Sobolev space

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.
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 formula_1 is an open subset of formula_2
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 formula_3 — 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 formula_3 (or formula_5, 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 formula_6-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.
The integration by parts formula yields that for every formula_7, where formula_8 is a natural number, and for all infinitely differentiable functions with compact support formula_9
where formula_11 is a multi-index of order formula_12 and we are using the notation:
The left-hand side of this equation still makes sense if we only assume formula_14 to be locally integrable. If there exists a locally integrable function formula_15, such that
then we call formula_15 the weak formula_11-th partial derivative of formula_14. If there exists a weak formula_11-th partial derivative of formula_14, 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 formula_7, then the classical and the weak derivative coincide. Thus, if formula_15 is a weak formula_11-th partial derivative of formula_14, we may denote it by formula_26.
For example, the function
is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function
satisfies the definition for being the weak derivative of formula_29 which then qualifies as being in the Sobolev space formula_30 (for any allowed formula_31, see definition below).
The Sobolev spaces formula_32 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 formula_33 for formula_34 is defined as the subset of functions formula_35 in formula_36 such that formula_35 and its weak derivatives up to order formula_8 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 formula_39-th derivative formula_40 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,
One can extend this to the case formula_42, with the norm then defined using the essential supremum by
Equipped with the norm formula_44 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
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:
The space formula_47 can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,
where formula_49 is the Fourier series of formula_50 and formula_51 denotes the 1-torus. As above, one can use the equivalent norm
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 formula_47 admits an inner product, like the space formula_54 In fact, the formula_47 inner product is defined in terms of the formula_6 inner product:
The space formula_47 becomes a Hilbert space with this inner product.
Other examples.
In one dimension, some other Sobolev spaces permit a simpler description. For example, formula_59 is the space of absolutely continuous functions on (or rather, equivalence classes of functions that are equal almost everywhere to such), while formula_60 is the space of Lipschitz functions on , for every interval . However, these properties are lost or not as simple for functions of more than one variable.
All spaces formula_61 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 formula_62 (E.g., functions behaving like |"x"|−1/3 at the origin are in formula_63 but the product of two such functions is not in formula_6).
Multidimensional case.
The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that formula_40 be the integral of formula_66 does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.
A formal definition now follows. Let formula_67 The Sobolev space formula_32 is defined to be the set of all functions formula_35 on formula_1 such that for every multi-index formula_11 with formula_72 the mixed partial derivative
exists in the weak sense and is in formula_74 i.e.
That is, the Sobolev space formula_32 is defined as
The natural number formula_8 is called the order of the Sobolev space formula_79
There are several choices for a norm for formula_79 The following two are common and are equivalent in the sense of equivalence of norms:
and
With respect to either of these norms, formula_32 is a Banach space. For formula_84 is also a separable space. It is conventional to denote formula_85 by formula_86 for it is a Hilbert space with the norm formula_87.
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 formula_88 can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If formula_31 is finite and formula_1 is open, then there exists for any formula_88 an approximating sequence of functions formula_92 such that:
If formula_1 has Lipschitz boundary, we may even assume that the formula_95 are the restriction of smooth functions with compact support on all of formula_2
Examples.
In higher dimensions, it is no longer true that, for example, formula_97 contains only continuous functions. For example, formula_98 where formula_99 is the unit ball in three dimensions. For "k" > "n"/"p" the space formula_32 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 formula_101 defined on the "n"-dimensional ball we have:
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 formula_103 If a function is in formula_104 then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in formula_105 is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in formula_106 Conversely, if the restriction of formula_35 to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient formula_108 exists almost everywhere, and formula_35 is in formula_110 provided formula_111 In particular, in this case the weak partial derivatives of formula_35 and pointwise partial derivatives of formula_35 agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see .
A stronger result holds when formula_114 A function in formula_110 is, after modifying on a set of measure zero, Hölder continuous of exponent formula_116 by Morrey's inequality. In particular, if formula_117 then the function is Lipschitz continuous.
Functions vanishing at the boundary.
The Sobolev space formula_118 is also denoted by formula_119 It is a Hilbert space, with an important subspace formula_120 defined to be the closure of the infinitely differentiable functions compactly supported in formula_1 in formula_119 The Sobolev norm defined above reduces here to
When formula_1 has a regular boundary, formula_120 can be described as the space of functions in formula_126 that vanish at the boundary, in the sense of traces (see below). When formula_127 if formula_128 is a bounded interval, then formula_129 consists of continuous functions on formula_130 of the form
where the generalized derivative formula_132 is in formula_133 and has 0 integral, so that formula_134
When formula_1 is bounded, the Poincaré inequality states that there is a constant formula_136 such that:
When formula_1 is bounded, the injection from formula_120 to formula_140 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 formula_141 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 formula_142, those boundary values are described by the restriction formula_143. However, it is not clear how to describe values at the boundary for formula_144, 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 formula_110 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 formula_148
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
where
In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in formula_110 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 formula_152 can equivalently be defined as
with the norm
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
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 formula_156 is the set of restrictions of functions from formula_157 to Ω equipped with the norm
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
the Bessel potential spaces formula_157 form a continuous scale between the Sobolev spaces formula_161 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
where:
Sobolev–Slobodeckij spaces.
Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the "Lp"-setting. For formula_164 and formula_165 the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by
Let be not an integer and set formula_167. Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space formula_168 is defined as
It is a Banach space for the norm
If formula_1 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
There are examples of irregular Ω such that formula_110 is not even a vector subspace of formula_168 for 0 < "s" < 1 (see Example 9.1 of )
From an abstract point of view, the spaces formula_168 coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:
Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases of Besov spaces.
Extension operators.
If formula_1 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 formula_1 to functions of formula_105 such that:
We will call such an operator "A" an extension operator for formula_182
Case of "p" = 2.
Extension operators are the most natural way to define formula_183 for non-integer "s" (we cannot work directly on formula_1 since taking Fourier transform is a global operation). We define formula_183 by saying that formula_186 if and only if formula_187 Equivalently, complex interpolation yields the same formula_183 spaces so long as formula_1 has an extension operator. If formula_1 does not have an extension operator, complex interpolation is the only way to obtain the formula_183 spaces.
As a result, the interpolation inequality still holds.
Extension by zero.
Like above, we define formula_192 to be the closure in formula_183 of the space formula_194 of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following
If formula_199 we may define its extension by zero formula_200 in the natural way, namely
For "f" ∈ "Lp"(Ω) its extension by zero,
is an element of formula_207 Furthermore,
In the case of the Sobolev space "W"1,p(Ω) for 1 ≤ p ≤ ∞, extending a function "u" by zero will not necessarily yield an element of formula_209 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
such that for each formula_211 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
We call "Eu" an extension of "u" to formula_2
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 formula_214 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 formula_61 is defined to be the Hölder space "C""n",α where "k" = "n" + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if formula_216 and formula_217 then
and the embedding is continuous. Moreover, if formula_219 and formula_220 then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich-Kondrachov theorem). Functions in formula_221 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 formula_105 . Sobolev embeddings on formula_105 that are not compact often have a related, but weaker, property of cocompactness.