一些函数空间的相关概念

This is a brief introduction to the concepts in functional analysis, which are used in the variational formulation in finite element methods to solve the PDE. The content below is from the page 11-12 and page 27-31 in the book Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing¹.

Function Spaces

Definition of partial derivatives

$$ D^{\mathbf{n}} u(x):=\frac{\partial^{|\mathbf{n}|} u(x)}{\partial x _1^{n _1} \cdots \partial x _d^{n _d}}=\partial _{x _1}^{n _1} \cdots \partial _{x _d}^{n _d} u(x), \quad x=\left(x _1, \ldots, x _d\right) \in G, $$

where $\mathbf{n}=\left(n _1, \ldots, n _d\right) \in \mathbb{N} _0^d$ is a multi-index. The order of the partial derivative is given by $|\mathbf{n}|=\sum _{i=1}^d n _i$.

Definition of $C^n(G)$ space

For any integer $n \in \mathbb{N} _0$, we define

$$ C^n(G)=\left\{u: D^{\mathbf{n}} u \text { exists and is continuous on } G \text { for }|\mathbf{n}| \leq n\right\}, $$

and set $C^{\infty}(G)=\bigcap _{n \geq 0} C^n(G)$. The support of $u$ is denoted by $\operatorname{supp} u$, and we define $C _0^n(G), C _0^{\infty}(G)$ consisting of all functions $u \in C^n(G), C^{\infty}(G)$ with compact support supp $u \Subset G$.

Definition of $L^p(G)$ space

We denote by $L^p(G), 1 \leq p \leq \infty$ the usual space which consists of all Lebesgue measurable functions $u: G \rightarrow \mathbb{R}$ with finite $L^p$-norm, where ess sup means the essential supremum disregarding values on nullsets. The case $p=2$ is of particular interest. The space $L^2(G)$ is a Hilbert space with respect to the inner product $(u, v)=\int _G u(x) v(x) \mathrm{d} x$.

Riesz representation theorem

Let $\mathcal{H}$ be a Hilbert space with the inner product $(\cdot, \cdot) _{\mathcal{H}}$ and norm $\|u\| _{\mathcal{H}}:=$ $(u, u) _{\mathcal{H}}^{1 / 2}$. We denote by $\mathcal{H}^*$ the dual space of $\mathcal{H}$ which consists of all bounded linear functionals $u^*: \mathcal{H} \rightarrow \mathbb{R}$ on $\mathcal{H}$. $\mathcal{H}^*$ can be identified with $\mathcal{H}$ by the Riesz representation theorem.

$$ \left\langle u^\ast, v\right\rangle_{\mathcal{H}^\ast, \mathcal{H}}=(u, v)_{\mathcal{H}} \quad \forall v \in \mathcal{H} . $$

The mapping $u^\ast \mapsto u$ is a linear isomorphism of $\mathcal{H}^*$ onto $\mathcal{H}$.

Definition of $L^p(J ; \mathcal{H})$ space

The theory of parabolic partial differential equations requires the introduction of Hilbert space-valued $L^p$-functions. As above, let $\mathcal{H}$ be a Hilbert space with the norm $\|\cdot\| _{\mathcal{H}}$. Denote by $J$ the interval $J:=(0, T)$ with $T>0$, and let $1 \leq p \leq \infty$. The space $L^p(J ; \mathcal{H})$ is defined by

$$ L^p(J ; \mathcal{H}):=\left\{u: \bar{J} \rightarrow \mathcal{H} \text { measurable }:\|u\| _{L^p(J ; \mathcal{H})}<\infty\right\}, $$$$ \|u\| _{L^p(J ; \mathcal{H})}:= \begin{cases}\left(\int _J\|u(t)\| _{\mathcal{H}}^p \,\mathrm{d} t\right)^{1 / p} & \text { if } 1 \leq p<\infty \\ \operatorname{ess\,sup} _J\|u(t)\| _{\mathcal{H}} & \text { if } p=\infty .\end{cases} $$

Furthermore, for $n \in \mathbb{N}_0$ let $C^n(J ; \mathcal{H})$ be the space of $\mathcal{H}$-valued functions that are of the class $C^n$ with respect to $t$.

Sobolev Spaces

We now introduce some particular Hilbert spaces, which consist of functions that are square integrable together with their partial derivatives up to a certain order.

Definition of weak derivatives

$$ \int _G u^{\prime} \varphi \mathrm{d} x=-\int _G u \varphi^{\prime} \mathrm{d} x, \quad \forall \varphi \in C _0^1(G) . $$$$ u^\ast(\varphi)=-\int _G u \varphi^{\prime} \mathrm{d} x, \quad \forall \varphi \in C _0^1(G) . $$$$ -\int _G u \varphi^{\prime} \mathrm{d} x=\int _G w \varphi \mathrm{d} x, \quad \forall \varphi \in C _0^1(G) . $$

We then say that the weak derivative belongs to $L^2(G)$ and write $u^{\prime}=w$. In particular, if $u \in C^1(\bar{G})$, the generalized derivative $u^{\prime}$ coincides with the classical derivative $u^{\prime}$. In a similar way, we can define weak derivatives $D^n u$ of higher order $n \in \mathbb{N}$.

$$ \int _G D^n u \varphi \mathrm{d} x=(-1)^n \int _G u D^n \varphi \mathrm{d} x, \quad \forall \varphi \in C _0^n(G) . $$

Definition of $H^m(G)$ space

We can now define the spaces $H^m(G)$.

$$ H^m(G)=\left\{u \in L^2(G): D^n u \in L^2(G) \text { for } n \leq m\right\} . $$$$ (u, v) _{H^m(G)}=\sum _{n=0}^m\left(D^n u, D^n v\right) _{L^2(G)}, $$$$ \|u \| _{H^m(G)}^2=(u, u) _{H^m(G)}=\sum _{n=0}^m\left \|D^n u\right \| _{L^2(G)}^2 . $$

We sometimes omit the $(G)$ if the domain is clear from the context. $H^m(G)$ is complete and thus a Hilbert space. The space $H^m(G)$ is an example of a more general class of function spaces, called Sobolev spaces.

Definition of $W^{m, p}(G)$

$$ W^{m, p}(G)=\left\{u \in L^p(G): D^n u \in L^p(G) \text { for } n \leq m\right\} . $$$$ \|u \| _{W^{m, p}(G)}^p=\sum _{n=0}^m\left \|D^n u\right \| _{L^p(G)}^p . $$

The normed space $W^{m, p}(G)$ is complete and hence a Banach space for $1 \leq p \leq \infty$. Functions $u \in W^{1, p}(G)$ are “essentially” continuous.

$$ \tilde{u}\left(x _2\right)-\tilde{u}\left(x _1\right)=\int _{x _1}^{x _2} u^{\prime}(\xi) \mathrm{d} \xi . $$

We will also need spaces with boundary conditions where we impose $u=0$ on $\partial G$.

Definition of $W _0^{1, p}(G)$

$$ W _0^{1, p}(G)=\overline{C _0^1(G)}{ }^{ \|\cdot \| _{W^{1, p}(G)}} . $$

The space $W _0^{1, p}(G) \subset W^{1, p}(G)$ is a closed linear subspace. In particular, $H _0^1(G):=W _0^{1,2}(G)$ is again a Hilbert space with the norm $ \|\cdot \| _{H^1(G)}$. We have the important Poincaré inequality.

Poincaré inequality

$$ \|u \| _{L^p(G)} \leq C\left \|u^{\prime}\right \| _{L^p(G)}, \quad \forall u \in W _0^{1, p}(G) . $$$$ W _\ast^{1, p}(G):=\left\{u \in W^{1, p}(G): \int _G u \,\mathrm{d} x=0\right\} . $$

Then, (1) holds also for all $u \in W _\ast^{1, p}(G)$, with different $C$.

$$ \int _J u^{\prime}(t) \varphi(t) \mathrm{d} t=-\int _J u(t) \varphi^{\prime}(t) \mathrm{d} t, \quad \forall \varphi \in C _0^1(J) . $$

Definition of $W^{1, p}(J ; \mathcal{H})$

$$ W^{1, p}(J ; \mathcal{H}):=\left\{u \in L^p(J ; \mathcal{H}): u^{\prime} \in L^p(J ; \mathcal{H})\right\}, $$

with the norm

$$ \|u\| _{W^{1, p}(J ; \mathcal{H})}:= \begin{cases}\left(\int _J\bigl(\|u(t)\| _{\mathcal{H}}^p+\left\|u^{\prime}(t)\right\| _{\mathcal{H}}^p\bigr) \,\mathrm{d} t\right)^{1 / p} & \text { if } 1 \leq p<\infty, \\ \operatorname{ess\,sup} _J\left(\|u(t)\| _{\mathcal{H}}+\left\|u^{\prime}(t)\right\| _{\mathcal{H}}\right) & \text { if } p=\infty .\end{cases} $$

We again denote by $H^1(J ; \mathcal{H}):=W^{1,2}(J ; \mathcal{H})$​.