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 $$\left(W^{k,q}_0(\Omega)\right)^*$$, 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 $$D'(\Omega)$$.

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, $$W^{-k,p}(\Omega)$$ 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
 * $$\bigl\|u\bigr\|_{H^s}^2=\bigl\|(1+|\xi|^2)^{s/2}\tilde u\bigr\|_{L_2}^2=\int_{\mathbb{R}^n}|(1+|\xi|^2)^s|\tilde u(\xi)|^2d\xi$$

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