User:Igny/Sobolev space

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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 $\backslash Omega\backslash subseteq\; \backslash mathbb\{R\}^n$ into $\backslash mathbb\{C\}$ is a generalization of the space of smooth functions, $C^k(\backslash Omega)$, by using a broader notion of weak derivatives. In some sense, Sobolev space is a completion of $C^k(\backslash Omega)$ under a suitable norm, see Meyers-Serrin Theorem below.

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

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

This is an original definition, used by Sergei Sobolev.

This space is a Banach space with a norm

:$\backslash bigl\backslash |u\backslash bigr\backslash |\_\{k,p,\backslash Omega\}^p=\backslash sum\_$

\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 $\backslash Omega\backslash subset\; R^n$, and for $p\backslash in[1,\backslash infty)$, $C^k(\backslash Omega)$ is dense in $W^\{k,p\}(\backslash Omega)$, that is the Sobolev spaces can alternatively be defined as closure of $C^k(\backslash Omega)$, because : Besides, $C^k(\backslash overline\; \backslash Omega)$ is dense in $W^\{k,p\}(\backslash Omega)$, if $\backslash Omega$ satisfies the so called segment property (in particular if it has Lipschitz boundary). Note that $C^k(\backslash Omega)$ is not dense in $W^\{k,\backslash infty\}(\backslash Omega)$ because : Sobolev spaces with negative index. For natural k, the Sobolev spaces $W^\{-k,p\}(\backslash Omega)$ are defined as dual spaces $\backslash left(W^\{k,q\}\_0(\backslash Omega)\backslash right)^*$, where q is conjugate to p, $\backslash frac\; 1p+\backslash frac\; 1q\; =1$. Their elements are no longer regular functions, but rather distributions. Alternative definition of Sobolev spaces with negative index is :$W^\{-k,p\}(\backslash Omega)=\backslash left\backslash \{u\backslash in\; D\text{'}(\backslash Omega):u=\backslash sum\_\{|\backslash alpha|\backslash leq\; k\}\backslash partial^\backslash alpha\; u\_\{\backslash alpha\},\; \{\backslash rm\backslash \; for\backslash \; some\backslash \; \}u\_\backslash alpha\backslash in\; L\_p(\backslash Omega)\backslash right\backslash \}$ Here all the derivatives are calculated in a sense of distributions in space $D\text{'}(\backslash Omega)$. These definitions are equivalent. For a natural k, $u\backslash in\; W^\{-k,p\}(\backslash Omega)$ defines a linear operator on $v\backslash in\; W^\{k,q\}\_0(\backslash Omega)$ and vice versa by :$\backslash bigl\backslash langle\; u,v\backslash bigr\backslash rangle=\backslash sum\_\{|\backslash alpha|\backslash leq\; k\}\backslash bigl\backslash langle\; \backslash partial^\backslash alpha\; u\_\{\backslash alpha\},v\backslash bigr\backslash rangle=\backslash sum\_\{|\backslash alpha|\backslash leq\; k\}(-1)^\{|\backslash alpha$

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

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

\langle u,v\rangle

{\|v\|_{k,q,\Omega}}

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

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

:$\backslash bigl\backslash langle\; u,v\backslash bigr\backslash rangle\_\{H^k\}=\backslash sum\_\{|\backslash alpha|\backslash leq\; k\}\backslash bigl\backslash langle\backslash partial^\backslash alpha\; u,\; \backslash partial^\backslash alpha\; v\backslash bigr\backslash rangle\_\{L\_2\}=$

\int_\Omega \sum_{|\alpha|\leq k} \partial ^\alpha u \,\overline{\partial^\alpha v}\, dx

Fourier transform The Sobolev space $H^s(\backslash mathbb\{R\}^n)$ can be defined for any real s by using the Fourier transform (in a sense of distributions). A distribution $u\backslash in\; D\text{'}(\backslash mathbb\{R\}^n)$ is said to belong to $H^s(\backslash mathbb\{R\}^n)$ if its Fourier transform $\backslash tilde\; u(\backslash xi)=\backslash mathcal\{F\}u$ is a regular function of $\backslash xi$ and $(1+|\backslash xi|^2)^\{s/2\}\backslash tilde\; u(\backslash xi)$ belongs to $L\_2(\backslash mathbb\{R\}^n)$. $H^s(\backslash mathbb\{R\}^n)$ is a Banach space with a norm

:$\backslash bigl\backslash |u\backslash bigr\backslash |\_\{H^s\}^2=\backslash bigl\backslash |(1+|\backslash xi|^2)^\{s/2\}\backslash tilde\; u\backslash bigr\backslash |\_\{L\_2\}^2=\backslash int\_\{\backslash mathbb\{R\}^n\}|(1+|\backslash xi|^2)^s|\backslash tilde\; u(\backslash xi)|^2d\backslash xi$

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

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

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, $H^\{-s\}(\backslash mathbb\{R\}^n)$ is dual to $H^s(\backslash mathbb\{R\}^n)$. Note that $H^0(\backslash mathbb\{R\}^n)=L\_2(\backslash mathbb\{R\}^n)$ is self-dual. In bra-ket notation, $u\backslash in\; H^\{-s\}(\backslash mathbb\{R\}^n)$ defines a linear operator on $v\backslash in\; H^s(\backslash mathbb\{R\}^n)$ by

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