User:Igny/Sobolev space

Introduction In mathematics, Sobolev spaces play important role in studying partial differential equations. They are named after Sergei Sobolev, who introduced them in 1930s along with a theory of generalized functions. Sobolev space of functions acting from $\Omega \subseteq \mathbb {R} ^{n}$ into $\mathbb {C}$ is a generalization of the space of smooth functions, $C^{k}(\Omega )$, by using a broader notion of weak derivatives. In some sense, Sobolev space is a completion of $C^{k}(\Omega )$ under a suitable norm, see Meyers-Serrin Theorem below.

Definition Sobolev spaces are subspaces of the space of integrable functions $L_{p}(\Omega )$ with a certain restriction on their smoothness, such that their weak derivatives up to a certain order are also integrable functions.

$W^{k,p}(\Omega )=\{u\in L_{p}(\Omega ):\partial ^{\alpha }u\in L_{p}(\Omega )$ for all multi-indeces $\alpha$ such that $|\alpha |\leq k\}$

This is an original definition, used by Sergei Sobolev.

This space is a Banach space with a norm

${\bigl \|}u{\bigr \|}_{k,p,\Omega }^{p}=\sum _{|\alpha |\leq k}{\bigl \|}\partial ^{\alpha }u{\bigr \|}_{L_{p}}^{p}=\int _{\Omega }\sum _{|\alpha |\leq k}|\partial ^{\alpha }u|^{p}dx$

Meyers-Serrin Theorem. For a Lipschitz domain $\Omega \subset R^{n}$, and for $p\in [1,\infty )$, $C^{k}(\Omega )$ is dense in $W^{k,p}(\Omega )$, that is the Sobolev spaces can alternatively be defined as closure of $C^{k}(\Omega )$, because

$W^{k,p}(\Omega )=\operatorname {cl} _{L_{p}(\Omega ),\|\cdot \|_{k,p,\Omega }}\left({\bigl \{}f\in C^{k}(\Omega ):\|f\|_{k,p,\Omega }<\infty {\bigr \}}\right)$

Besides, $C^{k}({\overline {\Omega }})$ is dense in $W^{k,p}(\Omega )$, if $\Omega$ satisfies the so called segment property (in particular if it has Lipschitz boundary).

Note that $C^{k}(\Omega )$ is not dense in $W^{k,\infty }(\Omega )$ because

$\operatorname {cl} _{L_{\infty }(\Omega ),\|\cdot \|_{k,\infty ,\Omega }}\left({\bigl \{}f\in C^{k}(\Omega ):\|f\|_{k,\infty ,\Omega }<\infty {\bigr \}}\right)=C^{k}(\Omega )$

Sobolev spaces with negative index. For natural k, the Sobolev spaces $W^{-k,p}(\Omega )$ are defined as dual spaces $\left(W_{0}^{k,q}(\Omega )\right)^{*}$, where q is conjugate to p, ${\frac {1}{p}}+{\frac {1}{q}}=1$. Their elements are no longer regular functions, but rather distributions. Alternative definition of Sobolev spaces with negative index is

$W^{-k,p}(\Omega )=\left\{u\in D'(\Omega ):u=\sum _{|\alpha |\leq k}\partial ^{\alpha }u_{\alpha },{\rm {\ for\ some\ }}u_{\alpha }\in L_{p}(\Omega )\right\}$

Here all the derivatives are calculated in a sense of distributions in space $D'(\Omega )$.

These definitions are equivalent. For a natural k, $u\in W^{-k,p}(\Omega )$ defines a linear operator on $v\in W_{0}^{k,q}(\Omega )$ and vice versa by

${\bigl \langle }u,v{\bigr \rangle }=\sum _{|\alpha |\leq k}{\bigl \langle }\partial ^{\alpha }u_{\alpha },v{\bigr \rangle }=\sum _{|\alpha |\leq k}(-1)^{|\alpha |}{\bigl \langle }u_{\alpha },\partial ^{\alpha }v{\bigr \rangle }=\sum _{|\alpha |\leq k}(-1)^{|\alpha |}\int _{\Omega }u_{\alpha }{\overline {\partial ^{\alpha }v}}dx$

Naturally, $W^{-k,p}(\Omega )$ is a Banach space with a norm

${\bigl \|}u{\bigr \|}_{-k,p,\Omega }=\sup _{v\in W^{k,q}(\Omega ),\|v\|_{k,q,\Omega }\not =0}{\frac {|\langle u,v\rangle |}{\|v\|_{k,q,\Omega }}}$

Now for any integer k, $\partial ^{\alpha }$ is a bounded operator from $W^{k,p}$ to $W^{k-|\alpha |,p}$

Special case p=2 . The space $H^{k}(\Omega )=W^{k,2}(\Omega )$ is in fact a separable Hilbert space with the inner product

${\bigl \langle }u,v{\bigr \rangle }_{H^{k}}=\sum _{|\alpha |\leq k}{\bigl \langle }\partial ^{\alpha }u,\partial ^{\alpha }v{\bigr \rangle }_{L_{2}}=\int _{\Omega }\sum _{|\alpha |\leq k}\partial ^{\alpha }u\,{\overline {\partial ^{\alpha }v}}\,dx$

Fourier transform The Sobolev space $H^{s}(\mathbb {R} ^{n})$ can be defined for any real s by using the Fourier transform (in a sense of distributions). A distribution $u\in D'(\mathbb {R} ^{n})$ is said to belong to $H^{s}(\mathbb {R} ^{n})$ if its Fourier transform ${\tilde {u}}(\xi )={\mathcal {F}}u$ is a regular function of $\xi$ and $(1+|\xi |^{2})^{s/2}{\tilde {u}}(\xi )$ belongs to $L_{2}(\mathbb {R} ^{n})$. $H^{s}(\mathbb {R} ^{n})$ is a Banach space with a norm

${\bigl \|}u{\bigr \|}_{H^{s}}^{2}={\bigl \|}(1+|\xi |^{2})^{s/2}{\tilde {u}}{\bigr \|}_{L_{2}}^{2}=\int _{\mathbb {R} ^{n}}|(1+|\xi |^{2})^{s}|{\tilde {u}}(\xi )|^{2}d\xi$

In fact, it is a Hilbert space with the inner product

${\bigl \langle }u,v{\bigr \rangle }_{H^{s}}=\int _{\mathbb {R} ^{n}}(1+|\xi |^{2})^{s}{\tilde {u}}(\xi ){\overline {{\tilde {v}}(\xi )}}d\xi$

It can be checked that for integer s these definitions of the space, norm, and the inner product are equivalent to the definitions in the previous sections.

Duality For any real s, $H^{-s}(\mathbb {R} ^{n})$ is dual to $H^{s}(\mathbb {R} ^{n})$. Note that $H^{0}(\mathbb {R} ^{n})=L_{2}(\mathbb {R} ^{n})$ is self-dual. In bra-ket notation, $u\in H^{-s}(\mathbb {R} ^{n})$ defines a linear operator on $v\in H^{s}(\mathbb {R} ^{n})$ by

${\bigl \langle }u,v{\bigr \rangle }=\int _{\mathbb {R} ^{n}}{\tilde {u}}(\xi ){\overline {{\tilde {v}}(\xi )}}d\xi$