Sobolev space

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In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

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Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.

Motivation

In this section and throughout the article ${\displaystyle \Omega }$ is an open subset of ${\displaystyle \mathbb {R} ^{n}.}$

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class ${\displaystyle C^{1}}$ see Differentiability classes). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space ${\displaystyle C^{1}}$ (or ${\displaystyle C^{2}}$, etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.

Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms, rather than the uniform norm. A typical example is measuring the energy of a temperature or velocity distribution by an ${\displaystyle L^{2}}$-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

The integration by parts formula yields that for every ${\displaystyle u\in C^{k}(\Omega )}$, where ${\displaystyle k}$ is a natural number, and for all infinitely differentiable functions with compact support ${\displaystyle \varphi \in C_{c}^{\infty }(\Omega ),}$

${\displaystyle \int _{\Omega }u\,D^{\alpha \!}\varphi \,dx=(-1)^{|\alpha |}\int _{\Omega }\varphi \,D^{\alpha \!}u\,dx,}$

where ${\displaystyle \alpha }$ is a multi-index of order ${\displaystyle |\alpha |=k}$ and we are using the notation:

${\displaystyle D^{\alpha \!}f={\frac {\partial ^{|\alpha |}\!f}{\partial x_{1}^{\alpha _{1}}\dots \partial x_{n}^{\alpha _{n}}}}.}$

The left-hand side of this equation still makes sense if we only assume ${\displaystyle u}$ to be locally integrable. If there exists a locally integrable function ${\displaystyle v}$, such that

${\displaystyle \int _{\Omega }u\,D^{\alpha \!}\varphi \;dx=(-1)^{|\alpha |}\int _{\Omega }v\,\varphi \;dx\qquad {\text{for all }}\varphi \in C_{c}^{\infty }(\Omega ),}$

then we call ${\displaystyle v}$ the weak ${\displaystyle \alpha }$-th partial derivative of ${\displaystyle u}$. If there exists a weak ${\displaystyle \alpha }$-th partial derivative of ${\displaystyle u}$, then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space. On the other hand, if ${\displaystyle u\in C^{k}(\Omega )}$, then the classical and the weak derivative coincide. Thus, if ${\displaystyle v}$ is a weak ${\displaystyle \alpha }$-th partial derivative of ${\displaystyle u}$, we may denote it by ${\displaystyle D^{\alpha }u:=v}$.

For example, the function

${\displaystyle u(x)={\begin{cases}1+x&-1

is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function

${\displaystyle v(x)={\begin{cases}1&-1

satisfies the definition for being the weak derivative of ${\displaystyle u(x),}$ which then qualifies as being in the Sobolev space ${\displaystyle W^{1,p}}$ (for any allowed ${\displaystyle p}$, see definition below).

The Sobolev spaces ${\displaystyle W^{k,p}(\Omega )}$ combine the concepts of weak differentiability and Lebesgue norms.

Sobolev spaces with integer k

One-dimensional case

In the one-dimensional case the Sobolev space ${\displaystyle W^{k,p}(\mathbb {R} )}$ for ${\displaystyle 1\leq p\leq \infty }$ is defined as the subset of functions ${\displaystyle f}$ in ${\displaystyle L^{p}(\mathbb {R} )}$ such that ${\displaystyle f}$ and its weak derivatives up to order ${\displaystyle k}$ have a finite Lp norm. As mentioned above, some care must be taken to define derivatives in the proper sense. In the one-dimensional problem it is enough to assume that the ${\displaystyle (k{-}1)}$-th derivative ${\displaystyle f^{(k-1)}}$ is differentiable almost everywhere and is equal almost everywhere to the Lebesgue integral of its derivative (this excludes irrelevant examples such as Cantor's function).

With this definition, the Sobolev spaces admit a natural norm,

${\displaystyle \|f\|_{k,p}=\left(\sum _{i=0}^{k}\left\|f^{(i)}\right\|_{p}^{p}\right)^{\frac {1}{p}}=\left(\sum _{i=0}^{k}\int \left|f^{(i)}(t)\right|^{p}\,dt\right)^{\frac {1}{p}}.}$

One can extend this to the case ${\displaystyle p=\infty }$, with the norm then defined using the essential supremum by

${\displaystyle \|f\|_{k,\infty }=\max _{i=0,\ldots ,k}\left\|f^{(i)}\right\|_{\infty }=\max _{i=0,\ldots ,k}\left({\text{ess}}\,\sup _{t}\left|f^{(i)}(t)\right|\right).}$

Equipped with the norm ${\displaystyle \|\cdot \|_{k,p},W^{k,p}}$ becomes a Banach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

${\displaystyle \left\|f^{(k)}\right\|_{p}+\|f\|_{p}}$

is equivalent to the norm above (i.e. the induced topologies of the norms are the same).

The case p = 2

Sobolev spaces with p = 2 are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case, since the space is a Hilbert space:

${\displaystyle H^{k}=W^{k,2}.}$

The space ${\displaystyle H^{k}}$ can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,

${\displaystyle H^{k}(\mathbb {T} )=\left\{f\in L^{2}(\mathbb {T} ):\sum _{n=-\infty }^{\infty }\left(1+n^{2}+n^{4}+\dots +n^{2k}\right)\left|{\widehat {f}}(n)\right|^{2}<\infty \right\}}$

where ${\displaystyle {\widehat {f}}}$ is the Fourier series of ${\displaystyle f,}$ and ${\displaystyle \mathbb {T} }$ denotes the 1-torus. As above, one can use the equivalent norm

${\displaystyle \|f\|_{k,2}^{2}=\sum _{n=-\infty }^{\infty }\left(1+|n|^{2}\right)^{k}\left|{\widehat {f}}(n)\right|^{2}.}$

Both representations follow easily from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in.

Furthermore, the space ${\displaystyle H^{k}}$ admits an inner product, like the space ${\displaystyle H^{0}=L^{2}.}$ In fact, the ${\displaystyle H^{k}}$ inner product is defined in terms of the ${\displaystyle L^{2}}$ inner product:

${\displaystyle \langle u,v\rangle _{H^{k}}=\sum _{i=0}^{k}\left\langle D^{i}u,D^{i}v\right\rangle _{L^{2}}.}$

The space ${\displaystyle H^{k}}$ becomes a Hilbert space with this inner product.

Other examples

In one dimension, some other Sobolev spaces permit a simpler description. For example, ${\displaystyle W^{1,1}(0,1)}$ is the space of absolutely continuous functions on (0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while ${\displaystyle W^{1,\infty }(I)}$ is the space of Lipschitz functions on I, for every interval I. However, these properties are lost or not as simple for functions of more than one variable.

All spaces ${\displaystyle W^{k,\infty }}$ are (normed) algebras, i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for ${\displaystyle p<\infty .}$ (E.g., functions behaving like |x|−1/3 at the origin are in ${\displaystyle L^{2},}$ but the product of two such functions is not in ${\displaystyle L^{2}}$).

Multidimensional case

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that ${\displaystyle f^{(k-1)}}$ be the integral of ${\displaystyle f^{(k)}}$ does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.

A formal definition now follows. Let ${\displaystyle k\in \mathbb {N} ,1\leqslant p\leqslant \infty .}$ The Sobolev space ${\displaystyle W^{k,p}(\Omega )}$ is defined to be the set of all functions ${\displaystyle f}$ on ${\displaystyle \Omega }$ such that for every multi-index ${\displaystyle \alpha }$ with ${\displaystyle |\alpha |\leqslant k,}$ the mixed partial derivative

${\displaystyle f^{(\alpha )}={\frac {\partial ^{|\alpha |\!}f}{\partial x_{1}^{\alpha _{1}}\dots \partial x_{n}^{\alpha _{n}}}}}$

exists in the weak sense and is in ${\displaystyle L^{p}(\Omega ),}$ i.e.

${\displaystyle \left\|f^{(\alpha )}\right\|_{L^{p}}<\infty .}$

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

${\displaystyle W^{k,p}(\Omega )=\left\{u\in L^{p}(\Omega ):D^{\alpha }u\in L^{p}(\Omega )\,\,\forall |\alpha |\leqslant k\right\}.}$

The natural number ${\displaystyle k}$ is called the order of the Sobolev space ${\displaystyle W^{k,p}(\Omega ).}$

There are several choices for a norm for ${\displaystyle W^{k,p}(\Omega ).}$ The following two are common and are equivalent in the sense of equivalence of norms:

${\displaystyle \|u\|_{W^{k,p}(\Omega )}:={\begin{cases}\left(\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{p}(\Omega )}^{p}\right)^{\frac {1}{p}}&1\leqslant p<\infty$ ;\\\max _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{\infty }(\Omega )}&p=\infty ;\end{cases}}}

and

${\displaystyle \|u\|'_{W^{k,p}(\Omega )}:={\begin{cases}\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{p}(\Omega )}&1\leqslant p<\infty$ ;\\\sum _{|\alpha |\leqslant k}\left\|D^{\alpha }u\right\|_{L^{\infty }(\Omega )}&p=\infty .\end{cases}}}

With respect to either of these norms, ${\displaystyle W^{k,p}(\Omega )}$ is a Banach space. For ${\displaystyle p<\infty ,W^{k,p}(\Omega )}$ is also a separable space. It is conventional to denote ${\displaystyle W^{k,2}(\Omega )}$ by ${\displaystyle H^{k}(\Omega )}$ for it is a Hilbert space with the norm ${\displaystyle \|\cdot \|_{W^{k,2}(\Omega )}}$. [1]

Approximation by smooth functions

It is rather hard to work with Sobolev spaces relying only on their definition. It is therefore interesting to know that by theorem of Meyers and Serrin a function ${\displaystyle u\in W^{k,p}(\Omega )}$ can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If ${\displaystyle p}$ is finite and ${\displaystyle \Omega }$ is open, then there exists for any ${\displaystyle u\in W^{k,p}(\Omega )}$ an approximating sequence of functions ${\displaystyle u_{m}\in C^{\infty }(\Omega )}$ such that:

${\displaystyle \left\|u_{m}-u\right\|_{W^{k,p}(\Omega )}\to 0.}$

If ${\displaystyle \Omega }$ has Lipschitz boundary, we may even assume that the ${\displaystyle u_{m}}$ are the restriction of smooth functions with compact support on all of ${\displaystyle \mathbb {R} ^{n}.}$ [2]

Examples

In higher dimensions, it is no longer true that, for example, ${\displaystyle W^{1,1}}$ contains only continuous functions. For example, ${\displaystyle |x|^{-1}\in W^{1,1}(\mathbb {B} ^{3})}$ where ${\displaystyle \mathbb {B} ^{3}}$ is the unit ball in three dimensions. For k > n/p the space ${\displaystyle W^{k,p}(\Omega )}$ will contain only continuous functions, but for which k this is already true depends both on p and on the dimension. For example, as can be easily checked using spherical polar coordinates for the function ${\displaystyle f:\mathbb {B} ^{n}\to \mathbb {R} \cup \{\infty \}}$ defined on the n-dimensional ball we have:

${\displaystyle f(x)=|x|^{-\alpha }\in W^{k,p}(\mathbb {B} ^{n})\Longleftrightarrow \alpha <{\tfrac {n}{p}}-k.}$

Intuitively, the blow-up of f at 0 "counts for less" when n is large since the unit ball has "more outside and less inside" in higher dimensions.

Absolutely continuous on lines (ACL) characterization of Sobolev functions

Let ${\displaystyle 1\leqslant p\leqslant \infty .}$ If a function is in ${\displaystyle W^{1,p}(\Omega ),}$ then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in ${\displaystyle \mathbb {R} ^{n}}$ is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in ${\displaystyle L^{p}(\Omega ).}$ Conversely, if the restriction of ${\displaystyle f}$ to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient ${\displaystyle \nabla f}$ exists almost everywhere, and ${\displaystyle f}$ is in ${\displaystyle W^{1,p}(\Omega )}$ provided ${\displaystyle f,|\nabla f|\in L^{p}(\Omega ).}$ In particular, in this case the weak partial derivatives of ${\displaystyle f}$ and pointwise partial derivatives of ${\displaystyle f}$ agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see ( Maz'ya 1985 , §1.1.3).

A stronger result holds when ${\displaystyle p>n.}$ A function in ${\displaystyle W^{1,p}(\Omega )}$ is, after modifying on a set of measure zero, Hölder continuous of exponent ${\displaystyle \gamma =1-{\tfrac {n}{p}},}$ by Morrey's inequality. In particular, if ${\displaystyle p=\infty ,}$ then the function is Lipschitz continuous.

Functions vanishing at the boundary

The Sobolev space ${\displaystyle W^{1,2}(\Omega )}$ is also denoted by ${\displaystyle H^{1}\!(\Omega ).}$ It is a Hilbert space, with an important subspace ${\displaystyle H_{0}^{1}\!(\Omega )}$ defined to be the closure of the infinitely differentiable functions compactly supported in ${\displaystyle \Omega }$ in ${\displaystyle H^{1}\!(\Omega ).}$ The Sobolev norm defined above reduces here to

${\displaystyle \|f\|_{H^{1}}=\left(\int _{\Omega }\!|f|^{2}\!+\!|\nabla \!f|^{2}\right)^{\!{\frac {1}{2}}}.}$

When ${\displaystyle \Omega }$ has a regular boundary, ${\displaystyle H_{0}^{1}\!(\Omega )}$ can be described as the space of functions in ${\displaystyle H^{1}\!(\Omega )}$ that vanish at the boundary, in the sense of traces (see below). When ${\displaystyle n=1,}$ if ${\displaystyle \Omega =(a,b)}$ is a bounded interval, then ${\displaystyle H_{0}^{1}(a,b)}$ consists of continuous functions on ${\displaystyle [a,b]}$ of the form

${\displaystyle f(x)=\int _{a}^{x}f'(t)\,\mathrm {d} t,\qquad x\in [a,b]}$

where the generalized derivative ${\displaystyle f'}$ is in ${\displaystyle L^{2}(a,b)}$ and has 0 integral, so that ${\displaystyle f(b)=f(a)=0.}$

When ${\displaystyle \Omega }$ is bounded, the Poincaré inequality states that there is a constant ${\displaystyle C=C(\Omega )}$ such that:

${\displaystyle \int _{\Omega }|f|^{2}\leqslant C^{2}\int _{\Omega }|\nabla f|^{2},\qquad f\in H_{0}^{1}(\Omega ).}$

When ${\displaystyle \Omega }$ is bounded, the injection from ${\displaystyle H_{0}^{1}\!(\Omega )}$ to ${\displaystyle L^{2}\!(\Omega ),}$ is compact. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an orthonormal basis of ${\displaystyle L^{2}(\Omega )}$ consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).

Traces

Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If ${\displaystyle u\in C(\Omega )}$, those boundary values are described by the restriction ${\displaystyle u|_{\partial \Omega }}$. However, it is not clear how to describe values at the boundary for ${\displaystyle u\in W^{k,p}(\Omega )}$, as the n-dimensional measure of the boundary is zero. The following theorem [2] resolves the problem:

Trace Theorem. Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator ${\displaystyle T:W^{1,p}(\Omega )\to L^{p}(\partial \Omega )}$ such that
{\displaystyle {\begin{aligned}Tu&=u|_{\partial \Omega }&&u\in W^{1,p}(\Omega )\cap C({\overline {\Omega }})\\\|Tu\|_{L^{p}(\partial \Omega )}&\leqslant c(p,\Omega )\|u\|_{W^{1,p}(\Omega )}&&u\in W^{1,p}(\Omega ).\end{aligned}}}

Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space ${\displaystyle W^{1,p}(\Omega )}$ for well-behaved Ω. Note that the trace operator T is in general not surjective, but for 1 < p < ∞ it maps continuously onto the Sobolev-Slobodeckij space ${\displaystyle W^{1-{\frac {1}{p}},p}(\partial \Omega ).}$

Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality

${\displaystyle W_{0}^{1,p}(\Omega )=\left\{u\in W^{1,p}(\Omega ):Tu=0\right\},}$

where

${\displaystyle W_{0}^{1,p}(\Omega ):=\left\{u\in W^{1,p}(\Omega ):\exists \{u_{m}\}_{m=1}^{\infty }\subset C_{c}^{\infty }(\Omega ),\ {\text{such that}}\ u_{m}\to u\ {\textrm {in}}\ W^{1,p}(\Omega )\right\}.}$

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in ${\displaystyle W^{1,p}(\Omega )}$ can be approximated by smooth functions with compact support.

Sobolev spaces with non-integer k

Bessel potential spaces

For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers [3] [4] ) that the space ${\displaystyle W^{k,p}(\mathbb {R} ^{n})}$ can equivalently be defined as

${\displaystyle W^{k,p}(\mathbb {R} ^{n})=H^{k,p}(\mathbb {R} ^{n}):=\left\{f\in L^{p}(\mathbb {R} ^{n}):{\mathcal {F}}^{-1}\left[(1+|\xi |^{2})^{\frac {k}{2}}{\mathcal {F}}f\right]\in L^{p}(\mathbb {R} ^{n})\right\},}$

with the norm

${\displaystyle \|f\|_{H^{k,p}(\mathbb {R} ^{n})}:=\left\|{\mathcal {F}}^{-1}\left[\left(1+|\xi |^{2}\right)^{\frac {k}{2}}{\mathcal {F}}f\right]\right\|_{L^{p}(\mathbb {R} ^{n})}.}$

This motivates Sobolev spaces with non-integer order since in the above definition we can replace k by any real number s. The resulting spaces

${\displaystyle H^{s,p}(\mathbb {R} ^{n}):=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{n}):{\mathcal {F}}^{-1}\left[\left(1+|\xi |^{2}\right)^{\frac {s}{2}}{\mathcal {F}}f\right]\in L^{p}(\mathbb {R} ^{n})\right\}}$

are called Bessel potential spaces [5] (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case p = 2.

For ${\displaystyle s\geq 0,H^{s,p}(\Omega )}$ is the set of restrictions of functions from ${\displaystyle H^{s,p}(\mathbb {R} ^{n})}$ to Ω equipped with the norm

${\displaystyle \|f\|_{H^{s,p}(\Omega )}:=\inf \left\{\|g\|_{H^{s,p}(\mathbb {R} ^{n})}:g\in H^{s,p}(\mathbb {R} ^{n}),g|_{\Omega }=f\right\}}$.

Again, Hs,p(Ω) is a Banach space and in the case p = 2 a Hilbert space.

Using extension theorems for Sobolev spaces, it can be shown that also Wk,p(Ω) = Hk,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform Ck-boundary, k a natural number and 1 < p < ∞. By the embeddings

${\displaystyle H^{k+1,p}(\mathbb {R} ^{n})\hookrightarrow H^{s',p}(\mathbb {R} ^{n})\hookrightarrow H^{s,p}(\mathbb {R} ^{n})\hookrightarrow H^{k,p}(\mathbb {R} ^{n}),\quad k\leqslant s\leqslant s'\leqslant k+1}$

the Bessel potential spaces ${\displaystyle H^{s,p}(\mathbb {R} ^{n})}$ form a continuous scale between the Sobolev spaces ${\displaystyle W^{k,p}(\mathbb {R} ^{n}).}$ From an abstract point of view, the Bessel potential spaces occur as complex interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms it holds that

${\displaystyle \left[W^{k,p}(\mathbb {R} ^{n}),W^{k+1,p}(\mathbb {R} ^{n})\right]_{\theta }=H^{s,p}(\mathbb {R} ^{n}),}$

where:

${\displaystyle 1\leqslant p\leqslant \infty ,\ 0<\theta <1,\ s=(1-\theta )k+\theta (k+1)=k+\theta .}$

Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting. [6] For ${\displaystyle 1\leqslant p<\infty ,\theta \in (0,1)}$ and ${\displaystyle f\in L^{p}(\Omega ),}$ the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by

${\displaystyle [f]_{\theta ,p,\Omega }:=\left(\int _{\Omega }\int _{\Omega }{\frac {|f(x)-f(y)|^{p}}{|x-y|^{\theta p+n}}}\;dx\;dy\right)^{\frac {1}{p}}.}$

Let s > 0 be not an integer and set ${\displaystyle \theta =s-\lfloor s\rfloor \in (0,1)}$. Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space [7] ${\displaystyle W^{s,p}(\Omega )}$ is defined as

${\displaystyle W^{s,p}(\Omega ):=\left\{f\in W^{\lfloor s\rfloor ,p}(\Omega ):\sup _{|\alpha |=\lfloor s\rfloor }[D^{\alpha }f]_{\theta ,p,\Omega }<\infty \right\}.}$

It is a Banach space for the norm

${\displaystyle \|f\|_{W^{s,p}(\Omega )}:=\|f\|_{W^{\lfloor s\rfloor ,p}(\Omega )}+\sup _{|\alpha |=\lfloor s\rfloor }[D^{\alpha }f]_{\theta ,p,\Omega }.}$

If ${\displaystyle \Omega }$ is suitably regular in the sense that there exist certain extension operators, then also the Sobolev–Slobodeckij spaces form a scale of Banach spaces, i.e. one has the continuous injections or embeddings

${\displaystyle W^{k+1,p}(\Omega )\hookrightarrow W^{s',p}(\Omega )\hookrightarrow W^{s,p}(\Omega )\hookrightarrow W^{k,p}(\Omega ),\quad k\leqslant s\leqslant s'\leqslant k+1.}$

There are examples of irregular Ω such that ${\displaystyle W^{1,p}(\Omega )}$ is not even a vector subspace of ${\displaystyle W^{s,p}(\Omega )}$ for 0 < s < 1 (see Example 9.1 of [8] )

From an abstract point of view, the spaces ${\displaystyle W^{s,p}(\Omega )}$ coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:

${\displaystyle W^{s,p}(\Omega )=\left(W^{k,p}(\Omega ),W^{k+1,p}(\Omega )\right)_{\theta ,p},\quad k\in \mathbb {N} ,s\in (k,k+1),\theta =s-\lfloor s\rfloor }$.

Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases of Besov spaces. [4]

Extension operators

If ${\displaystyle \Omega }$ is a domain whose boundary is not too poorly behaved (e.g., if its boundary is a manifold, or satisfies the more permissive "cone condition") then there is an operator A mapping functions of ${\displaystyle \Omega }$ to functions of ${\displaystyle \mathbb {R} ^{n}}$ such that:

1. Au(x) = u(x) for almost every x in ${\displaystyle \Omega }$ and
2. ${\displaystyle A:W^{k,p}(\Omega )\to W^{k,p}(\mathbb {R} ^{n})}$ is continuous for any 1 ≤ p ≤ ∞ and integer k.

We will call such an operator A an extension operator for ${\displaystyle \Omega .}$

Case of p = 2

Extension operators are the most natural way to define ${\displaystyle H^{s}(\Omega )}$ for non-integer s (we cannot work directly on ${\displaystyle \Omega }$ since taking Fourier transform is a global operation). We define ${\displaystyle H^{s}(\Omega )}$ by saying that ${\displaystyle u\in H^{s}(\Omega )}$ if and only if ${\displaystyle Au\in H^{s}(\mathbb {R} ^{n}).}$ Equivalently, complex interpolation yields the same ${\displaystyle H^{s}(\Omega )}$ spaces so long as ${\displaystyle \Omega }$ has an extension operator. If ${\displaystyle \Omega }$ does not have an extension operator, complex interpolation is the only way to obtain the ${\displaystyle H^{s}(\Omega )}$ spaces.

As a result, the interpolation inequality still holds.

Extension by zero

Like above, we define ${\displaystyle H_{0}^{s}(\Omega )}$ to be the closure in ${\displaystyle H^{s}(\Omega )}$ of the space ${\displaystyle C_{c}^{\infty }(\Omega )}$ of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following

Theorem. Let ${\displaystyle \Omega }$ be uniformly Cm regular, ms and let P be the linear map sending u in ${\displaystyle H^{s}(\Omega )}$ to
${\displaystyle \left.\left(u,{\frac {du}{dn}},\dots ,{\frac {d^{k}u}{dn^{k}}}\right)\right|_{G}}$
where d/dn is the derivative normal to G, and k is the largest integer less than s. Then ${\displaystyle H_{0}^{s}}$ is precisely the kernel of P.

If ${\displaystyle u\in H_{0}^{s}(\Omega )}$ we may define its extension by zero${\displaystyle {\tilde {u}}\in L^{2}(\mathbb {R} ^{n})}$ in the natural way, namely

${\displaystyle {\tilde {u}}(x)={\begin{cases}u(x)&x\in \Omega \\0&{\text{else}}\end{cases}}}$
Theorem. Let ${\displaystyle s>{\tfrac {1}{2}}.}$ The map ${\displaystyle u\mapsto {\tilde {u}}}$ is continuous into ${\displaystyle H^{s}(\mathbb {R} ^{n})}$ if and only if s is not of the form ${\displaystyle n+{\tfrac {1}{2}}}$ for n an integer.

For f  Lp(Ω) its extension by zero,

${\displaystyle Ef:={\begin{cases}f&{\textrm {on}}\ \Omega ,\\0&{\textrm {otherwise}}\end{cases}}}$

is an element of ${\displaystyle L^{p}(\mathbb {R} ^{n}).}$ Furthermore,

${\displaystyle \|Ef\|_{L^{p}(\mathbb {R} ^{n})}=\|f\|_{L^{p}(\Omega )}.}$

In the case of the Sobolev space W1,p(Ω) for 1  p  ∞, extending a function u by zero will not necessarily yield an element of ${\displaystyle W^{1,p}(\mathbb {R} ^{n}).}$ But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C1), then for any bounded open set O such that Ω⊂⊂O (i.e. Ω is compactly contained in O), there exists a bounded linear operator [2]

${\displaystyle E:W^{1,p}(\Omega )\to W^{1,p}(\mathbb {R} ^{n}),}$

such that for each ${\displaystyle u\in W^{1,p}(\Omega ):Eu=u}$ a.e. on Ω, Eu has compact support within O, and there exists a constant C depending only on p, Ω, O and the dimension n, such that

${\displaystyle \|Eu\|_{W^{1,p}(\mathbb {R} ^{n})}\leqslant C\|u\|_{W^{1,p}(\Omega )}.}$

We call Eu an extension of u to ${\displaystyle \mathbb {R} ^{n}.}$

Sobolev embeddings

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives (i.e. large k) result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.

Write ${\displaystyle W^{k,p}}$ for the Sobolev space of some compact Riemannian manifold of dimension n. Here k can be any real number, and 1  p  ∞. (For p = ∞ the Sobolev space ${\displaystyle W^{k,\infty }}$ is defined to be the Hölder space Cn where k = n + α and 0 < α  1.) The Sobolev embedding theorem states that if ${\displaystyle k\geqslant m}$ and ${\displaystyle k-{\tfrac {n}{p}}\geqslant m-{\tfrac {n}{q}}}$ then

${\displaystyle W^{k,p}\subseteq W^{m,q}}$

and the embedding is continuous. Moreover, if ${\displaystyle k>m}$ and ${\displaystyle k-{\tfrac {n}{p}}>m-{\tfrac {n}{q}}}$ then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich-Kondrachov theorem). Functions in ${\displaystyle W^{m,\infty }}$ have all derivatives of order less than m continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.

There are similar variations of the embedding theorem for non-compact manifolds such as ${\displaystyle \mathbb {R} ^{n}}$( Stein 1970 ). Sobolev embeddings on ${\displaystyle \mathbb {R} ^{n}}$ that are not compact often have a related, but weaker, property of cocompactness.

Notes

1. Evans 1998 , Chapter 5.2
3. Bessel potential spaces with variable integrability have been independently introduced by Almeida & Samko (A. Almeida and S. Samko, "Characterization of Riesz and Bessel potentials on variable Lebesgue spaces", J. Function Spaces Appl. 4 (2006), no. 2, 113–144) and Gurka, Harjulehto & Nekvinda (P. Gurka, P. Harjulehto and A. Nekvinda: "Bessel potential spaces with variable exponent", Math. Inequal. Appl. 10 (2007), no. 3, 661–676).
4. In the literature, fractional Sobolev-type spaces are also called Aronszajn spaces, Gagliardo spaces or Slobodeckij spaces, after the names of the mathematicians who introduced them in the 1950s: N. Aronszajn ("Boundary values of functions with finite Dirichlet integral", Techn. Report of Univ. of Kansas 14 (1955), 77–94), E. Gagliardo ("Proprietà di alcune classi di funzioni in più variabili", Ricerche Mat. 7 (1958), 102–137), and L. N. Slobodeckij ("Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations", Leningrad. Gos. Ped. Inst. Učep. Zap. 197 (1958), 54–112).
5. "Hitchhikerʼs guide to the fractional Sobolev spaces". Bulletin des Sciences Mathématiques. 136 (5): 521–573. 2012-07-01. doi:10.1016/j.bulsci.2011.12.004. ISSN   0007-4497.

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In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.

In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function, but can be every intersection of the graph itself with a hyperplane parallel to a fixed x-axis and to the y-axis.

In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded.

In mathematical analysis, Trudinger's theorem or the Trudinger inequality is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger.

In complex analysis, functional analysis and operator theory, a Bergman space is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions in D for which the p-norm is finite:

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α>0, such that

In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent. Friedrichs's inequality is a general case of the Poincaré–Wirtinger inequality which deals with the case k = 1.

In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.

In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on.

In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions, where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions.

In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function by a polynomial of order at most in terms of derivatives of of order . Both the error of the approximation and the derivatives of are measured by norms on a bounded domain in . This is similar to classical numerical analysis, where, for example, the error of linear interpolation can be bounded using the second derivative of . However, the Bramble–Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of are measured by more general norms involving averages, not just the maximum norm.

In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where is allowed to range over . It is written as

In mathematics, the Besov space is a complete quasinormed space which is a Banach space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.

In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.

In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or semi-inner product in the sense of Lumer. which is an inner product not required to be conjugate symmetric. It was formulated by Günter Lumer, for the purpose of extending Hilbert space type arguments to Banach spaces in functional analysis. Fundamental properties were later explored by Giles.

In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.

In mathematics, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya. The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions. There is an analogous inequality for functions of three real variables, but the exponents are slightly different; much of the difficulty in establishing existence and uniqueness of solutions to the three-dimensional Navier–Stokes equations stems from these different exponents. Ladyzhenskaya's inequality is one member of a broad class of inequalities known as interpolation inequalities.

In mathematics the symmetrization methods are algorithms of transforming a set to a ball with equal volume and centered at the origin. B is called the symmetrized version of A, usually denoted . These algorithms show up in solving the classical isoperimetric inequality problem, which asks: Given all two-dimensional shapes of a given area, which of them has the minimal perimeter. The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method. From this many other isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball. Another problem is that the Newtonian capacity of a set A is minimized by and this was proved by Polya and G. Szego (1951) using circular symmetrization.

In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-value function is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers