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

Definition Sobolev spaces are subspaces of the space of integrable functions Lp(Ω) 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 α 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 open set \Omega\subset R^n, and for p\in[1,\infty), Ck(Ω) is dense in Wk,p(Ω), that is the Sobolev spaces can alternatively be defined as closure of Ck(Ω), 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 Wk,p(Ω), if Ω satisfies the so called segment property (in particular if it has Lipschitz boundary).

Note that Ck(Ω) 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(Ω) are defined as dual spaces \left(W^{k,q}_0(\Omega)\right)^*, where q is conjugate to p, \frac 1p+\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}(\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'(Ω).

These definitions are equivalent. For a natural k, u\in W^{-k,p}(\Omega) defines a linear operator on v\in W^{k,q}_0(\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(Ω) 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 Wk,p to Wk − | α | ,p

Special case p=2 . The space Hk(Ω) = Wk,2(Ω) 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 ξ 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)|^2d\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