| x \|_2 = \left( \sum_{i=1}^n x_i^2 \right)^{1/2} \qquad \| x \|_\infty = \sup_{i} |x_i|\]

Definition: Scalar product

Any positive definite symmetric bilinear form is called scalar product on \(E\).

\(\quad<.,.>\) : \(E\times E \rightarrow \RR\) is therefore a scalar product on \(E\) if it verifies :

S1

\[\forall\; x,y \in E, \quad <x,y> = <y,x>\]

S2

\[\forall\; x_1,x_2,y \in E, \quad <x_1+x_2,y> = <x_1,y>+ <x_2,y>\]

S3

\[\forall\; x,y \in E, \, \forall\, \lambda\in\RR,\quad <\lambda x,y> = \lambda <x,y>\]

S4

\[\forall\; x \in E, x\ne 0, \quad <x,x>\; \gt 0\]

From a scalar product, an induced norm can be defined:

\[\| x \| = \sqrt{<x,x>}\]

Then, according to (N3), we have the Cauchy-Schwarz inequality:

\[{ | <x,y> | \le \| x \| \; \| y \| }\]

For \(E=\RR^n\), we define the scalar product

\[<x,y> = \sum_{i=1}^n x_i \, y_i.\]

Its induced norm is \(\| . \|_2\) defined above.

Definition: Normed space

A vector space with a norm is called normed space.

Definition: Prehilbert space

A vector space with a scalar product is called a pre-Hilbert space. In particular, it is a normed space for the norm norm.

1.2. Cauchy sequences and complete spaces

Definition: Cauchy sequences

Let \(E\) be a vector space and \((x_n)_n\) a sequence of \(E\). \((x_n)_n\) is a Cauchy sequence if and only if

\[\forall \varepsilon > 0,\;\; \exists N / \forall p>N, \forall q>N, \quad \|x_p - x_q \| < \varepsilon\]

Theorem: Cauchy’s theorem

Every convergent sequence is Cauchy while the reciprocal of Theorem: Cauchy’s theorem is false.

Definition: Complete space

A vector space is complete if every Cauchy sequence y is convergent.

Definition: Banach space

A complete normed space is a Banach space.

Definition: Hilbert space

A complete pre-Hilbert space is a Hilbert space.

Definition: Euclidean space

A finite-dimensional Hilbert space is called a Euclidean Euclidean space.

2. Functional spaces

Definition: Functional space

A functional space is a vector space whose elements are elements are functions.

\({\cal C}^p([a;b\))]

\({\cal C}^p([a;b\))] denotes the space of functions defined on the interval \([a,b\)], all derivatives of which up to the order \(p\) exist and are continuous on \([a,b\)].

In the following, functions will be defined on a subset of \(\RR^n\) (most often an open denoted \(\Omega\)), with values in \(\RR\) or \(\RR^p\).

Temperature

The temperature \(T(x,y,z,t)\) at any point of an object \(\bar{\Omega}\subset \RR^3\) is a function of \( \bar{\Omega} \times \RR \longrightarrow \RR\).

The simplest usual norms for functional spaces are the norms \(\bf L^p\) defined by :

\[\| u \|_{L^p} = \left ( \int_{\Omega } |u|^p \right) ^{1/p} \quad ,\; p\in [1,+ \infty[ , \qquad \hbox{and}\qquad \| u \|_{L^\infty} = {\hbox{Sup}}_{\Omega } |u|\]

As we shall see, these \(L^p\) forms are not necessarily standards. And when they are, the functional spaces with these norms are not necessarily are not necessarily Banach spaces. For example, the forms \(L^\infty\) and \(L^1\) are norms on the space the space \({\cal C}^0([a;b\))], and this space is complete if we with the norm \(L^\infty\), but not with the norm \(L^\infty\). the \(L^1\) norm.

For this reason, we define the spaces \({\cal L}^p(\Omega)(p\in [1,+ \infty[\)) by

\[{\cal L}^p(\Omega) = \left\{ u : \Omega \rightarrow \RR, \hbox{ measurable, and such that }\int_\Omega |u|^p<\infty \right\}\]

A function \(u\) is measurable if \(\{ x / |u(x)|<r \}\) is measurable \(\forall r>0\).

On these spaces \({\cal L}^p(\Omega)\), the forms \(L^p\) are not norms. Indeed, \(\| u \|_{L^p} = 0\) implies that \(u\) is zero almost everywhere in \({\cal L}^p(\Omega)\), not \(u=0\). This is why we’ll define the spaces \(\bf L^p(\Omega)\) :

Definition: Equality almost everywhere

\(L^p(\Omega)\) is the equivalence class of functions of \({\cal L}^p(\Omega)\) for the equivalence relation equality almost everywhere everywhere. In other words, we’ll confuse two functions whenever they are are equal almost everywhere, i.e. they differ only on a set of zero measure. a set of zero measure.

Theorem: \(L^p(\Omega)\)

The form \(L^p\) is a norm on \(L^p(\Omega)\), and \(L^p(\Omega)\) equipped with the norm \(L^p\) is a Banach space (i.e. is complete).

A very important special case is \(p=2\). In this case the functional space \(L^2(\Omega)\), i.e. the space of the space of summable square functions on \(\Omega\) (at the equivalence relation equality almost everywhere). To the norm \(L^2\) : \(\| u \|_{L^2} = \left( \int_\Omega u^2 \right)^{1/2} \), we can associate the bilinear form the bilinear form \((u,v)_{L^2} = \int_\Omega u\, v\). This is a scalar product from which the norm \(L^2\) is derived.

Hence the following theorem that characterizes the space.

Theorem: \(L^2(\Omega)\)

\(L^2(\Omega)\) is a Hilbert space.

3. Notion of generalized derivative

We’ve just defined complete functional spaces, which provide a good framework for demonstrating the existence and uniqueness of solutions to partial differential equations, as we’ll see later with the Lax-Milgram theorem.

However, we have seen that the elements of these \(L^p\) spaces are not necessarily very regular functions.

Consequently, the partial derivatives of such functions are not necessarily defined everywhere.

To overcome this problem, we’re going to extend the notion of derivation.

The real tool to be introduced for this is the notion of distribution, due to L. Schwartz (1950).

For lack of time in this course, we’ll confine ourselves here to giving a very simplified idea, with the notion of generalized derivative.

The latter has much more limited properties than distributions, but allows us to get a "feel" for the aspects necessary to the variational formulation.

In the following, \(\Omega\) will be an open (not necessarily bounded) of \(\RR^n\).

3.1. Test functions

Let \(\varphi : \Omega \rightarrow \RR\).

Definition: support of \(\varphi\)

We call support of \(\bf \varphi\) the adherence of \(\{ x \in \Omega / \varphi(x) \ne 0 \}\).

Example: Test function

For \(\Omega = ]-1,1]\), and \(\varphi\) the constant function equal to 1, \(\hbox{supp}\, \varphi = [-1,1]\).

Definition: Space for test functions

Let \({\cal D}(\Omega)\) be the space of functions from \(\Omega\) to \(\RR\), of class \({\cal C}^\infty\), and with compact support included in \(\Omega\).

\({\cal D}(\Omega)\) is sometimes called test function space.

Example: Test function space

The most classic example in the 1-D case is the function

\[\varphi(x) = \left\{ \begin{array}{ll} { e^{- \frac{1}{1-x^2}} } & \hbox{si } |x|<1 \\ 0 & \hbox{si } |x| \ge 1 \end{array} \right.\]

\(\varphi\) is a function of \({\cal D}(\)a,b\()\) for all \(a < -1 < 1 < b\).

This example can easily be extended to the multi-dimensional case (\(n>1\)).

Let \(a\in\Omega\) and \(r>0\) be such that the closed ball of center \(a\) and radius \(r\) is included in \(\Omega\).

We then pose :

\[ \varphi(x) = \left\{ \begin{array}{ll} { e^{- \frac{1}{r^2-|x-a|^2}} } & \hbox{si } |x-a| < r\\ 0 & \hbox{ otherwise } \end{array} \right.\]

\(\varphi\) thus defined is an element of \({\cal D}(\Omega)\).

Theorem: Adherence of \(\overline{{\cal D}(\Omega)}\)

\(\overline{{\cal D}(\Omega) } = L^2(\Omega)\)

3.2. Generalized derivative

Let \(u\in {\cal C}^1(\Omega)\) and \(\varphi \in {\cal D}(\Omega)\).

By integration by parts (appendix [sec:green]), we have :

\[\int_\Omega \partial_i u\; \varphi = - \int_\Omega u \; \partial_i\varphi + \int_{\partial \Omega} u \; \varphi \; {\bf e}_i.{\bf n}\]

This last term (integral on the edge of \(\Omega\)) is null because \(\varphi\) is compactly supported (hence null on \(\partial \Omega\)).

But \(\int_\Omega u \; \partial_i\varphi\) makes sense as soon as \(u\in L^2(\Omega)\).

So \(\int_\Omega \partial_i u\; \varphi\) also makes sense, without \(u\) necessarily being of class \({\cal C}^1\).

This makes it possible to define \(\partial_i u\) even in this case.

Definition: Generalized derivative

1-D case \(\quad\) Let \(I\) be an interval of \(\RR\), not necessarily bounded. not necessarily bounded.

We say that \(u\in L^2(I)\) admits a generalized derivative in \(L^2(I)\) if \(\exists u_1\in L^2(I)\) such that

\( \forall \varphi\in {\cal D}(I), \quad \int_I u_1\;\varphi = - \int_I u \varphi' \)

Example: Generalized derivative

Let \(I=]a,b[\) be a bounded interval, and \(c\) be a point of \(I\). point of \(I\). Consider a function \(u\) formed by two branches of class \({\cal C}^1\), one on one on \(]a,c[\), the other on \(]c,b[\), and connecting continuously to \(c\). Then \(u\) has a generalized derivative defined by \(u_1(x)=u'(x)\quad \forall x\ne c\). Indeed :

\[\forall \varphi\in {\cal D}(]a,b[)\qquad \int_a^b u \varphi' = \int_a^c + \int_c^b = - \int_a^c u' \varphi - \int_c^b u'\varphi + \underbrace{(u(c^-)-u(c^+))}_{=0} \, \varphi(c)\]

by integration by parts. The value \(u_1(c)\) doesn’t matter: we end up with the same the same function as \(L^2(I)\), since it is defined as the equivalence class of the equivalence relation equivalence relation equality almost everywhere.

We now consider the definition of higher order generalized derivatives.

Definition: Generalized derivative of order \(k\)

By iterating, we say that \(u\) admits a generalized derivative of order \(\bf k\) in \(L^2(I)\), denoted by \(u_k\), ssi

\[{\forall \varphi\in {\cal D}(I), \quad \int_I u_k\;\varphi = (- 1)^k \; \int_I u \varphi^{(k)} }\]

These definitions extend naturally to the definition of generalized partial derivatives, in the case \(n>1\).

Theorem: Uniqueness of the generalized derivative

When it exists, the generalized derivative is unique.

Theorem: generalized derivative and classical derivative

When \(u\) is of class \({\cal C}^1(\bar{\Omega})\), the generalized derivative is equal to the classical derivative.

4. Sobolev spaces

4.1. \(H^m\) spaces

Definition: \(H^1(\Omega)\)

\[{ H^1(\Omega) = \left\{ u \in L^2(\Omega)\; / \; \partial_i u \; \in L^2(\Omega), \; 1 \leq i \leq n \right\} }\]

where \(\partial_i u\) is defined in the sense of the generalized derivative.

\(H^1(\Omega)\) is called Sobolev space of order 1.

Definition: \(H^m(\Omega)\)

For any integer \(m\ge 1\),

\[H^m(\Omega) = \left\{ u \in L^2(\Omega) \; / \; \partial^\alpha u \; \in L^2(\Omega) \quad \forall \alpha =(\alpha_1,\ldots,\alpha_n) \in \NN^n\hbox{ such that}\; |\alpha|= \alpha_1+\cdots+\alpha_n \le m \right\}\]

\(H^m(\Omega)\) is called *Sobolev space of order \(\bf m\).

By extension, we also see that \(H^0(\Omega)=L^2(\Omega)\).

In the case of dimension 1, it’s simpler to write \(I\) open from \(\RR\) :

\( H^m(I) = \left\{ u \in L^2(I) \; / \; u', \ldots, u^{(m)} \in L^2(I) \right\} \)

Theorem: \(H^1(\Omega)\) is a Hilbert space

\(H^1(\Omega)\) is a Hilbert space for the scalar product \((u,v)_1 = \int_\Omega u \, v\, + \sum_{i=1}^n \; \int_\Omega \partial_i u \; \partial_i v = (u,v)_0 + \sum_{i=1}^n (\partial_i u, \partial_i v )_0\)

noting \((.,.)_0\) the scalar product \(L^2\). Let \(\|.\|_1\) be the norm associated with \((.,.)_1\).

Similarly, we define a scalar product and a norm on \(H^m(\Omega)\) by

\[(u,v)_m = \sum_{|alpha| \le m} ( \partial^\alpha u , \partial^\alpha v )_0 \qquad \hbox{ and }\qquad \| u \|_m = (u,u)_m^{1/2}]\]

Theorem: \(H^m(\Omega)\) are Hilbert spaces

\(H^m(\Omega)\) with scalar product \((.,.)_m\) is a Hilbert space. is a Hilbert space [thr:8].

Theorem: \(H^m(\Omega)\) and \({\cal C}^k(\bar{\Omega})\)

If \(\Omega\) is an open of \(\RR^n\) with boundary \(\partial\Omega\). "sufficiently regular" \(\partial\Omega\) (for example example \({\cal C}^1\)), we have the inclusion : \(H^m(\Omega) \subset {\cal C}^k(\bar{\Omega})\) for \({ k < m-\frac{n}{2} }\)

Example: \(H^1(\Omega)\) and \({\cal C}^0(\bar{\Omega})\)

In particular, we see that for an interval \(I\) of \(\RR\), we have \(H^1(I) \subset {\cal C}^0(\bar{I})\), i.e. in 1-D, any \(H^1\) function is continuous.

The example of \(u(x) = x\, \sin\frac{1}{x}\) for \(x\in]0,1]\) and \(u(0)=0\) shows that the converse is false.

The example of \(u(x,y) = | \ln (x^2+y^2) |^k\) for \(0<k<1/2\) shows that in dimensions greater than 1 there are discontinuous \(H^1\) functions.

4.2. Trace of a function

To be able to perform integrations by parts, which will be useful for for variational formulation, you need to be able to define the extension extension (trace) of a function on the edge of the open \(\Omega\).

\(n=1\) (case 1-D)

we consider an open interval \(I=]a,b[\) bounded. We have seen that \(H^1(I) \subset {\cal C}^0(\bar{I})\). Therefore, for \(u\in H^1(I)\), \(u\) is continuous on \(\[a,b]\), and \(u(a)\) and \(u(b)\) are well-defined.

\(n>1\)

we no longer have \(H^1(\Omega) \subset {\cal C}^0(\bar{\Omega})\). How can define the trace? Here’s how:

We define the space \({\cal C}^1(\bar{\Omega}) = \left\{ \varphi : \Omega \rightarrow \RR \;/\; \exists O \hbox{ open containing } \bar{\Omega},\; \exists \psi \in {\cal C}^1(O),\; \psi_{|\Omega} = \varphi \right\}\) In other words, \({\cal C}^1(\bar{\Omega})\) is the space of functions \({\cal C}^1\) on \(\Omega\), extendable by continuity on \(\partial\Omega\) and whose gradient gradient can also be extended by continuity. There is therefore no to define the trace of such functions.
We show that, if \(\partial\Omega\) is a bounded open boundary \(\partial\Omega\) "regular enough", then \({\cal C}^1(\bar{\Omega})\) is dense in \(H^1(\Omega)\).
The continuous linear application, which to any function \(u\) of \({\cal C}^1(\bar{\Omega})\) associates its trace on \(\partial\Omega\), then extends into a continuous continuous linear application of \(H^1(\Omega)\) in \(L^2(\partial\Omega)\), denoted \(\gamma_0\), which we call trace application. trace application*. We say that \(\gamma_0(u)\) is the trace of \(u\) on \(\partial\Omega\).

For a function \(u\) of \(H^1(\Omega)\) which is at the same time continuous on \(\bar{\Omega}\), we obviously have \(\gamma_0(u) = u_{|\partial\Omega}\). This is why \(u_{|\partial\Omega}\) is often simply noted rather than \(\gamma_0(u)\).

Analogously, we can define \(\gamma_1\), an application which extends the usual definition of the normal derivative on \(\partial\O\). For \(u\in H^2(\Omega)\), we have \(\partial_i u \in H^1(\Omega)\), \(\forall i=1,\ldots,n\), and we can therefore define \(\gamma_0(\partial_i u)\). The boundary \(\partial\Omega\) being "fairly regular" (for example, ideally, of class \({\cal C}^1\)), we can define the normal \(n=\left( \begin{array}{l} n_1 \\ \vdots n_n \end{array} \right)\) at any point of \(\partial\Omega\). We then pose \({\gamma_1(u) = \sum_{i=1}^n \gamma_0(\partial_i u) n_i}\). This continuous application \(\gamma_1\) of \(H^2(\Omega)\) into \(L^2(\partial\Omega)\) thus extends the usual definition of the normal derivative. In the case where \(u\) is a function of \(H^2(\Omega)\) which is at the same time in \({\cal C}^1(\bar{\Omega})\)], the normal normal derivative in the usual sense of \(u\) exists, and \(\gamma_1(u)\) is obviously equal to it. This is why \(\partial_n u\) rather than \(\gamma_1(u)\). \(\gamma_1(u)\).

4.3. Space \(H^1_0(\Omega)\)

Definition: \(H^1_0(\Omega)\)

Let \(\Omega\) be open from \(\RR^n\). The space \(H^1_0(\Omega)\) is defined as the adherence of \({\cal D}(\Omega)\) to the norm \(\|.\|_1\) of \(H^1(\Omega)\). (Recall that \({\cal D}(\Omega)\) is the space of \({\cal C}^\infty\) functions on \(\Omega\) with compact support, also known as the space of test functions).

Theorem: \(H^1_0(\Omega)\) is a Hilbert space

By construction \(H^1_0(\Omega)\) is a complete space. It is a Hilbert space for the norm \(\|.\|_1\)

If \(n=1\) (case 1-D)}: consider a bounded open interval \(I=]a,b[\). Then

\[H^1_0(]a,b[) = \left\{ u \in H^1(]a,b[),\; u(a)=u(b)=0 \right\}\]

If \(n>1\): If \(\Omega\) is a bounded open with a "fairly regular" boundary (for example piecewise \({\cal C}^1\)), then \(H^1_0(\Omega) = \ker \gamma_0\). \(H^1_0(\Omega)\) is therefore the subspace of functions of \(H^1(\Omega)\) with zero trace on the \(\partial\Omega\) boundary.

Definition

For any \(u\) function of \(H^1(\Omega)\), we can define :

\[{ |u|_1 = \left( \sum_{i=1}^n \| \partial_i u \|_0^2 \right)^{1/2} = \left( \int_\Omega \sum_{i=1}^n \left( \partial_i u \right)^2 dx \right)^{1/2} }\]

Theorem: Poincaré’s inequality

If \(\Omega\) is bounded in at least one direction, then there exists a constant \(C(\Omega)\) such that

\[\forall u \in H^1_0(\Omega), \; \|u\|_0 \le C(\Omega)\; |u|_1.\]

We deduce that \(|.|_1\) is a norm on \(H^1_0(\Omega)\), equivalent to the norm \(\|.\|_1\).

The previous result extends to the case where we have a null Dirichlet condition only on a part of \(\partial\Omega\), if \(\Omega\) is connected.

We assume that \(\Omega\) is a connected bounded open, of boundary piecewise \({\cal C}^1\).

Let \(V=\left\{ v\in H^1(\Omega),\, v=0 \hbox{ on }\Gamma_0 \right\}\) be where \(\Gamma_0\) is a part of \(\partial\Omega\) of non-zero measure.

Then there exists a constant \(C(\Omega)\) such that \(\forall u \in V, \; \|u\|_{0,V} \le C(\Omega)\; |u|_{1,V}\), where \(\|.\|_{0,V}\) and \(|.|_{1,V}\) denote the norm and semi-norm induced on \(V\).

We deduce that \(|.|_{1,V}\) is a norm on \(V\), equivalent to the norm \(\|.\|_{1,V}\).

Theorem: Trace theorem

Let D be an open with type edge. There exists a constant \(c\) such that \(\forall g \in H^{\frac{1}{2}}(\partial \Omega)\) there exists \(u_g \in H^1(\Omega)\) satisfying

\[\gamma_0(u_g) = g \mbox{ and } \|u\|_{H^1} \leq c \|g\|_{H^\frac{1}{2}}\]

\(u_g\) is called a lifting of \(g\) in \(H^1(\Omega)\).

5. Exercises

Exercise:

Show that the functions defined by ([eq:test-function1]) and ([eq:function-test2]) are \({\cal C}^\infty\) with support support.
Show that \({\cal C}^0([a,b])\) is a complete space for the norm \(L^\infty\).
Show that this is not the case for \(L^1\). (show a non-convergent Cauchy sequence in \({\cal C}^0([a,b])\)).
Show that, when it exists, the generalized derivative is unique.
Show that, for a function of class \({\cal C}^1\), the generalized derivative is equal to the classical derivative.
Let be a function from \([a,b]\) to \(\RR\), formed by two branches of class \({\cal C}^1\) on \([a,c[\) and \(]c,b]\), and discontinuous in \(c\). Show that it has no generalized derivative. (you use the notion of distribution to derive this function).
Show that \(|.|_1\) is a norm on \(H^1_0(\Omega)\), equivalent to the norm \(\|.\|_1\)