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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
into
is a generalization of the space of smooth functions,
, by using a broader notion of weak derivatives. In some sense, Sobolev space is a completion of
under a suitable norm, see Meyers-Serrin Theorem below.
Definition Sobolev spaces are subspaces of the space of integrable functions
with a certain restriction on their smoothness, such that their weak derivatives up to a certain order are also integrable functions.
for all multi-indeces
such that ![{\displaystyle |\alpha |\leq k\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2227ed3c8a043f0c18833a1361aaca533b7748d9)
This is an original definition, used by Sergei Sobolev.
This space is a Banach space with a norm
![{\displaystyle {\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2855462c6da695c532658e689678202edf5aec)
Meyers-Serrin Theorem.
For a Lipschitz domain
, and for
,
is dense in
, that is the Sobolev spaces can alternatively be defined as closure of
, because
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/373213a09d4533424791cdb6db4d7f3a38298e57)
Besides,
is dense in
, if
satisfies the so called segment property (in particular if it has Lipschitz boundary).
Note that
is not dense in
because
![{\displaystyle \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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e98e50955973ca3d394b38727c772be3d3b0e270)
Sobolev spaces with negative index. For natural k, the Sobolev spaces
are defined as dual spaces
, where q is conjugate to p,
. Their elements are no longer regular functions, but rather distributions. Alternative definition
of Sobolev spaces with negative index is
![{\displaystyle 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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641f5ed6c721a734af47082258303752d6622229)
Here all the derivatives are calculated in a sense of distributions in space
.
These definitions are equivalent. For a natural k,
defines a linear operator on
and vice versa by
![{\displaystyle {\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0670c5ffd3a64158db6d8f2590b69093c5f790b2)
Naturally,
is a Banach space with a norm
![{\displaystyle {\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 }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db4a20af0a5ffccbd446413e679df0563e60ef05)
Now for any integer k,
is a bounded operator from
to
Special case p=2 . The space
is in fact a separable Hilbert space with the inner product
![{\displaystyle {\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89b3bfe4366fd1adeb50eec5347626a7b846a56f)
Fourier transform The Sobolev space
can be defined for any real s by using the Fourier transform (in a sense of distributions). A distribution
is said to belong to
if its Fourier transform
is a regular function of
and
belongs to
.
is a Banach space with a norm
![{\displaystyle {\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc8a472e6c86bc4311fe08966c4c046a8e17947)
In fact, it is a Hilbert space with the inner product
![{\displaystyle {\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d444eb432fba0914875fc09f237c9eebfbbdff2)
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,
is dual to
. Note that
is self-dual. In bra-ket notation,
defines a linear operator on
by
![{\displaystyle {\bigl \langle }u,v{\bigr \rangle }=\int _{\mathbb {R} ^{n}}{\tilde {u}}(\xi ){\overline {{\tilde {v}}(\xi )}}d\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2847d45e141dc54bf30a53e81555cceb44b2fd16)