# Calculus in Banach Spaces: Application to classical functionals

The first two examples presented in the begining of these notes show us that the space of modelling of a PDE is quite related to its formulation. So far we have presented two types of problems: the classical Dirichlet problem and the weak variational problem. The former involves a pointwise statement and the later an integral equation. In order two present the energy functionals that matters for these two problems it is necessary to consider some functional spaces, before this it is necessary to introduce some notation.

Definition 1 (Multiindex Notation) Let ${u}$ be a real valued infinite differentiable function at some point ${x\in {\mathbb R}^N}$. We say that ${\alpha=(\alpha_1,\dots, \alpha_N)\in ({\mathbb N}\cup\{0\})^N}$ is a multiindex with length ${|\alpha|:=\sum_{i=1}^{N}\alpha_i}$ and we define the ${\alpha}$-derivative of $u$ as follows:

$\displaystyle \partial^\alpha u(x):=\frac{\partial^{\alpha_N}\dots \partial^{\alpha_1}u}{\partial x_N^{\alpha_N}\dots \partial x_1^{\alpha_1}}(x), \ \ \ \ \ (1)$

with ${\partial^0 u(x):=u(x)}$.

Remark 1 For derivatives of low order (first and second order) we will keep the notation:

$\displaystyle \partial_i u(x):=\frac{\partial_i u}{\partial x_i}(x) \qquad \partial_{ji} u(x):=\frac{\partial^2 u}{\partial x_j \partial x_i}(x) \qquad \forall i,j\in \{1,\dots, N\}. \ \ \ \ \ (2)$

Definition 2 (${C^k(\overline{\Omega})}$ Spaces) Let ${\Omega}$ be an open subset of ${{\mathbb R}^N}$ we define the space of ${k}$-differentiable real valued functions:

$\displaystyle C^k(\overline{\Omega}):=\{u\in C(\overline{\Omega})|\, \, \partial^\alpha u \, \,\text{is continuously extended to}\, \, \, \overline{\Omega}, \, \, \, \forall |\alpha|\leq k \}, \ \ \ \ \ (3)$

endowed with the norm ${|| u||_{C^k(\overline{\Omega})}:=\sum_{|\alpha|\leq k} || \partial^\alpha u ||_{L^{\infty}(\overline{\Omega})}.}$

In the same spirit, we define the spaces:

${C^k_0(\overline{\Omega}):=\{u\in C^k(\overline{\Omega})|supp(u) \, \, \, \text{compact}\}}$,

${C^{\infty}(\overline{\Omega}):=\bigcap_{k\in {\mathbb N}} C^k(\overline{\Omega}),}$

${C^{\infty}_0(\overline{\Omega}):=\bigcap_{k\in {\mathbb N}} C^k_0(\overline{\Omega}).}$

Remark 2 It is a straightforward exercise show that ${C^k_0(\overline{\Omega})}$ and ${C^k(\overline{\Omega})}$ are Banach spaces endowed with the norm ${|| .||_{C^k(\overline{\Omega})}}$. However, the topologies of ${C^{\infty}(\overline{\Omega})}$ and ${C^{\infty}_0(\overline{\Omega})}$ are subtler.

The following spaces constitute an intermediate step between continuity and differentiability:

Definition 3 ( Hölder continuity) Let ${\Omega}$ be an open subset of ${{\mathbb R}^N}$ we say that ${u:\overline{\Omega}\rightarrow {\mathbb R}}$ is ${\alpha}$-Hölder continuous for ${\gamma \in (0,1]}$ if there exist a constant ${k>0}$ such that:

$\displaystyle |u(x)-u(y)|\leq k|x-y|^\gamma. \ \ \ \ \ (4)$

Under this consideration, for ${k\in {\mathbb N}\cup \{0\}}$ and for ${\gamma \in (0,1]}$ we define the following spaces:

${C^{k,\gamma}(\overline{\Omega}):=\{u\in C^k(\overline{\Omega})|\, \, \partial^\alpha u \, \,}$ is ${\gamma}$-Hölder continuous on ${\, \, \, \overline{\Omega}, \, \, \, \forall |\alpha|=k \}}$,

${C_0^{k,\gamma }(\overline{\Omega}):=C^{k,\gamma}(\overline{\Omega})\cap C_0^{k}(\overline{\Omega})}$,

both endowed with the norm:

$\displaystyle || u||_{C^{k,\gamma}}(\Omega):=|| u||_{C^k(\overline{\Omega})}+\sum_{|\gamma|=k}\, \,\, \sup_{x,y\in \overline{\Omega}\atop x\neq y}\frac{|\partial^\gamma u(x)-\partial^\gamma u(y)|}{|x-y|^\gamma} . \ \ \ \ \ (5)$

Remark 3 It is clear (exercise!) that these spaces are nested in a continous way, i.e. for ${k\in {\mathbb N}\cup \{0\}}$ and for ${1\geq\gamma>\gamma'>0 }$ the following injections are continuous:

$\displaystyle C^{k+1}(\overline{\Omega}) \hookrightarrow C^{k,\gamma}(\overline{\Omega})\hookrightarrow C^{k,\gamma}(\overline{\Omega})\hookrightarrow C^{k} (\overline{\Omega}). \ \ \ \ \ (6)$

On the other hand, the case when ${\gamma=1}$ is the well-known Lipschitz case and for ${\gamma>1}$ it easy to see that (${\textbf{exercise}!}$) the functions ${\gamma}$-H\”{o}lder continuous are locally constant (constant on each connected component of ${\overline{\Omega}}$).

It is temptative to think that ${C^{k,1}(\overline{\Omega})=C^{k+1}(\overline{\Omega})}$, however it is quite easy to find Lipschitz functions with edges and this cleary rules out the differentiability of them. In spite of that, Lipschitz functions are differentiable in some sense:

Theorem 4 (Rademacher’s theorem) Let ${\Omega}$ be an open subset of ${{\mathbb R}^N}$ and let ${u:\Omega \rightarrow {\mathbb R}^M}$ then ${u}$ is Lipschitz if and only if ${u}$ is differentiable a. e. on ${\Omega}$ and ${Du\in L^\infty (\Omega)}$.

Proof: See Federer’s book of geometric measure theory. $\Box$

This type of relaxation in the differentiability of the functions is captured (in some sense) by the Sobolev spaces presented below.

Definition 5 (Sobolev Spaces) Let ${\Omega}$ be an open subset of ${{\mathbb R}^N}$, we say that ${u\in L^p(\Omega)}$ (for ${1\leq p\leq \infty}$) has a weak derivative in the direction ${i\in \{1,\dots ,N\}}$ on ${\Omega}$ given by the measurable function ${u_i}$ if:

$\displaystyle \int_{\Omega}u\frac{\partial\phi}{\partial x_i} =-\int_{\Omega} u_i \phi, \qquad \forall \phi\in C_0^1(\Omega). \ \ \ \ \ (7)$

For ${k\in {\mathbb N}}$ with ${k>1}$ we define the Sobolev spaces inductively as follows:

${W^{1,p}(\Omega):=\{u\in L^p(\Omega)|u_i\in L^p(\Omega),\,\, \forall i=1,\dots N \}.}$

${W^{k,p}(\Omega):=\{u\in W^{k,p}(\Omega)| u_i\in W^{k-1,p}(\Omega),\, \, \forall i=1,\dots N \}}$,

Remark 4 Another important motivation for the definition of Sobolev spaces comes from the classical Dirichlet problem:

$\displaystyle \begin{cases} -\Delta u=f \quad \text{in} \; \Omega \, ,\\ u = 0 \qquad \hspace{1.5mm} \enspace \text{on} \: \partial \Omega. \end{cases} \ \ \ \ \ (8)$

Where ${\Omega}$ is a bounded and open subset of ${{\mathbb R}^N}$, ${\Delta}$ is the Laplacian operator and ${f}$ is initially a continuous function.

If we suppose that ${u\in C^2(\overline{\Omega})}$ is a solution of (8) multiplying both sides by a function ${\phi \in C_0^1(\Omega)}$ and integrating by parts (Exercise: Consider the vector field ${\phi \nabla u}$ and apply to it Divergence Theorem, also see the appendix of Evans) we get the following integral equation:

$\displaystyle \int_{\Omega}\nabla{u}\cdot\nabla \phi=\int_{\Omega} f\phi, \hspace{3mm} \forall v\in C_0^1(\Omega). \ \ \ \ \ (9)$

It is clear that a solution of the classical problem (8) is also a solution of the weak variational problem (9), under certain conditions on the data ${f}$ and on the domain ${\Omega}$ both problems are equivalent. The theory that studies the issues of the nontrivial implication of this equivalence is called regularity theory, see Gilbarg and Trudinger’s book.

As we discussed at the begining of these notes, the classical functional analysis provide us powerful tools to approach this kind of linear integral equations. For this particular case it is necessary to change the modeling space (or test space) ${C_0^1(\Omega)}$ because the completeness issues of this space with respect to the integration (Exercise: Show that ${C_0^1(\Omega)}$ is not complete endowed with any ${L^p}$ norm).

Hence, it is natural to postulate a Sobolev space as the suitable modeling space for this problem, in this case ${H^1(\Omega):=W^{1,2}(\Omega)}$. However, this space does not take in account the boundary condition implicit in ${C_0^1(\Omega)}$ and it is not obvious how can we impose a pointwise condition on a space of functions that are defined almost everywhere.

This remark motivates the following definition.

Definition 6 (Sobolev Spaces ${W_0^{1,p}}$) Let ${\Omega}$ be an open subset of ${{\mathbb R}^N}$ and let ${1\leq p<\infty}$, let us define:

$\displaystyle W_0^{1,p}(\Omega):={\overline{C_0^1(\Omega)}}^{|| . ||_{W^{1,p}(\Omega)}}. \ \ \ \ \ (10)$

Sobolev spaces are an important branch of study in PDE, there are many good references in this topic, see for example the textbooks of Adams,L. Evans and of H. Brezis. We will summarize the required results about this spaces in the following lemma.

Lemma 7 (Some properties about Sobolev spaces) Let ${\Omega}$ be an open subset of ${{\mathbb R}^N}$ then:

1. Poincaré’s inequality. Let ${1\leq p<\infty}$, if ${\Omega}$ has finite measure or if ${\Omega}$ has bounded projection on some axis then there exist a constant ${C}$ only deppending on ${p}$ and ${\Omega}$ such that:

$\displaystyle || u||_{L^p(\Omega)}^p\leq C || \nabla u||_{L^p(\Omega)}^p \quad \forall u\in W_0^{1,p}(\Omega). \ \ \ \ \ (11)$

Particularly, for ${p=2}$ we have that ${C=\frac{1}{\lambda_1}}$ where ${\lambda_1}$ is the first eigenvalue of the Laplacian on ${\Omega}$ with homogeneus Dirichlet boundary condition.

2. Sobolev embeddings. Suppose that ${\Omega}$ is bounded with ${C^1}$-boundary then we have the following:
1. Continuous injections
2. Compact injections

Where ${p*=\frac{Np}{N-p}}$ is the Sobolev critical exponent for ${p.

Proof: See Brezis’ book. The optimal constant for the case ${p=2}$ in the Poincaré’s inequality will be discussed later in this notes. $\Box$

Remark 5 The concept of smooth boundary will be discussed later in the section of Manifolds, however we can think by the moment in ${\Omega}$ a domain with ${C^k}$ boundary as the preimage of a ${C^k}$ function ${\gamma:{\mathbb R}^N\rightarrow {\mathbb R}}$ in the following sense:

${\Omega=\gamma^{-1}(0,\infty),}$

${\partial \Omega=\gamma^{-1}\{0\}.}$

For example, (exercise) to show that the ball ${B_1(0)\subset {\mathbb R}^N}$ is an ${C^{\infty}}$-domain we can consider function given by ${\gamma(x):=1-|| x||^2}$.

We will see later on this notes that this is, in fact, the local model for these sets ( modulus some details).

Example 1 Let ${\Omega}$ be a bounded open subset of ${{\mathbb R}^N}$.

Let us consider again the weak variational problem presented before:

$\displaystyle \int_{\Omega}\nabla{u}\cdot\nabla v=\int_{\Omega} fv, \hspace{3mm} \forall v\in H_0^1(\Omega):=W_0^{1,2}(\Omega). \ \ \ \ \ (12)$

With ${f\in L^2(\Omega)}$. Associated to this problem we can define the following energy functional.

${J : H_0^1(\Omega)\rightarrow {\mathbb R}}$

${u \rightarrow \frac{1}{2}\int_{\Omega} |\nabla u|^2-\int_{\Omega}fu.}$

By Poincaré’s inequality (11) the first term of this functional is defined by a bounded symmetric bilinear form, and by the Cauchy-Schwartz inequality the second term is a bounded linear function, therefore applying the last results we get that:

$\displaystyle DJ(u)(v)=\int_{\Omega}\nabla{u}\cdot\nabla v-\int_{\Omega} fv. \ \ \ \ \ (13)$

With this little example we have shown that the weak solutions for this PDE are exactly the critical points of an energy functional. We will discuss sistematically these kind of relationships between finding critical points of functions and finding solutions of PDEs later in the part of ${\,}$ Variational Methods.

Remark 6 The last example gives us an interesting but not surprising result. Indeed, the weak variational problem presented before can be thought as a linear equation of the form ${x-b=0}$, clearly the unique solution of this simple linear equation corresponds whith the unique minimum of the quadratic function ${\frac{1}{2}x^2-bx}$.

Example 2 Let ${\Omega}$ be a bounded open subset of ${{\mathbb R}^N}$ and let ${\phi \in C^1({\mathbb R})}$ a real valued function, let us consider the function:

${J: C(\overline{\Omega})\rightarrow C(\overline{\Omega})}$

${ u \rightarrow \phi(u).}$

Let ${h}$ in the unit ball of ${C(\overline{\Omega})}$, applying the fundamental theorem of calculus we get:

$\displaystyle J(u+h)-J(u)-\phi'(u)h=\int_{0}^{1}[\phi'(u+th)-\phi'(u)]hdt \ \ \ \ \ (14)$

Using the fact that ${\phi'}$ is uniformly continuous on the compact set

$\displaystyle [\min_{x\in \overline{\Omega}} u(x)-1,\max_{x\in \overline{\Omega}}u(x)+1] \ \ \ \ \$

and using Lebegue’s Dominate convergence theorem we get:

$\displaystyle \frac{|| J(u+h)-J(u)-\phi'(u)h||_{C(\overline{\Omega})}}{|| h||_{C(\overline{\Omega})} }\leq \int_{0}^{1}|| \phi'(u+th)-\phi'(u)||_{C(\overline{\Omega})}dt\rightarrow 0, \ \ \ \ \ (15)$

as ${|| h||_{C(\overline{\Omega})}\rightarrow 0}$.

Finally, ${DJ(u)h=\phi'(u)h}$.

Example 3 Given ${r\in {\mathbb N}}$ let us define ${r^*:=|\{\alpha\in ({\mathbb N}\cup \{0\})^N||\alpha|\leq r\}}$ (the cardinality of this set), and let us define the vector of functions ${D^ru:=(\partial ^\alpha u)_{|\alpha|\leq r}}$ of length ${r^*}$.

Let ${\Omega}$ be a bounded open subset of ${{\mathbb R}^N}$, suppose that ${\phi\in {\mathbb C}^\infty(\overline{\Omega}\times {\mathbb R}^{r^*})}$. Fix ${m\in {\mathbb N}}$ with ${m>r}$ and ${\gamma \in (0,1]}$, let us consider the functions:

${J: C^m(\overline{\Omega})\rightarrow C^{m-r}(\overline{\Omega}) }$

${u \rightarrow \phi(.,D^ru),}$

${I: C^{m,\gamma}(\overline{\Omega})\rightarrow C^{m-r,\gamma}(\overline{\Omega})}$

${u \rightarrow \phi(.,D^ru).}$

Following the same idea than in the last example it can be shown (exercise!) that these functions are differentiable and their derivative is given by:

$\displaystyle DI(u)h(x)=DJ(u)h(x)=\sum_{|\alpha|\leq r} \phi_u(x,D^ru(x))\partial^\alpha h(x) \ \ \ \ \ (16)$

For any ${h\in C^m,\gamma(\overline{\Omega})}$, respectively ${h\in C^m(\overline{\Omega})}$ . Here ${\phi_\alpha}$ is the partial derivative of ${\phi}$ with respect to the entry that contains ${\partial^\alpha u}$.

Example 4 Keeping the same notation of the last example, let us consider the function:

${F: C^m(\overline{\Omega})\rightarrow {\mathbb R}}$

${u \rightarrow \int_{\Omega} \phi(x,D^ru(x))dx.}$

We can see this function as the composition of the last function and a bounded linear operator, as follows:

$\displaystyle C^m(\overline{\Omega})\xrightarrow{J} C^{m-r}(\overline{\Omega})\xrightarrow{\int_{\Omega}} {\mathbb R}, \ \ \ \ \ (17)$

Therefore, applying the chain rule we get:

$\displaystyle DF(u)h=\int_{\Omega}\sum_{|\alpha|\leq r} \phi_u(x,D^ru(x))\partial^\alpha h(x)dx \ \ \ \ \ (18)$

For ${h\in C^m(\overline{\Omega})}$.

Example 5 (Dirichlet problem for the Laplacian) Let us consider again the nonlinear problem presented in our introduction to the nonlinear equations, for the case ${L=\Delta}$, ${N>2}$ and ${\Omega}$ bounded:

$\displaystyle \begin{cases} -\Delta u=f(u) \quad \text{in} \; \Omega \, ,\\ u = 0 \qquad \hspace{2mm} \enspace \text{on} \: \partial \Omega. \end{cases} \ \ \ \ \ (19)$

we formulate an associated weak variational problem as we discussed in the remark 4.

$\displaystyle \int_{\Omega}\nabla{u}\cdot\nabla v=\int_{\Omega} f(u)v, \hspace{3mm} \forall v\in H_0^1(\Omega). \ \ \ \ \ (20)$

In order to guarantee the well-definition of this expression it is necessary to impose some regularity conditions on ${f}$ i. e., ${f:{\mathbb R}\rightarrow {\mathbb R}}$ is continuous and there exist positive constants ${a}$ and ${b}$ such that ${|f(t)|\leq a|t|^q+b}$ for ${2\leq q\leq 2^*-1=\frac{N+2}{N-2}}$.

Under these conditions it is easy to see that the antiderivative ${F(t):=\int_{0}^{t}f(s)ds}$ satisfies:

$\displaystyle |F(t)|\leq a' |t|^{q+1}+b', \ \ \ \ \ (21)$

for ${a'}$ and ${b'}$ positive constants, therefore in virtue of Vainberg’s lemma and by the Sobolev inequalities the following energy functional is well-defined:

${J : H_0^1(\Omega)\rightarrow {\mathbb R}}$

${ u \rightarrow \frac{1}{2}\int_{\Omega} |\nabla u|^2-\int_{\Omega}F(u)}$.

As we discussed before, Poincaré’s inequality (11) guarantee the differentiability of the first term, hence it is enough to compute the derivative of the second term. Let ${h\in H_0^1(\Omega)}$, by the fundamental theorem of calculus we have:

$\displaystyle \int_{\Omega}(F(u+h)-F(u)-f(u)h)dx=\int_{\Omega}\int_{0}^{1}(f(u+th)-f(u))hdtdx. \ \ \ \ \ (22)$

Applying Fubini’s theorem and Hölder’s inequality with the conjugate exponents ${2^*=\frac{2N}{N-2}}$ and ${s:=\frac{2N}{N+2}}$ we get:

$\displaystyle \int_{\Omega}\int_{0}^{1}(f(u+th)-f(u))hdtdx\leq \int_{0}^{1}|| f(u+th)-f(u)||_{L^s(\Omega)}dt || h||_{L^{2^*}(\Omega)}. \ \ \ \ \ (23)$

By Vainberg’s lemma and by the Sobolev inequalities, the Nemytskii operator associated to ${f}$, ${\mathcal{N}_f:H_0^1(\Omega)\rightarrow L^\frac{2^*}{q}(\Omega)}$ is continuous. Using the fact that ${s<2^*}$, the critical continuous embedding of the Sobolev inequalities and Lebesgue’s dominated convergence theorem it follows:

$\displaystyle \frac{ \Bigg|\int_{\Omega}(F(u+h)-F(u)-f(u)h)dx\Bigg|}{|| h||_{H_0^1(\Omega)}}\leq \int_{0}^{1}|| f(u+th)-f(u)||_{L^s(\Omega)}dt\rightarrow 0 , \ \ \ \ \ (24)$

as ${|| h||_{H_0^1(\Omega)}\rightarrow 0}$.

Finally ${DJ(u)h=\int_{\Omega}f(u)h}$.

Example 6 (Dirichlet problem for the ${p}$-Laplacian) In this example we generalize the last example for more general operator. Let ${u:{\mathbb R}^N\rightarrow {\mathbb R}}$ a smooth function. For ${1\leq 1 <\infty}$ let us define the ${p}$-Laplacian:

$\displaystyle \Delta_pu:=\text{div}(| \nabla u|^ {p-2}\nabla u) \ \ \ \ \ (25)$

Clearly this operator generalize the Laplacian operator (${p=2}$) and also generalize some other important operators in differential geometry like the mean curvature operator (${p=1}$). We will address the geometric interpretation of this operator later in this notes.

The case ${1\leq p <2}$ is called singular because the singularity in the set ${\nabla u=0}$. The case ${2 is called degenerated, in this case the ${p}$-Laplacian belongs too a general family of operators called quasilinear elliptic operators (see Gilbarg and Trudinger’s book). In this part we are interested in this last case.

For ${p>2}$ and for ${\Omega}$ open and bounded in ${{\mathbb R}^N}$ we can define (analogously to the last example) the following weak variational problem:

$\displaystyle \int_{\Omega}|\nabla u|^ {{p-2}} \nabla{u}\cdot\nabla v=\int_{\Omega} f(u)v, \hspace{3mm} \forall v\in W_0^{1,p}(\Omega). \ \ \ \ \ (26)$

In order to define an energy functional associated to this problem it is necessary to impose some regularity conditions on ${f}$. These conditions comes naturally from imitating the last example and being aware of the critical Sobolev exponent for this case, i. e. ${p^*:=\frac{Np}{N-p}}$.

It is left as an exercise to find these conditions on ${f}$ to guarantee the differentiability of the part of the energy functional associated to ${f}$. On the other hand, it is another important exercise, show that the (first) term associated to the ${p}$-Laplacian given by

${J : W_0^{1,p}(\Omega)\rightarrow {\mathbb R}}$

${u \rightarrow \frac{1}{p}\int_{\Omega} |\nabla u|^p}$,

satisifies ${J\in C^2(W_0^{1,p}(\Omega))}$ and its derivatives are given by:

${DJ(u)v=\int_{\Omega} |\nabla u|^{p-2}\nabla u\cdot \nabla v }$,

${DJ(u)(v,w)=\int_{\Omega} |\nabla u|^{p-2}\nabla v\cdot \nabla w+(p-2)\int_{\Omega} |\nabla u|^{p-4}(\nabla u\cdot \nabla v )(\nabla u\cdot \nabla w) .}$

For  ${v,w\in W_0^{1,p}(\Omega)}$.

To provide some extra fun for the reader at the end of this (maybe a little heavy) section we leave the following exercise:

Suppose that ${\Omega}$ is a bounded open subset of ${{\mathbb R}^N}$. Compute the first two derivatives of the following functions:

1. ${J:X\rightarrow {\mathbb R}}$ with ${X:=C_0^1(\overline{\Omega},{\mathbb R}^N)}$ defined by:

$\displaystyle J(u):=\int_{\Omega} \det(Du) \ \ \ \ \ (27)$,

where ${Du}$ is the matrix of first derivatives of ${u}$.

2. ${J:X\rightarrow {\mathbb R}}$ with ${X:=C_0^1(\overline{\Omega},{\mathbb R})}$ defined by:

$\displaystyle J(u):=\int_{\Omega} \sqrt{1+|\nabla u|^2} \ \ \ \ \ (28)$.