{"history":[],"article":{"id":"5f8efc48-59c3-4cc9-be4e-13c3d3553981","title":"Sobolev space","slug":"sobolev-space","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **Sobolev space** is a [vector space](https://bliptext.com/articles/vector-space) of functions equipped with a [norm](https://bliptext.com/articles/normed-space) that is a combination of Lp-[norm](https://bliptext.com/articles/normed-space)s of the function together with its derivatives up to a given order.\nThe derivatives are understood in a suitable [weak sense](https://bliptext.com/articles/weak-derivative) to make the space [complete](https://bliptext.com/articles/complete-metric-space), i.e. a [Banach space](https://bliptext.com/articles/banach-space).\nIntuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as [partial differential equations](https://bliptext.com/articles/partial-differential-equation), and equipped with a norm that measures both the size and regularity of a function.\nSobolev spaces are named after the Russian [mathematician](https://bliptext.com/articles/mathematician) [Sergei Sobolev](https://bliptext.com/articles/sergei-lvovich-sobolev).\nTheir importance comes from the fact that [weak solutions](https://bliptext.com/articles/weak-solution) of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of [continuous functions](https://bliptext.com/articles/continuous-function) with the [derivatives](https://bliptext.com/articles/derivative) understood in the classical sense.\n\n# Motivation\nIn this section and throughout the article \\Omega is an [open subset](https://bliptext.com/articles/open-subset) of \\R^n.\nThere are many criteria for smoothness of [mathematical functions](https://bliptext.com/articles/mathematical-function).\nThe most basic criterion may be that of [continuity](https://bliptext.com/articles/continuous-function).\nA stronger notion of smoothness is that of [differentiability](https://bliptext.com/articles/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](https://bliptext.com/articles/differentiability-classes)).\nDifferentiable functions are important in many areas, and in particular for [differential equations](https://bliptext.com/articles/differential-equation).\nIn 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.\nThe Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.\nQuantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms.\nA typical example is measuring the energy of a temperature or velocity distribution by an L^2-norm.\nIt is therefore important to develop a tool for differentiating [Lebesgue space](https://bliptext.com/articles/lp-space) functions.\nThe [integration by parts](https://bliptext.com/articles/integration-by-parts) formula yields that for every, where k is a [natural number](https://bliptext.com/articles/natural-number), and for all infinitely differentiable functions with [compact support](https://bliptext.com/articles/compact-support)\nwhere \\alpha is a [multi-index](https://bliptext.com/articles/multi-index) of order |\\alpha|=k and we are using the notation:\nThe left-hand side of this equation still makes sense if we only assume u to be [locally integrable](https://bliptext.com/articles/locally-integrable).\nIf there exists a locally integrable function v, such that\nthen we call v the [weak \\alpha-th partial derivative](https://bliptext.com/articles/weak-derivative) of u. If there exists a [weak \\alpha-th partial derivative](https://bliptext.com/articles/weak-derivative) of u, then it is uniquely defined [almost everywhere](https://bliptext.com/articles/almost-everywhere), and thus it is uniquely determined as an element of a [Lebesgue space](https://bliptext.com/articles/lp-space).\nOn the other hand, if, then the classical and the weak derivative coincide.\nThus, if v is a weak \\alpha-th partial derivative of u, we may denote it by.\nFor example, the function\nis not continuous at zero, and not differentiable at −1, 0, or 1.\nYet the function\nsatisfies the definition for being the weak derivative of u(x), which then qualifies as being in the Sobolev space W^{1,p} (for any allowed p, see definition below).\nThe Sobolev spaces combine the concepts of weak differentiability and [Lebesgue norms](https://bliptext.com/articles/lp-norm).\n\n# Sobolev spaces with integer k\n\n## One-dimensional case\nIn the one-dimensional case the Sobolev space W^{k,p}(\\R) for is defined as the subset of functions f in [L^p(\\R)](https://bliptext.com/articles/lp-space) such that f and its [weak derivatives](https://bliptext.com/articles/weak-derivative) up to order k have a finite [[Lp norm|\nLp\nnorm]].\nAs mentioned above, some care must be taken to define derivatives in the proper sense.\nIn the one-dimensional problem it is enough to assume that the (k{-}1)-th derivative f^{(k-1)} is differentiable almost everywhere and is equal almost everywhere to the [Lebesgue integral](https://bliptext.com/articles/lebesgue-integration) of its derivative (this excludes irrelevant examples such as [Cantor's function](https://bliptext.com/articles/cantor-function)).\nWith this definition, the Sobolev spaces admit a natural [norm](https://bliptext.com/articles/normed-vector-space),\nOne can extend this to the case p = \\infty, with the norm then defined using the [essential supremum](https://bliptext.com/articles/essential-supremum-and-essential-infimum) by\nEquipped with the norm becomes a [Banach space](https://bliptext.com/articles/banach-space).\nIt turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by\nis equivalent to the norm above (i.e. the [induced topologies](https://bliptext.com/articles/normed-vector-space) of the norms are the same).\n\n### The case \n\np {{=}} 2\nSobolev spaces with\nare especially important because of their connection with [Fourier series](https://bliptext.com/articles/fourier-series) and because they form a [Hilbert space](https://bliptext.com/articles/hilbert-space).\nA special notation has arisen to cover this case, since the space is a Hilbert space:\nThe space H^k can be defined naturally in terms of [Fourier series](https://bliptext.com/articles/fourier-series) whose coefficients decay sufficiently rapidly, namely,\nwhere \\widehat{f} is the Fourier series of f, and \\mathbb{T} denotes the 1-torus.\nAs above, one can use the equivalent norm\nBoth representations follow easily from [Parseval's theorem](https://bliptext.com/articles/parseval-s-theorem) and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in.\nFurthermore, the space H^k admits an [inner product](https://bliptext.com/articles/inner-product-space), like the space H^0 = L^2.\nIn fact, the H^k inner product is defined in terms of the L^2 inner product:\nThe space H^k becomes a Hilbert space with this inner product.\n\n### Other examples\nIn one dimension, some other Sobolev spaces permit a simpler description.\nFor example, is the space of [absolutely continuous functions](https://bliptext.com/articles/absolute-continuity) on\n(0, 1)\n(or rather, equivalence classes of functions that are equal almost everywhere to such), while is the space of bounded [Lipschitz functions](https://bliptext.com/articles/lipschitz-continuity) on I, for every interval I. However, these properties are lost or not as simple for functions of more than one variable.\nAll spaces are (normed) [algebras](https://bliptext.com/articles/algebra-over-a-field), i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for p<\\infty.\n(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).\n\n## Multidimensional case\nThe transition to multiple dimensions brings more difficulties, starting from the very definition.\nThe requirement that f^{(k-1)} be the integral of f^{(k)} does not generalize, and the simplest solution is to consider derivatives in the sense of [distribution theory](https://bliptext.com/articles/distribution-mathematics).\nA formal definition now follows.\nLet The Sobolev space is defined to be the set of all functions f on \\Omega such that for every [multi-index](https://bliptext.com/articles/multi-index) \\alpha with the mixed [partial derivative](https://bliptext.com/articles/partial-derivative)\nexists in the [weak](https://bliptext.com/articles/weak-derivative) sense and is in i.e.\nThat is, the Sobolev space is defined as\nThe [natural number](https://bliptext.com/articles/natural-number) k is called the order of the Sobolev space\nThere are several choices for a norm for The following two are common and are equivalent in the sense of [equivalence of norms](https://bliptext.com/articles/norm-mathematics):\nand\nWith respect to either of these norms, is a Banach space.\nFor is also a [separable space](https://bliptext.com/articles/separable-space).\nIt is conventional to denote by H^k(\\Omega) for it is a [Hilbert space](https://bliptext.com/articles/hilbert-space) with the norm.\n\n### Approximation by smooth functions\nIt is rather hard to work with Sobolev spaces relying only on their definition.\nIt is therefore interesting to know that by the [Meyers–Serrin theorem](https://bliptext.com/articles/meyers-serrin-theorem) a function can be approximated by [smooth functions](https://bliptext.com/articles/smooth-functions).\nThis fact often allows us to translate properties of smooth functions to Sobolev functions.\nIf p is finite and \\Omega is open, then there exists for any an approximating sequence of functions such that:\nIf \\Omega has [Lipschitz boundary](https://bliptext.com/articles/lipschitz-boundary), we may even assume that the u_m are the restriction of smooth functions with compact support on all of \\R^n.\n\n### Examples\nIn higher dimensions, it is no longer true that, for example, W^{1,1} contains only continuous functions.\nFor example, where is the [unit ball](https://bliptext.com/articles/unit-ball) in three dimensions.\nFor k > n/p, the space will contain only continuous functions, but for which k this is already true depends both on p and on the dimension.\nFor example, as can be easily checked using [spherical polar coordinates](https://bliptext.com/articles/spherical-polar-coordinates) for the function defined on the n-dimensional ball we have:\nIntuitively, 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.\n\n### Absolutely continuous on lines (ACL) characterization of Sobolev functions\nLet If a function is in then, possibly after modifying the function on a set of measure zero, the restriction to [almost every](https://bliptext.com/articles/almost-every) line parallel to the coordinate directions in \\R^n is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous); what's more, the classical derivative along the lines that are parallel to the coordinate directions are in Conversely, if the restriction of f to [almost every](https://bliptext.com/articles/almost-every) line parallel to the coordinate directions is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous), then the pointwise gradient \\nabla f exists [almost every](https://bliptext.com/articles/almost-every)where, and f is in provided In particular, in this case the weak partial derivatives of f and pointwise partial derivatives of f agree [almost every](https://bliptext.com/articles/almost-every)where.\nThe ACL characterization of the Sobolev spaces was established by [Otto M. Nikodym](https://bliptext.com/articles/otto-m-nikodym) ([1933](https://bliptext.com/articles/)); see.\nA stronger result holds when p>n.\nA function in is, after modifying on a set of measure zero, [Hölder continuous](https://bliptext.com/articles/h-lder-continuous) of exponent by [Morrey's inequality](https://bliptext.com/articles/sobolev-inequality).\nIn particular, if p=\\infty and \\Omega has Lipschitz boundary, then the function is [Lipschitz continuous](https://bliptext.com/articles/lipschitz-continuous).\n\n### Functions vanishing at the boundary\nThe Sobolev space is also denoted by It is a Hilbert space, with an important subspace defined to be the closure of the infinitely differentiable functions compactly supported in \\Omega in The Sobolev norm defined above reduces here to\nWhen \\Omega has a regular boundary, can be described as the space of functions in that vanish at the boundary, in the sense of traces ([see below](https://bliptext.com/articles/)).\nWhen n=1, if is a bounded interval, then H^1_0(a,b) consists of continuous functions on [a,b] of the form\nwhere the generalized derivative f' is in L^2(a,b) and has 0 integral, so that\nWhen \\Omega is bounded, the [Poincaré inequality](https://bliptext.com/articles/poincar-inequality) states that there is a constant such that:\nWhen \\Omega is bounded, the injection from to is [compact](https://bliptext.com/articles/compact-operator).\nThis fact plays a role in the study of the [Dirichlet problem](https://bliptext.com/articles/dirichlet-problem), and in the fact that there exists an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of L^2(\\Omega) consisting of eigenvectors of the [Laplace operator](https://bliptext.com/articles/laplace-operator) (with [Dirichlet boundary condition](https://bliptext.com/articles/dirichlet-boundary-condition)).\n\n# Traces\nSobolev spaces are often considered when investigating partial differential equations.\nIt is essential to consider boundary values of Sobolev functions.\nIf, those boundary values are described by the restriction However, it is not clear how to describe values at the boundary for as the n-dimensional measure of the boundary is zero.\nThe following theorem resolves the problem:\nTu is called the trace of u.\nRoughly speaking, this theorem extends the restriction operator to the Sobolev space for well-behaved Ω.\nNote that the [trace operator](https://bliptext.com/articles/trace-operator) T is in general not surjective, but for 1 < p < ∞ it maps continuously onto the Sobolev–Slobodeckij space\nIntuitively, taking the trace costs 1/p of a derivative.\nThe functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality\nwhere\nIn other words, for Ω bounded with Lipschitz boundary, trace-zero functions in can be approximated by smooth functions with compact support.\n\n# Sobolev spaces with non-integer k\n\n## Bessel potential spaces\nFor a natural number k and\n1 < p < ∞\none can show (by using [Fourier multipliers](https://bliptext.com/articles/multiplier-fourier-analysis) ) that the space can equivalently be defined as\nwith the norm\nThis motivates Sobolev spaces with non-integer order since in the above definition we can replace k by any real number s.\nThe resulting spaces\nare called Bessel potential spaces (named after [Friedrich Bessel](https://bliptext.com/articles/friedrich-bessel)).\nThey are Banach spaces in general and Hilbert spaces in the special case p = 2.\nFor is the set of restrictions of functions from to Ω equipped with the norm\nAgain, Hs,p(Ω) is a Banach space and in the case p = 2 a Hilbert space.\nUsing 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\n1 < p < ∞\n. By the [embeddings](https://bliptext.com/articles/embedding)\nthe Bessel potential spaces form a continuous scale between the Sobolev spaces From an abstract point of view, the Bessel potential spaces occur as complex [interpolation spaces](https://bliptext.com/articles/interpolation-space) of Sobolev spaces, i.e. in the sense of equivalent norms it holds that\nwhere:\n\n## Sobolev–Slobodeckij spaces\nAnother approach to define fractional order Sobolev spaces arises from the idea to generalize the [Hölder condition](https://bliptext.com/articles/h-lder-condition) to the Lp-setting.\nFor and the **Slobodeckij seminorm** (roughly analogous to the Hölder seminorm) is defined by\nLet\ns > 0\nbe not an integer and set.\nUsing the same idea as for the [Hölder spaces](https://bliptext.com/articles/holder-space), the **Sobolev–Slobodeckij space** is defined as\nIt is a Banach space for the norm\nIf \\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](https://bliptext.com/articles/embedding)\nThere are examples of irregular Ω such that is not even a vector subspace of for 0 < s < 1 (see Example 9.1 of )\nFrom an abstract point of view, the spaces coincide with the real [interpolation spaces](https://bliptext.com/articles/interpolation-space) of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:\nSobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions.\nThey are special cases of [Besov spaces](https://bliptext.com/articles/besov-space).\nThe constant arising in the characterization of the fractional Sobolev space can be characterized through the Bourgain-Brezis-Mironescu formula:\nand the condition\ncharacterizes those functions of that are in the first-order Sobolev space.\n\n# Extension operators\nIf \\Omega is a [domain](https://bliptext.com/articles/domain-mathematical-analysis) whose boundary is not too poorly behaved (e.g., if its boundary is a manifold, or satisfies the more permissive \"[cone condition](https://bliptext.com/articles/cone-condition)\") then there is an operator A mapping functions of \\Omega to functions of \\R^n such that:\nWe will call such an operator A an extension operator for \\Omega.\n\n## Case of p = 2\nExtension 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).\nWe define H^s(\\Omega) by saying that if and only if Equivalently, complex interpolation yields the same H^s(\\Omega) spaces so long as \\Omega has an extension operator.\nIf \\Omega does not have an extension operator, complex interpolation is the only way to obtain the H^s(\\Omega) spaces.\nAs a result, the interpolation inequality still holds.\n\n## Extension by zero\nLike [above](https://bliptext.com/articles/), we define to be the closure in H^s(\\Omega) of the space of infinitely differentiable compactly supported functions.\nGiven the definition of a trace, above, we may state the following\nIf we may define its **extension by zero** in the natural way, namely\nFor\nf ∈ Lp(Ω)\nits extension by zero,\nis an element of L^p(\\R^n).\nFurthermore,\nIn the case of the Sobolev space W1,p(Ω) for\n1 ≤ p ≤ ∞\n, extending a function u by zero will not necessarily yield an element of 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\nsuch that for each 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\nWe call Eu an extension of u to \\R^n.\n\n# Sobolev embeddings\nIt is a natural question to ask if a Sobolev function is continuous or even continuously differentiable.\nRoughly speaking, sufficiently many weak derivatives (i.e. large k) result in a classical derivative.\nThis idea is generalized and made precise in the [Sobolev embedding theorem](https://bliptext.com/articles/sobolev-inequality).\nWrite W^{k,p} for the Sobolev space of some compact [Riemannian manifold](https://bliptext.com/articles/riemannian-manifold) of dimension n.\nHere k can be any real number, and 1 ≤ p ≤ ∞.\n(For p = ∞ the Sobolev space is defined to be the [Hölder space](https://bliptext.com/articles/h-lder-space) Cn,α where k = n + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if and then\nand the embedding is continuous.\nMoreover, if k > m and then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the [Rellich–Kondrachov theorem](https://bliptext.com/articles/rellich-kondrachov-theorem)).\nFunctions in have all derivatives of order less than m continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous.\nInformally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.\nThere are similar variations of the embedding theorem for non-compact manifolds such as \\R^n.\nSobolev embeddings on \\R^n that are not compact often have a related, but weaker, property of [cocompactness](https://bliptext.com/articles/cocompact-embedding).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T20:10:37.309Z","updated_at":"2025-01-25T20:10:37.309Z","search_vector":"'sobolev':1 'space':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **Sobolev space** is a [vector space](https://bliptext.com/articles/vector-space) of functions equipped with a [norm](https://bliptext.com/articles/normed-space) that is a combination of Lp-[norm](https://bliptext.com/articles/normed-space)s of the function together with its derivatives up to a given order.\nThe derivatives are understood in a suitable [weak sense](https://bliptext.com/articles/weak-derivative) to make the space [complete](https://bliptext.com/articles/complete-metric-space), i.e. a [Banach space](https://bliptext.com/articles/banach-space).\nIntuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as [partial differential equations](https://bliptext.com/articles/partial-differential-equation), and equipped with a norm that measures both the size and regularity of a function.\nSobolev spaces are named after the Russian [mathematician](https://bliptext.com/articles/mathematician) [Sergei Sobolev](https://bliptext.com/articles/sergei-lvovich-sobolev).\nTheir importance comes from the fact that [weak solutions](https://bliptext.com/articles/weak-solution) of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of [continuous functions](https://bliptext.com/articles/continuous-function) with the [derivatives](https://bliptext.com/articles/derivative) understood in the classical sense.\n\n# Motivation\nIn this section and throughout the article \\Omega is an [open subset](https://bliptext.com/articles/open-subset) of \\R^n.\nThere are many criteria for smoothness of [mathematical functions](https://bliptext.com/articles/mathematical-function).\nThe most basic criterion may be that of [continuity](https://bliptext.com/articles/continuous-function).\nA stronger notion of smoothness is that of [differentiability](https://bliptext.com/articles/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](https://bliptext.com/articles/differentiability-classes)).\nDifferentiable functions are important in many areas, and in particular for [differential equations](https://bliptext.com/articles/differential-equation).\nIn 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.\nThe Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.\nQuantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms.\nA typical example is measuring the energy of a temperature or velocity distribution by an L^2-norm.\nIt is therefore important to develop a tool for differentiating [Lebesgue space](https://bliptext.com/articles/lp-space) functions.\nThe [integration by parts](https://bliptext.com/articles/integration-by-parts) formula yields that for every, where k is a [natural number](https://bliptext.com/articles/natural-number), and for all infinitely differentiable functions with [compact support](https://bliptext.com/articles/compact-support)\nwhere \\alpha is a [multi-index](https://bliptext.com/articles/multi-index) of order |\\alpha|=k and we are using the notation:\nThe left-hand side of this equation still makes sense if we only assume u to be [locally integrable](https://bliptext.com/articles/locally-integrable).\nIf there exists a locally integrable function v, such that\nthen we call v the [weak \\alpha-th partial derivative](https://bliptext.com/articles/weak-derivative) of u. If there exists a [weak \\alpha-th partial derivative](https://bliptext.com/articles/weak-derivative) of u, then it is uniquely defined [almost everywhere](https://bliptext.com/articles/almost-everywhere), and thus it is uniquely determined as an element of a [Lebesgue space](https://bliptext.com/articles/lp-space).\nOn the other hand, if, then the classical and the weak derivative coincide.\nThus, if v is a weak \\alpha-th partial derivative of u, we may denote it by.\nFor example, the function\nis not continuous at zero, and not differentiable at −1, 0, or 1.\nYet the function\nsatisfies the definition for being the weak derivative of u(x), which then qualifies as being in the Sobolev space W^{1,p} (for any allowed p, see definition below).\nThe Sobolev spaces combine the concepts of weak differentiability and [Lebesgue norms](https://bliptext.com/articles/lp-norm).\n\n# Sobolev spaces with integer k\n\n## One-dimensional case\nIn the one-dimensional case the Sobolev space W^{k,p}(\\R) for is defined as the subset of functions f in [L^p(\\R)](https://bliptext.com/articles/lp-space) such that f and its [weak derivatives](https://bliptext.com/articles/weak-derivative) up to order k have a finite [[Lp norm|\nLp\nnorm]].\nAs mentioned above, some care must be taken to define derivatives in the proper sense.\nIn the one-dimensional problem it is enough to assume that the (k{-}1)-th derivative f^{(k-1)} is differentiable almost everywhere and is equal almost everywhere to the [Lebesgue integral](https://bliptext.com/articles/lebesgue-integration) of its derivative (this excludes irrelevant examples such as [Cantor's function](https://bliptext.com/articles/cantor-function)).\nWith this definition, the Sobolev spaces admit a natural [norm](https://bliptext.com/articles/normed-vector-space),\nOne can extend this to the case p = \\infty, with the norm then defined using the [essential supremum](https://bliptext.com/articles/essential-supremum-and-essential-infimum) by\nEquipped with the norm becomes a [Banach space](https://bliptext.com/articles/banach-space).\nIt turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by\nis equivalent to the norm above (i.e. the [induced topologies](https://bliptext.com/articles/normed-vector-space) of the norms are the same).\n\n### The case \n\np {{=}} 2\nSobolev spaces with\nare especially important because of their connection with [Fourier series](https://bliptext.com/articles/fourier-series) and because they form a [Hilbert space](https://bliptext.com/articles/hilbert-space).\nA special notation has arisen to cover this case, since the space is a Hilbert space:\nThe space H^k can be defined naturally in terms of [Fourier series](https://bliptext.com/articles/fourier-series) whose coefficients decay sufficiently rapidly, namely,\nwhere \\widehat{f} is the Fourier series of f, and \\mathbb{T} denotes the 1-torus.\nAs above, one can use the equivalent norm\nBoth representations follow easily from [Parseval's theorem](https://bliptext.com/articles/parseval-s-theorem) and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in.\nFurthermore, the space H^k admits an [inner product](https://bliptext.com/articles/inner-product-space), like the space H^0 = L^2.\nIn fact, the H^k inner product is defined in terms of the L^2 inner product:\nThe space H^k becomes a Hilbert space with this inner product.\n\n### Other examples\nIn one dimension, some other Sobolev spaces permit a simpler description.\nFor example, is the space of [absolutely continuous functions](https://bliptext.com/articles/absolute-continuity) on\n(0, 1)\n(or rather, equivalence classes of functions that are equal almost everywhere to such), while is the space of bounded [Lipschitz functions](https://bliptext.com/articles/lipschitz-continuity) on I, for every interval I. However, these properties are lost or not as simple for functions of more than one variable.\nAll spaces are (normed) [algebras](https://bliptext.com/articles/algebra-over-a-field), i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for p n/p, the space will contain only continuous functions, but for which k this is already true depends both on p and on the dimension.\nFor example, as can be easily checked using [spherical polar coordinates](https://bliptext.com/articles/spherical-polar-coordinates) for the function defined on the n-dimensional ball we have:\nIntuitively, 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.\n\n### Absolutely continuous on lines (ACL) characterization of Sobolev functions\nLet If a function is in then, possibly after modifying the function on a set of measure zero, the restriction to [almost every](https://bliptext.com/articles/almost-every) line parallel to the coordinate directions in \\R^n is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous); what's more, the classical derivative along the lines that are parallel to the coordinate directions are in Conversely, if the restriction of f to [almost every](https://bliptext.com/articles/almost-every) line parallel to the coordinate directions is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous), then the pointwise gradient \\nabla f exists [almost every](https://bliptext.com/articles/almost-every)where, and f is in provided In particular, in this case the weak partial derivatives of f and pointwise partial derivatives of f agree [almost every](https://bliptext.com/articles/almost-every)where.\nThe ACL characterization of the Sobolev spaces was established by [Otto M. Nikodym](https://bliptext.com/articles/otto-m-nikodym) ([1933](https://bliptext.com/articles/)); see.\nA stronger result holds when p>n.\nA function in is, after modifying on a set of measure zero, [Hölder continuous](https://bliptext.com/articles/h-lder-continuous) of exponent by [Morrey's inequality](https://bliptext.com/articles/sobolev-inequality).\nIn particular, if p=\\infty and \\Omega has Lipschitz boundary, then the function is [Lipschitz continuous](https://bliptext.com/articles/lipschitz-continuous).\n\n### Functions vanishing at the boundary\nThe Sobolev space is also denoted by It is a Hilbert space, with an important subspace defined to be the closure of the infinitely differentiable functions compactly supported in \\Omega in The Sobolev norm defined above reduces here to\nWhen \\Omega has a regular boundary, can be described as the space of functions in that vanish at the boundary, in the sense of traces ([see below](https://bliptext.com/articles/)).\nWhen n=1, if is a bounded interval, then H^1_0(a,b) consists of continuous functions on [a,b] of the form\nwhere the generalized derivative f' is in L^2(a,b) and has 0 integral, so that\nWhen \\Omega is bounded, the [Poincaré inequality](https://bliptext.com/articles/poincar-inequality) states that there is a constant such that:\nWhen \\Omega is bounded, the injection from to is [compact](https://bliptext.com/articles/compact-operator).\nThis fact plays a role in the study of the [Dirichlet problem](https://bliptext.com/articles/dirichlet-problem), and in the fact that there exists an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of L^2(\\Omega) consisting of eigenvectors of the [Laplace operator](https://bliptext.com/articles/laplace-operator) (with [Dirichlet boundary condition](https://bliptext.com/articles/dirichlet-boundary-condition)).\n\n# Traces\nSobolev spaces are often considered when investigating partial differential equations.\nIt is essential to consider boundary values of Sobolev functions.\nIf, those boundary values are described by the restriction However, it is not clear how to describe values at the boundary for as the n-dimensional measure of the boundary is zero.\nThe following theorem resolves the problem:\nTu is called the trace of u.\nRoughly speaking, this theorem extends the restriction operator to the Sobolev space for well-behaved Ω.\nNote that the [trace operator](https://bliptext.com/articles/trace-operator) T is in general not surjective, but for 1 0\nbe not an integer and set.\nUsing the same idea as for the [Hölder spaces](https://bliptext.com/articles/holder-space), the **Sobolev–Slobodeckij space** is defined as\nIt is a Banach space for the norm\nIf \\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](https://bliptext.com/articles/embedding)\nThere are examples of irregular Ω such that is not even a vector subspace of for 0 m and then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the [Rellich–Kondrachov theorem](https://bliptext.com/articles/rellich-kondrachov-theorem)).\nFunctions in have all derivatives of order less than m continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous.\nInformally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.\nThere are similar variations of the embedding theorem for non-compact manifolds such as \\R^n.\nSobolev embeddings on \\R^n that are not compact often have a related, but weaker, property of [cocompactness](https://bliptext.com/articles/cocompact-embedding).","revisionCount":0},"hasMore":false,"nextCursor":null}

History