Sobolev space
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
, 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. Sobolev spaces are named after the Russian mathematician
Sergei Sobolev Prof Sergei Lvovich Sobolev (russian: Серге́й Льво́вич Со́болев) HFRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduced ...
. 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
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s understood in the classical sense.


Motivation

In this section and throughout the article \Omega is an open subset of \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 C^1 — see
Differentiability classes In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
). Differentiable functions are important in many areas, and in particular for
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s. In the twentieth century, however, it was observed that the space C^1 (or 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 L^2-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions. The
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
formula yields that for every u\in C^k(\Omega), where k is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, and for all infinitely differentiable functions with compact support \varphi \in C_c^(\Omega), : \int_\Omega u\,D^\varphi\,dx=(-1)^\int_\Omega \varphi\, D^ u\,dx, where \alpha is a multi-index of order , \alpha, =k and we are using the notation: :D^f = \frac. The left-hand side of this equation still makes sense if we only assume u to be locally integrable. If there exists a locally integrable function v, such that : \int_\Omega u\,D^\varphi\;dx=(-1)^\int_\Omega v\,\varphi \;dx \qquad\text\varphi\in C_c^\infty(\Omega), then we call v the weak \alpha-th partial derivative of u. If there exists a weak \alpha-th partial derivative of 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 u\in C^k(\Omega), then the classical and the weak derivative coincide. Thus, if v is a weak \alpha-th partial derivative of u, we may denote it by D^\alpha u := v. For example, the function :u(x)=\begin 1+x & -1 is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function :v(x)=\begin 1 & -1 satisfies the definition for being the weak derivative of u(x), which then qualifies as being in the Sobolev space W^ (for any allowed p, see definition below). The Sobolev spaces W^(\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 W^(\R) for 1 \le p \le \infty is defined as the subset of functions f in L^p(\R) such that f and its weak derivatives up to order k have a finite 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 (k1)-th derivative f^ 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, :\, f\, _ = \left (\sum_^k \left \, f^ \right \, _p^p \right)^ = \left (\sum_^k \int \left , f^(t) \right , ^p\,dt \right )^. One can extend this to the case p = \infty , with the norm then defined using the essential supremum by :\, f\, _ = \max_ \left \, f^ \right \, _\infty = \max_ \left(\text\, \sup_t \left , f^(t) \right , \right). Equipped with the norm \, \cdot\, _, W^ 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 :\left \, f^ \right \, _p + \, f\, _p is equivalent to the norm above (i.e. the induced topologies of the norms are the same).


The case

Sobolev spaces with 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: :H^k = W^. The space H^k can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely, :H^k(\mathbb) = \Big \, where \widehat is the Fourier series of f, and \mathbb denotes the 1-torus. As above, one can use the equivalent norm :\, f\, ^2_=\sum_^\infty \left (1 + , n, ^ \right )^k \left , \widehat(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 H^k admits an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, like the space H^0 = L^2. In fact, the H^k inner product is defined in terms of the L^2 inner product: :\langle u,v\rangle_ = \sum_^k \left \langle D^i u,D^i v \right \rangle_. The space 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, W^(0,1) is the space of absolutely continuous functions on (or rather, equivalence classes of functions that are equal almost everywhere to such), while W^(I) is the space of bounded Lipschitz functions on , for every interval . However, these properties are lost or not as simple for functions of more than one variable. All spaces W^ are (normed)
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
s, i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for p<\infty. (E.g., functions behaving like , ''x'', −1/3 at the origin are in L^2, but the product of two such functions is not in L^2).


Multidimensional case

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that f^ be the integral of f^ does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory. A formal definition now follows. Let k \in \N, 1 \leqslant p \leqslant \infty. The Sobolev space W^(\Omega) is defined to be the set of all functions f on \Omega such that for every multi-index \alpha with , \alpha, \leqslant k, the mixed partial derivative :f^ = \frac exists in the
weak Weak may refer to: Songs * "Weak" (AJR song), 2016 * "Weak" (Melanie C song), 2011 * "Weak" (SWV song), 1993 * "Weak" (Skunk Anansie song), 1995 * "Weak", a song by Seether from '' Seether: 2002-2013'' Television episodes * "Weak" (''Fear t ...
sense and is in L^p(\Omega), i.e. :\left \, f^ \right \, _ < \infty. That is, the Sobolev space W^(\Omega) is defined as :W^(\Omega) = \left \. The
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
k is called the order of the Sobolev space W^(\Omega). There are several choices for a norm for W^(\Omega). The following two are common and are equivalent in the sense of equivalence of norms: :\, u \, _ := \begin \left( \sum_ \left \, D^u \right \, _^p \right)^ & 1 \leqslant p < \infty; \\ \max_ \left \, D^u \right \, _ & p = \infty; \end and :\, u \, '_ := \begin \sum_ \left \, D^u \right \, _ & 1 \leqslant p < \infty; \\ \sum_ \left \, D^u \right \, _ & p = \infty. \end With respect to either of these norms, W^(\Omega) is a Banach space. For p<\infty, W^(\Omega) is also a separable space. It is conventional to denote W^(\Omega) by H^k(\Omega) for it is a Hilbert space with the norm \, \cdot \, _.


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 the
Meyers–Serrin theorem In functional analysis the Meyers–Serrin theorem, named after James Serrin James Burton Serrin (1 November 1926, Chicago, Illinois – 23 August 2012, Minneapolis, Minnesota) was an American mathematician, and a professor at University of Mi ...
a function u \in W^(\Omega) can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If p is finite and \Omega is open, then there exists for any u \in W^(\Omega) an approximating sequence of functions u_m \in C^(\Omega) such that: : \left \, u_m - u \right \, _ \to 0. If \Omega has
Lipschitz boundary In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. Th ...
, we may even assume that the u_m are the restriction of smooth functions with compact support on all of \R^n.


Examples

In higher dimensions, it is no longer true that, for example, W^ contains only continuous functions. For example, , x, ^ \in W^(\mathbb^3) where \mathbb^3 is the unit ball in three dimensions. For ''k'' > ''n''/''p'' the space W^(\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 f : \mathbb^n \to \R \cup \ defined on the ''n''-dimensional ball we have: :f(x) = , x , ^ \in W^(\mathbb^n) \Longleftrightarrow \alpha < \tfrac - 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 1\leqslant p \leqslant \infty. If a function is in W^(\Omega), then, possibly after modifying the function on a set of measure zero, the restriction to
almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
line parallel to the coordinate directions in \R^n is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in L^p(\Omega). Conversely, if the restriction of f to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient \nabla f exists almost everywhere, and f is in W^(\Omega) provided f, , \nabla f, \in L^p(\Omega). In particular, in this case the weak partial derivatives of f and pointwise partial derivatives of f agree almost everywhere. The ACL characterization of the Sobolev spaces was established by
Otto M. Nikodym Otto Marcin Nikodym (3 August 1887 – 4 May 1974) (also Otton Martin Nikodým) was a Polish mathematician. Education and career Nikodym studied mathematics at the University of Jan Kazimierz (UJK) in Lvov (today's University of Lviv). Imm ...
(
1933 Events January * January 11 – Sir Charles Kingsford Smith makes the first commercial flight between Australia and New Zealand. * January 17 – The United States Congress votes in favour of Philippines independence, against the wis ...
); see . A stronger result holds when p>n. A function in W^(\Omega) is, after modifying on a set of measure zero,
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
of exponent \gamma = 1 - \tfrac, by Morrey's inequality. In particular, if p=\infty and \Omega has Lipschitz boundary, then the function is Lipschitz continuous.


Functions vanishing at the boundary

The Sobolev space W^(\Omega) is also denoted by H^1\!(\Omega). It is a Hilbert space, with an important subspace H^1_0\!(\Omega) defined to be the closure of the infinitely differentiable functions compactly supported in \Omega in H^1\!(\Omega). The Sobolev norm defined above reduces here to :\, f\, _ = \left ( \int_\Omega \! , f, ^2 \!+\! , \nabla\! f, ^2 \right)^. When \Omega has a regular boundary, H^1_0\!(\Omega) can be described as the space of functions in H^1\!(\Omega) that vanish at the boundary, in the sense of traces ( see below). When n=1, if \Omega = (a,b) is a bounded interval, then H^1_0(a,b) consists of continuous functions on ,b/math> of the form :f(x) = \int_a^x f'(t) \, \mathrmt, \qquad x \in , b/math> where the generalized derivative f' is in L^2(a,b) and has 0 integral, so that f(b) = f(a) = 0. When \Omega is bounded, the Poincaré inequality states that there is a constant C= C(\Omega) such that: :\int_\Omega , f, ^2 \leqslant C^2 \int_\Omega , \nabla f, ^2, \qquad f \in H^1_0(\Omega). When \Omega is bounded, the injection from H^1_0\!(\Omega) to L^2\!(\Omega), is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of 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 u\in C(\Omega), those boundary values are described by the restriction u, _. However, it is not clear how to describe values at the boundary for u\in W^(\Omega), as the ''n''-dimensional measure of the boundary is zero. The following theorem resolves the problem: ''Tu'' is called the trace of ''u''. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W^(\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 W^(\partial\Omega). Intuitively, taking the trace costs 1/''p'' of a derivative. The functions ''u'' in ''W''1,p(Ω) with zero trace, i.e. ''Tu'' = 0, can be characterized by the equality : W_0^(\Omega)= \left \, where : W_0^(\Omega):= \left \. In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in W^(\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 one can show (by using Fourier multipliers) that the space W^(\R^n) can equivalently be defined as : W^(\R^n) = H^(\R^n) := \Big \, with the norm :\, f\, _ := \left\, \mathcal^ \Big \xi, ^2\big)^ \mathcalf \Big\right\, _. 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 :H^(\R^n) := \left \ are called Bessel potential spaces (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case ''p'' = 2. For s \geq 0, H^(\Omega) is the set of restrictions of functions from H^(\R^n) to Ω equipped with the norm :\, f\, _ := \inf \left \ . 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 . By the embeddings : H^(\R^n) \hookrightarrow H^(\R^n) \hookrightarrow H^(\R^n) \hookrightarrow H^(\R^n), \quad k \leqslant s \leqslant s' \leqslant k+1 the Bessel potential spaces H^(\R^n) form a continuous scale between the Sobolev spaces W^(\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 : \left W^(\R^n), W^(\R^n) \right \theta = H^(\R^n), where: :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 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 : , f(x) - f(y) , \leq C ...
to the ''Lp''-setting. For 1 \leqslant p < \infty, \theta \in (0, 1) and f \in L^p(\Omega), the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by : :=\left(\int_ \int_ \frac \; dx \; dy \right )^. Let be not an integer and set \theta = s - \lfloor s \rfloor \in (0,1). Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space W^(\Omega) is defined as :W^(\Omega) := \left\. It is a Banach space for the norm :\, f \, _ := \, f\, _ + \sup_ ^\alpha f. If \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 : W^(\Omega) \hookrightarrow W^(\Omega) \hookrightarrow W^(\Omega) \hookrightarrow W^(\Omega), \quad k \leqslant s \leqslant s' \leqslant k+1. There are examples of irregular Ω such that W^(\Omega) is not even a vector subspace of W^(\Omega) for 0 < ''s'' < 1 (see Example 9.1 of ) From an abstract point of view, the spaces W^(\Omega) coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds: : W^(\Omega) = \left (W^(\Omega), W^(\Omega) \right)_ , \quad k \in \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 space In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B^s_(\mathbf) is a complete quasinormed space which is a Banach space when . These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize mor ...
s.


Extension operators

If \Omega is a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
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 \Omega to functions of \R^n such that: # ''Au''(''x'') = ''u''(''x'') for almost every ''x'' in \Omega and # A : W^(\Omega) \to W^(\R^n) is continuous for any 1 ≤ ''p'' ≤ ∞ and integer ''k''. We will call such an operator ''A'' an extension operator for \Omega.


Case of ''p'' = 2

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


Extension by zero

Like above, we define H^s_0(\Omega) to be the closure in H^s(\Omega) of the space C^\infty_c(\Omega) of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following If u\in H^s_0(\Omega) we may define its extension by zero \tilde u \in L^2(\R^n) in the natural way, namely :\tilde u(x)= \begin u(x) & x \in \Omega \\ 0 & \text \end For its extension by zero, :Ef := \begin f & \textrm \ \Omega, \\ 0 & \textrm \end is an element of L^p(\R^n). Furthermore, : \, Ef \, _= \, f \, _. In the case of the Sobolev space ''W''1,p(Ω) for , extending a function ''u'' by zero will not necessarily yield an element of W^(\R^n). But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is ''C''1), 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 : E: W^(\Omega)\to W^(\R^n), such that for each u\in W^(\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 :\, Eu \, _\leqslant C \, u\, _. We call Eu an extension of u to \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 In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the ...
. Write W^ 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 W^ is defined to be the
Hölder space Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a mo ...
''C''''n'',α where ''k'' = ''n'' + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if k \geqslant m and k - \tfrac \geqslant m - \tfrac then :W^\subseteq W^ and the embedding is continuous. Moreover, if k > m and k - \tfrac > m - \tfrac then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich–Kondrachov theorem). Functions in W^ 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 \R^n . Sobolev embeddings on \R^n that are not compact often have a related, but weaker, property of cocompactness.


See also

*
Sobolev mapping In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations, including the ...


Notes


References

* . * . * * * * *. *. *. *. *. *. *; translation of Mat. Sb., 4 (1938) pp. 471–497. *. *. *. *.


External links


Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci (2011). "Hitchhiker's guide to the fractional Sobolev spaces".
{{DEFAULTSORT:Sobolev Space Sobolev spaces Fourier analysis Fractional calculus Function spaces