Functional analysis tools
1. Basic definitions
1.1. Norms and scalar products
Let \(E\) be a vector space.
For \(E=\RR^n\) and \(x=(x_1,\ldots,x_n) \in\RR^n\), we define the norms
From a scalar product, an induced norm can be defined:
Then, according to (N3), we have the CauchySchwarz inequality:
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. 
2. Functional spaces
In the following, functions will be defined on a subset of \(\RR^n\) (most often an open denoted \(\Omega\)), with values in \(\RR\) or \(\RR^p\).
The simplest usual norms for functional spaces are the norms \(\bf L^p\) defined by :
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
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 \(\bf L^p(\Omega)\) :
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.
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 LaxMilgram 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 \(\RR^n\).
3.2. Generalized derivative
Let \(u\in {\cal C}^1(\Omega)\) and \(\varphi \in {\cal D}(\Omega)\).
By integration by parts (appendix [sec:green]), we have :
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.
We now consider the definition of higher order generalized derivatives.
These definitions extend naturally to the definition of generalized partial derivatives, in the case \(n>1\).
4. Sobolev spaces
4.1. \(H^m\) spaces
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 \(\RR\) : \( H^m(I) = \left\{ u \in L^2(I) \; / \; u', \ldots, u^{(m)} \in L^2(I) \right\} \) 
Similarly, we define a scalar product and a norm on \(H^m(\Omega)\) by
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 1D)

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 welldefined.
 \(n>1\)

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 \(H^1_0(\Omega)\)
 If \(n=1\) (case 1D)}

consider a bounded open interval \(I=]a,b[\). Then
 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.
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 nonzero 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 seminorm induced on \(V\).
We deduce that \(._{1,V}\) is a norm on \(V\), equivalent to the norm \(\.\_{1,V}\).