Sobolev room

A Sobolev space , also Sobolew space (after Sergei Lwowitsch Sobolew , with a transliteration and in English transcription Sobolev) is in mathematics a function space of weakly differentiable functions , which is also a Banach space . The concept was significantly advanced by the systematic theory of the calculus of variations at the beginning of the 20th century . This minimizes functional over functions. Today Sobolev spaces form the basis of the solution theory of partial differential equations .

Sobolev spaces of integer order

Definition as a function space of weak derivatives

Be open and not empty . Be and . ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle 1\leq p\leq \infty }$ ${\displaystyle k\in \mathbb {N} }$

Then the Sobolev space is defined as: ${\displaystyle W^{k,p}(\Omega )}$

${\displaystyle W^{k,p}(\Omega )=\left\{u\in L^{p}(\Omega ):\,\,\forall \alpha \in \mathbb {N} ^{n}\,\,{\text{mit}}\,\,|\alpha |\leq k\,\,{\text{existieren}}\,\,D^{\alpha }u\in L^{p}(\Omega )\right\}}$

Denotes the weak derivatives of . ${\displaystyle D^{\alpha }u}$ ${\displaystyle u}$

In other words, the Sobolev space is the space of those real-valued functions whose mixed partial weak derivatives are up to the order in Lebesgue space . ${\displaystyle u\in L^{p}(\Omega )}$ ${\displaystyle k}$ ${\displaystyle L^{p}(\Omega )}$

The spelling is also common for. ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle W_{p}^{k}(\Omega )}$

Sobolev norm

For functions to define the norm by ${\displaystyle u\in W^{k,p}(\Omega )}$ ${\displaystyle W^{k,p}(\Omega )}$

${\displaystyle \|u\|_{W^{k,p}(\Omega )}={\begin{cases}\left(\sum _{|\alpha |\leq k}\|\partial ^{\alpha }u\|_{L^{p}(\Omega )}^{p}\right)^{1/p},&{\text{falls }}p<\infty ,\\\max _{|\alpha |\leq k}\|\partial ^{\alpha }u\|_{L^{\infty }(\Omega )},&{\text{falls }}p=\infty .\end{cases}}}$

There is a multi-index with and . Furthermore is . ${\displaystyle \alpha }$ ${\displaystyle \alpha =(\alpha _{1},\cdots ,\alpha _{n})}$ ${\displaystyle \alpha _{i}\in \mathbb {N} _{0}}$ ${\displaystyle \textstyle \partial ^{\alpha }u:=\left({\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}\right)u}$ ${\displaystyle \textstyle |\alpha |=\sum _{i=1}^{n}\alpha _{i}}$

The mentioned Sobolev standard is provided as a standard equivalent to the sum of the norms of all possible combinations of partial derivatives up to the th order. The Sobolev space with respect to the respective Sobolev standard completely , so a Banach space. ${\displaystyle L^{p}}$ ${\displaystyle k}$ ${\displaystyle W^{k,p}(\Omega )}$

Definition as a topological closure

Let us now consider the space of the functions, whose partial derivatives lie up to the degree in , and denote this function space with . As different functions each other never -equivalent (see L ^p -space ) are, one can in embedding, and it is the following inclusion ${\displaystyle C^{\infty }(\Omega )}$ ${\displaystyle k}$ ${\displaystyle L^{p}(\Omega )}$ ${\displaystyle C^{k,p}(\Omega )}$ ${\displaystyle C^{k,p}}$ ${\displaystyle L^{p}}$ ${\displaystyle C^{k,p}(\Omega )}$ ${\displaystyle L^{p}(\Omega )}$

${\displaystyle C^{k,p}(\Omega )\subset W^{k,p}(\Omega )\subset L^{p}(\Omega ).}$

The space is not complete with regard to the standard. Rather, its completion is straight . The partial derivatives up to order k can be used as continuous operators room Sobolev clearly on these continued steadily be. These continuations are just the weak derivatives . ${\displaystyle C^{k,p}(\Omega )}$ ${\displaystyle W^{k,p}}$ ${\displaystyle W^{k,p}(\Omega )}$

This gives an alternative definition of Sobolev spaces. According to the Meyers-Serrin theorem , it is equivalent to the above definition.

properties

As already mentioned, the norm is a complete vector space , thus a Banach space . For he is even reflexive . ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle \|{\cdot }\|_{W^{k,p}(\Omega )}}$ ${\displaystyle 1<p<\infty }$

For becomes the norm by the scalar product ${\displaystyle p=2}$

${\displaystyle (u,v)_{W^{k,2}(\Omega )}:=\sum _{|\alpha |\leq k}(\partial ^{\alpha }u,\partial ^{\alpha }v)_{L^{2}(\Omega )}}$

induced . is therefore a Hilbert space , and one also writes . ${\displaystyle W^{k,2}(\Omega )}$ ${\displaystyle H^{k}(\Omega ):=W^{k,2}(\Omega )}$

Boundary value problems

The weak derivative or the Sobolev spaces were developed to solve partial differential equations. However, there is still one difficulty in solving boundary value problems . The weakly differentiable functions, like the functions, are not defined on zero sets. The expression for and doesn't make sense at first. For this problem, the restriction mapping has been generalized to the track operator. ${\displaystyle L^{p}}$ ${\displaystyle f|_{\partial \Omega }=g}$ ${\displaystyle f\in W^{q,p}(\Omega )}$ ${\displaystyle g\in C(\partial \Omega )}$ ${\displaystyle f\mapsto f|_{\partial \Omega }}$

Trace operator

Be a restricted area with Rand, . Then there is a bounded, linear operator ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle C^{m}}$ ${\displaystyle m\in \mathbb {N} }$

${\displaystyle T:W^{m,p}(\Omega )\to W^{m-1,q}(\partial \Omega ),}$

so that

${\displaystyle Tu=u|_{\partial \Omega }}$ if ${\displaystyle u\in C^{m}({\overline {\Omega }})}$

and

${\displaystyle \|Tu\|_{W^{m-1,q}(\partial \Omega )}\leq C\|u\|_{W^{m,p}(\Omega )}}$ for all ${\displaystyle u\in W^{m,p}(\Omega )}$

applies. It is

${\displaystyle q=(n-1)p/(n-p)}$ if ${\displaystyle p<n}$

${\displaystyle q<\infty }$ if ${\displaystyle p=n}$

${\displaystyle q=\infty }$ if ${\displaystyle p>n}$

The constant depends only on , , and starting. The operator 's track operator . A similar statement can also be proven for Lipschitz areas : ${\displaystyle C}$ ${\displaystyle p}$ ${\displaystyle \Omega }$ ${\displaystyle m}$ ${\displaystyle q}$ ${\displaystyle T}$

Trace operator for Lipschitz areas

Let be a restricted Lipschitz area, i.e. with -Rand. Then there is a bounded linear operator ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle C^{0,1}}$

${\displaystyle T:W^{1,p}(\Omega )\to L^{q}(\partial \Omega ),}$

so that

${\displaystyle Tu=u|_{\partial \Omega }}$ if ${\displaystyle u\in C^{\infty }({\overline {\Omega }})}$

and

${\displaystyle \|Tu\|_{L^{q}(\partial \Omega )}\leq C\|u\|_{W^{1,p}(\Omega )}}$ for all ${\displaystyle u\in W^{1,p}(\Omega )}$

applies. It is

${\displaystyle q=(n-1)p/(n-p)}$if , ${\displaystyle p<n}$

${\displaystyle q<\infty }$if , ${\displaystyle p=n}$

${\displaystyle q=\infty }$if . ${\displaystyle p>n}$

The constant depends solely on , and . ${\displaystyle C}$ ${\displaystyle p}$ ${\displaystyle \Omega }$ ${\displaystyle q}$

Sobolev space with zero boundary conditions

With denotes the termination of the test function space in . This means if and only if there is a sequence with in ${\displaystyle W_{0}^{k,p}(\Omega )}$ ${\displaystyle C_{c}^{\infty }(\Omega )}$ ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle u\in W_{0}^{k,p}(\Omega )}$ ${\displaystyle (u_{m})_{m\in \mathbb {N} }\subset C_{c}^{\infty }(\Omega )}$ ${\displaystyle u_{m}\to u}$ ${\displaystyle W^{k,p}(\Omega ).}$

For one can prove that this set are exactly the Sobolev functions with zero boundary conditions. So if it has a Lipschitz edge , then applies if and only if applies in the sense of traces. ${\displaystyle k=1}$ ${\displaystyle \Omega }$ ${\displaystyle u\in W_{0}^{1,p}(\Omega )}$ ${\displaystyle u|_{\partial \Omega }=0}$

Embedding sets

Sobolev number

Any Sobolev space with assigns a single number to which is important in connection with embedding sets. One sets ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$

${\displaystyle \gamma :=k-{\frac {n}{p}}}$

and calls this number the Sobolev number. ${\displaystyle \gamma }$

Embedding theorem by Sobolev

There are several interrelated statements that are called Sobolev's embedding theorem, Sobolev's embedding theorem, or Sobolev's lemma. Be open and bounded subset of , , and the Sobolev number to . There is a continuous embedding for ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle \gamma }$ ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle \gamma >m}$

${\displaystyle W^{k,p}(\Omega )\hookrightarrow C^{m}(\Omega )\subset C(\Omega ),}$

where respectively are equipped with the supremum norm . In other words, each equivalence class has a representative in . If, on the other hand , applies , then one can at least continuously embed in the space for all , whereby one sets. ${\displaystyle C^{m}(\Omega )}$ ${\displaystyle C(\Omega )}$ ${\displaystyle f\in W^{k,p}(\Omega )}$ ${\displaystyle C^{m}(\Omega )}$ ${\displaystyle \gamma \leq 0}$ ${\displaystyle W^{k,p}(\Omega )}$ ${\displaystyle L^{q}(\Omega )}$ ${\displaystyle 1\leq q<{\tfrac {np}{n-kp}}}$ ${\displaystyle {\tfrac {np}{0}}:=\infty }$

From Sobolev's embedding theorem it can be concluded that it is for a continuous embedding ${\displaystyle (k-m)p\leq n}$

${\displaystyle W^{k,p}(\Omega )\hookrightarrow W^{m,q}(\Omega )}$

for everyone there. ${\displaystyle 1\leq q\leq {\tfrac {np}{n-(k-m)p}}}$

Embedding theorem by Rellich

Be open and limited and . Then is the embedding ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle 1\leq p<\infty }$

${\displaystyle \operatorname {Id} \colon W_{0}^{k,p}(\Omega )\hookrightarrow W_{0}^{k-1,p}(\Omega )}$

a linear compact operator . The identical figure denotes . ${\displaystyle \operatorname {Id} }$

Sobolev's embedding theorems in the R ^d

From now on, if a fixed space dimension is given, then the embedding is ${\displaystyle d\geqslant 1}$

${\displaystyle W^{1,p}(\mathbb {R} ^{d})\subseteq L^{q}(\mathbb {R} ^{d})}$		(1)

steadily provided the conditions

${\displaystyle 1\leqslant p\leqslant q\leqslant \infty ,\quad {\frac {d}{p}}-1\leqslant {\frac {d}{q}},\quad {\text{und}}\quad (p,q)\notin \left\{\left(d,\infty \right),\left(1,{\frac {d}{d-1}}\right)\right\}}$

are fulfilled, d. that is, there is a constant , so the following estimate holds ${\displaystyle C=C(d,p,q)>0}$

${\displaystyle \left\|u\right\|_{L^{q}(\mathbb {R} ^{d})}\leqslant C\left\|u\right\|_{W^{1,p}(\mathbb {R} ^{d})}\quad \forall \,u\in W^{1,p}(\mathbb {R} ^{d}).}$		(2)

This result follows from the Hardy-Littlewood-Sobolev inequality for fractional integrations. The endpoint cases are to be examined separately. ${\displaystyle (p,q)\in \left\{\left(d,\infty \right),\left(1,{\frac {d}{d-1}}\right)\right\}}$

The first endpoint case is embedding ${\displaystyle (p,q)=\left(1,{\frac {d}{d-1}}\right)}$

${\displaystyle W^{1,1}(\mathbb {R} ^{d})\subseteq L^{\frac {d}{d-1}}(\mathbb {R} ^{d})}$		(3)

also steady, whereby we set in the case . Therefore there is again a constant , so that the following estimate holds ${\displaystyle {\frac {1}{0}}:=\infty }$ ${\displaystyle d=1}$ ${\displaystyle C=C(d)>0}$

${\displaystyle \left\|u\right\|_{L^{\frac {d}{d-1}}(\mathbb {R} ^{d})}\leqslant C\left\|u\right\|_{W^{1,1}(\mathbb {R} ^{d})}\quad \forall \,u\in W^{1,1}(\mathbb {R} ^{d}).}$		(4)

This result follows from the Loomis-Whitney inequality , which goes back to Gagliardo and Nirenberg .

The second endpoint case is embedding ${\displaystyle (p,q)=(d,\infty )}$

${\displaystyle W^{1,d}(\mathbb {R} ^{d})\subseteq L^{\infty }(\mathbb {R} ^{d})}$		(5)

only for fulfilled and steady. This follows, for example, from the fundamental theorem of analysis . For , the embedding (5) is fundamentally wrong and therefore not fulfilled. As a counterexample, consider the function for , and . Overall, there is therefore only one constant with regard to (5) , so that the following estimate applies ${\displaystyle d=1}$ ${\displaystyle d\geqslant 2}$ ${\displaystyle f(x)=\sum _{n=1}^{N}\phi \left(2^{n}x\right)}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle \phi \in C_{0}^{\infty }(\mathbb {R} ^{d})}$ ${\displaystyle \mathrm {supp} \,\phi \subseteq \{x\in \mathbb {R} ^{d}\mid 1\leqslant |x|\leqslant 2\}}$ ${\displaystyle d=1}$ ${\displaystyle C>0}$

${\displaystyle \left\|u\right\|_{L^{\infty }(\mathbb {R} )}\leqslant C\left\|u\right\|_{W^{1,1}(\mathbb {R} )}\quad \forall \,u\in W^{1,1}(\mathbb {R} ).}$		(6)

The embeddings (3) and (5) are called Sobolev endpoint embeddings and the estimates (4) and (6) are called Sobolev endpoint inequalities .

More generally we even get that embedding

${\displaystyle W^{k,p}(\mathbb {R} ^{d})\subseteq W^{l,q}(\mathbb {R} ^{d})}$		(7)

is continuous if one of the following cases is true

${\displaystyle {\text{(i)}}\;0\leqslant l\leqslant k,\quad 1<p<q\leqslant \infty ,\quad {\frac {d}{p}}-k<{\frac {d}{q}}-l,}$

${\displaystyle {\text{(ii)}}\;0\leqslant l\leqslant k,\quad 1<p\leqslant q<\infty ,\quad {\frac {d}{p}}-k\leqslant {\frac {d}{q}}-l,}$

d. that is, there is again a constant , so that the following estimate holds ${\displaystyle C=C(d,p,q,k,l)>0}$

${\displaystyle \left\|u\right\|_{W^{l,q}(\mathbb {R} ^{d})}\leqslant C\left\|u\right\|_{W^{k,p}(\mathbb {R} ^{d})}\quad \forall \,u\in W^{k,p}(\mathbb {R} ^{d}).}$		(8th)

This result can be shown by complete induction using (1) . The embedding (7) is called the Sobolev embedding and the estimate (8) is called the Sobolev inequality . Note that the embedding is generally not fulfilled in this case . Conditions (i) and (ii) show very nicely to what extent the associated Sobolev numbers and are related to each other. Note that this version of Sobolev's embedding theorem does not require the additional and very restrictive condition compared to the above version . The evidence of these statements can be found in terrytao.wordpress.com (Thm. 3, Ex. 20, Lem. 4, Ex. 24 and Ex. 25) and unfortunately cannot be obtained directly from the standard sources (under these weak conditions) . ${\displaystyle q<p}$ ${\displaystyle {\frac {d}{p}}-k}$ ${\displaystyle {\frac {d}{q}}-l}$ ${\displaystyle (k-l)p\leqslant d}$

In addition, the following embedding result applies:

The embedding

${\displaystyle W^{d,1}(\mathbb {R} ^{d})\subseteq C_{\mathrm {b} }(\mathbb {R} ^{d})}$		(9)

is continuous for all , i.e. that is, there is a constant , so the following estimate holds ${\displaystyle d\geqslant 1}$ ${\displaystyle C=C(d)>0}$

${\displaystyle \left\|u\right\|_{\infty }\leqslant C\left\|u\right\|_{W^{d,1}(\mathbb {R} ^{d})}\quad \forall \,u\in W^{d,1}(\mathbb {R} ^{d}).}$		(10)

Here denotes the set of continuous and limited functions and the supreme norm on the . ${\displaystyle C_{\mathrm {b} }(\mathbb {R} ^{d})}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \left\|\cdot \right\|_{\infty }}$ ${\displaystyle \mathbb {R} ^{d}}$

Sobolev space of real-valued order

definition

Sobolev spaces with real exponents are also often used. In the whole-space case, these are defined via the Fourier transform of the function involved. The Fourier transformation is referred to here with . For a function is an element of , if ${\displaystyle s}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle s\in \mathbb {R} ,s\geq 0}$ ${\displaystyle f\in L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle H^{s}(\mathbb {R} ^{n})}$

${\displaystyle \zeta \mapsto (1+|\zeta |^{2})^{\frac {s}{2}}\cdot {\mathcal {F}}(f)(\zeta )\in L^{2}(\mathbb {R} ^{n})}$

applies. Due to the identity , these are for the same rooms that were already defined in the first section. With ${\displaystyle {\mathcal {F}}(\partial ^{\alpha }f)=(i\zeta )^{\alpha }{\mathcal {F}}(f)}$ ${\displaystyle s\in \mathbb {N} }$

${\displaystyle (f,g)_{H^{s}(\mathbb {R} ^{n})}:=\int _{\mathbb {R} ^{n}}(1+|k|^{2})^{s}({\mathcal {F}}(f))(k)\cdot {\overline {({\mathcal {F}}(g))(k)}}dk}$

becomes a Hilbert dream . The norm is given by ${\displaystyle H^{s}(\mathbb {R} ^{n})}$

${\displaystyle \|f\|_{H^{s}(\mathbb {R} ^{n})}:=\|(1+|\cdot |^{2})^{\frac {s}{2}}\cdot {\mathcal {F}}(f)\|_{L^{2}(\mathbb {R} ^{n})}}$.

For a smoothly bounded, limited area , space is defined as the set of all that can be continued to a ( defined) function in . ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle H^{s}(\Omega )\subset L^{2}(\Omega )}$ ${\displaystyle f\in L^{2}(\Omega )}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle H^{s}(\mathbb {R} ^{n})}$

For one can also define Sobolev spaces. To do this, however, the theory of distributions must be used. Be the space of tempered distributions , then everyone is through ${\displaystyle s<0}$ ${\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}$ ${\displaystyle H^{s}(\mathbb {R} ^{n})}$ ${\displaystyle s\in \mathbb {R} }$

${\displaystyle H^{s}(\mathbb {R} ^{n}):=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{n}):(1+|\zeta |^{2})^{\frac {s}{2}}\cdot {\mathcal {F}}(f)(\zeta )\in L^{2}(\mathbb {R} ^{n})\right\}}$

Are defined.

Dual and Hilbert space

If one considers the Banach space with the -Scalar product , then its dual space is . However, one can calculate the space with the help of the scalar product ${\displaystyle H^{s}}$ ${\displaystyle L^{2}}$ ${\displaystyle \textstyle (u,v):=\int u(x){\overline {v(x)}}\mathrm {d} x}$ ${\displaystyle H^{-s}}$ ${\displaystyle H^{s}}$

${\displaystyle (u,v)_{H^{s}}={\frac {1}{(2\pi )^{n}}}\int {\mathcal {F}}(u)(\xi ){\mathcal {F}}(v)(\xi )(1+|\xi |^{2})^{s}\mathrm {d} \xi }$

understand it as a Hilbert space . Since Hilbert spaces are dual to themselves, to and to (with regard to different products) is now dual. One can and with the help of isometric isomorphism ${\displaystyle H^{s}}$ ${\displaystyle H^{s}}$ ${\displaystyle H^{-s}}$ ${\displaystyle H^{s}}$ ${\displaystyle H^{-s}}$

${\displaystyle {\begin{aligned}v\mapsto &{\mathcal {F}}^{-1}\left((1+|\xi |^{2})^{s}{\mathcal {F}}(v)(\xi )\right)(x)\\=&{\mathcal {F}}^{-1}{\mathcal {F}}((1+|D|^{2})^{s}v(\xi ))(x)\\=&(1+|D|^{2})^{s}v(x)\end{aligned}}}$

identify. In an analogous way, the spaces and through the isometric isomorphism ${\displaystyle H^{s}}$ ${\displaystyle H^{s-l}}$

${\displaystyle v\mapsto (1+|D|^{2})^{\frac {l}{2}}v}$

identify with each other.

Applications

Sobolev spaces are used in the theory of partial differential equations . The solutions of the weak formulation of a partial differential equation typically lie in a Sobolev space.

The theory of partial differential equations thus also provides numerical solution methods. The finite element method is based on the weak formulation of the partial differential equations and thus on Sobolev space theory.

Sobolev spaces also play a role in the optimal control of partial differential equations.

See also

• Locally poorly differentiable function
• Besov room

literature

• H.-W. Old: linear functional analysis . 5th edition. Springer, 2006, ISBN 3-540-34186-2
• RA Adams, JJF Fournier: Sobolev Spaces . 2nd edition. Academic Press, 2003, ISBN 0-12-044143-8
• M. Dobrowolsky: Applied Functional Analysis . 2nd Edition. Springer, 2010, ISBN 978-3-642-15268-9
• LC Evans: Partial Differential Equations . American Mathematical Society, 1998, ISBN 0-8218-0772-2
• LC Evans, RF Gariepy: Measure Theory and Fine Properties of Functions . CRC, 1991, ISBN 0-8493-7157-0
• V. Mazja: Sobolev Spaces . Springer, 1985, ISBN 3-540-13589-8
• WP Ziemer: Weakly Differentiable Functions . Springer, 1989, ISBN 0-387-97017-7

Individual evidence

1. ^ Mathematik.uni-wuerzburg.de (PDF) Theorem 3.15
2. ↑ M. Dobrowolsky: Applied functional analysis . 2nd Edition. Springer, 2010, ISBN 978-3-642-15268-9 , sentence 6.15
3. ↑ M. Dobrowolsky: Applied functional analysis . 2nd Edition. Springer, 2010, ISBN 978-3-642-15268-9 , sentence 6.17