# Sobolev space

A Sobolev space is a vector space of functions on a domain in [Euclidean space](https://www.edgepedia.org/euclidean-space) that possesses weak derivatives up to a given order, all of which belong to an L^p (Lebesgue) space. The spaces are named after Sergei L'vovich Sobolev (1908–1989), who introduced them in the 1930s together with the notion of generalized function, now called a distribution<sup>[1](https://mathshistory.st-andrews.ac.uk//Biographies/Sobolev/)</sup>. They are the standard setting for weak solutions of partial differential equations (PDEs), the calculus of variations, and the numerical analysis of PDEs<sup>[2](https://epubs.siam.org/doi/10.1137/1.9781611973488.ch7)</sup>.

| Key fact | Detail |
|---|---|
| Definition | \( W^{k,p}(\Omega) \) contains the functions in \( L^p(\Omega) \) whose weak derivatives \( \partial^{\alpha} u \) of order \( |\alpha| \le k \) also lie in \( L^p(\Omega) \)<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup> |
| Norm | \( \|u\|_{W^{k,p}} = \big(\sum_{|\alpha|\le k}\|\partial^{\alpha} u\|_{L^p}^p\big)^{1/p} \) for \( 1 \le p < \infty \)<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup> |
| Hilbert case | \( H^k(\Omega) := W^{k,2}(\Omega) \), a Hilbert space with inner product \( \sum_{|\alpha|\le k}(\partial^{\alpha} v, \partial^{\alpha} w)_{L^2} \)<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup> |
| Embedding | For \( p < d \), \( W^{1,p}(\mathbb{R}^d) \hookrightarrow L^q \) up to \( p^* = pd/(d-p) \); for \( p > d \), Morrey's embedding gives \( L^{\infty} \cap C^{0,\alpha} \) with \( \alpha = 1 - d/p \)<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup> |
| Compactness | Rellich–Kondrachov: a bounded sequence in \( W^{1,p}(\Omega) \) on a finite-measure extension domain has a subsequence converging in \( L^p(\Omega) \)<sup>[5](https://www.ams.org/publications/authors/books/postpub/gsm-229-appendix-sobolev.pdf)</sup> |
| Boundary values | Trace theorems assign boundary values to \( W^{k,p} \) functions on Lipschitz domains, with trace spaces described explicitly for \( W^{k,2} \)<sup>[6](https://arxiv.org/abs/1910.03971)</sup> |
| Origin | Introduced by Sobolev in the 1930s; his 1950 book *Applications of Functional Analysis in Mathematical Physics* appeared in English translation by the American Mathematical Society in 1963<sup>[1](https://mathshistory.st-andrews.ac.uk//Biographies/Sobolev/)</sup> |

## Weak derivatives

The spaces rest on a generalized notion of differentiation. If \( u \in C^k(\Omega) \) and \( \phi \in C^{\infty}_c(\Omega) \) (a smooth function with compact support in \( \Omega \)), integration by parts gives \( \int_{\Omega} \partial^{\alpha} u \, \phi \, dx = (-1)^{|\alpha|} \int_{\Omega} u \, \partial^{\alpha} \phi \, dx \) for every multiindex \( \alpha \) with \( |\alpha| \le k \). The right-hand side still makes sense when \( u \) is only locally integrable, which motivates the definition: \( u_{\alpha} \in L^1_{\mathrm{loc}}(\Omega) \) is the \( \alpha \)-th weak derivative of \( u \) if \( \int_{\Omega} u_{\alpha} \phi \, dx = (-1)^{|\alpha|} \int_{\Omega} u \, \partial^{\alpha} \phi \, dx \) holds for all \( \phi \in C^{\infty}_c(\Omega) \)<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>.

A weak derivative, when it exists, is unique up to changes on sets of measure zero<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>. The definition agrees with the classical derivative for smooth functions, but it admits functions that no classical derivative can: on \( \Omega = (-1,1) \), the weak derivative of \( u(x) = |x| \) is the Heaviside-type function equal to \( -1 \) on \( (-1,0) \) and \( 1 \) on \( [0,1) \)<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>. Calculus rules such as the product rule, chain rule, and fundamental theorem of calculus carry over to weak derivatives<sup>[7](https://www.youtube.com/watch?v=0TV1MV0582o)</sup>.

## Definition and basic structure

For an open set \( \Omega \subset \mathbb{R}^n \), integers \( k \ge 0 \), and \( 1 \le p \le \infty \), the Sobolev space is

\[ W^{k,p}(\Omega) := \{ u \in L^p(\Omega) \mid \partial^{\alpha} u \text{ exists and belongs to } L^p(\Omega) \text{ for all } |\alpha| \le k \}, \]

with the derivatives understood weakly<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup><sup> • </sup><sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>. Equipped with the norm above, \( W^{k,p}(\Omega) \) is a [Banach space](https://www.edgepedia.org/banach-space); for \( p = 2 \) the space \( H^k(\Omega) := W^{k,2}(\Omega) \) is a [Hilbert space](https://www.edgepedia.org/hilbert-space) under the inner product \( (v,w)_{H^k} = \sum_{|\alpha|\le k} \int_{\Omega} \partial^{\alpha} v \, \partial^{\alpha} w \, dx \)<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>.

The subspace \( W^{k,p}_0(\Omega) \) is defined as the closure of \( C^{\infty}_c(\Omega) \) in \( W^{k,p}(\Omega) \); it plays the role of functions vanishing on the boundary in a generalized sense<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>. Its dual for \( k = 1 \), \( p = 2 \), is written \( H^{-1}(\Omega) := (H^1_0(\Omega))' \)<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>. On \( W^{1,p}_0(\Omega) \) over a bounded domain, the Poincaré inequality \( \|u\|_{L^p(\Omega)} \le C_P \|\nabla u\|_{L^p(\Omega)} \) holds with a constant \( C_P \) depending on \( p \), \( n \), and \( \Omega \), so the gradient seminorm alone is equivalent to the full norm<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>. The Meyers–Serrin theorem, which shows that smooth functions are dense in \( W^{k,p} \) for \( p < \infty \), is among the standard structural results of the theory<sup>[8](https://www.math.uni-hamburg.de/home/schmidt/lectures/2019_PDG_II.html.en)</sup>.

## Embeddings

Sobolev embedding theorems trade differentiability for integrability: functions in \( W^{s,p} \) with \( s > 0 \) belong to Lebesgue spaces \( L^q \) with \( q > p \), and when \( s \) is large enough they have continuous, even Hölder continuous, representatives<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>. In dimension \( d \ge 2 \) on \( \mathbb{R}^d \), the Gagliardo–Nirenberg–Sobolev theorem gives \( W^{1,p}(\mathbb{R}^d) \hookrightarrow L^q(\mathbb{R}^d) \) for all \( q \in [p, p^*] \) when \( p < d \), with critical exponent \( p^* = pd/(d-p) \); if \( p = d \) the embedding holds for every finite \( q \ge d \); and if \( p > d \), Morrey's embedding gives \( W^{1,p}(\mathbb{R}^d) \hookrightarrow L^{\infty}(\mathbb{R}^d) \cap C^{0,\alpha}(\mathbb{R}^d) \) with Hölder exponent \( \alpha = 1 - d/p \)<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>. The exchange rate is \( 1/p^* = 1/p - s/d \) for \( s \) derivatives in \( L^p \)<sup>[9](https://www.buttenschoen.ca/MATH725/distributions/embeddings/)</sup>.

The critical case \( p = d \) is sharp in a precise sense: the formula predicts \( p^* = \infty \), but the embedding into \( L^{\infty} \) fails, with logarithmic counterexamples<sup>[9](https://www.buttenschoen.ca/MATH725/distributions/embeddings/)</sup>. In one dimension the situation is simpler, since \( W^{1,1}(\mathbb{R}) \hookrightarrow C^0(\mathbb{R}) \)<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>. Embeddings extend from \( \mathbb{R}^d \) to any open set with the \( (s,p) \)-extension property, meaning there is a bounded linear extension operator \( E : W^{s,p}(D) \to W^{s,p}(\mathbb{R}^d) \) that agrees with \( u \) on \( D \); every bounded Lipschitz set has this property<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>.

## Compactness and boundary traces

**Rellich–Kondrachov compactness.** On an extension domain of finite measure, every bounded sequence in \( W^{1,p}(\Omega) \) has a subsequence converging strongly in \( L^p(\Omega) \), and the limit lies in \( W^{1,p}(\Omega) \) when \( p > 1 \)<sup>[5](https://www.ams.org/publications/authors/books/postpub/gsm-229-appendix-sobolev.pdf)</sup>. In the Hilbert case, the inclusion \( H^1_0(\Omega) \hookrightarrow L^2(\Omega) \) is compact for every bounded open \( \Omega \subset \mathbb{R}^n \), a result attributed to [Franz Rellich](https://www.edgepedia.org/franz-rellich)<sup>[10](https://www.ams.org/bookstore/pspdf/gsm-116-prev.pdf)</sup>. Compactness is what turns weak convergence of minimizing or approximating sequences into strong convergence, which is needed to prove existence of solutions and minimizers<sup>[9](https://www.buttenschoen.ca/MATH725/distributions/embeddings/)</sup>.

**Traces.** A function in \( W^{k,p}(\Omega) \) is only defined almost everywhere, so its restriction to the boundary is not immediately meaningful. Trace theorems supply a continuous linear operator that assigns boundary values, with the trace landing in a Besov-type space on \( \partial\Omega \); for Lipschitz boundaries the trace spaces of \( W^{k,2}(\Omega) \) admit explicit descriptions, including characterizations on planar polygons in the tradition of P. Grisvard<sup>[6](https://arxiv.org/abs/1910.03971)</sup>. Compactness results for the trace operator and counterexamples showing that surjectivity can fail are also part of the theory<sup>[11](https://arxiv.org/abs/1704.06344)</sup>.

## Fractional and related spaces

The scale extends to non-integer smoothness \( s \in (0,1) \) through the Gagliardo seminorm

\[ |v|_{W^{s,p}(D)} := \left( \int_D \int_D \frac{|v(x) - v(y)|^p}{\|x - y\|^{sp+d}} \, dx \, dy \right)^{1/p}, \]

with \( W^{s,p}(D) \) the set of \( L^p \) functions for which this seminorm is finite; for \( s > 1 \) the spaces are defined recursively through the highest derivatives<sup>[4](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)</sup>. Fractional-order Sobolev spaces are closely related to Besov spaces, Bessel or Riesz potential spaces, and Triebel–Lizorkin spaces, and the unifying fractional framework has become a working tool in both linear and nonlinear PDE theory<sup>[12](https://pijms.com/index.php/pijms/article/view/5)</sup>. Recent work treats variable-exponent and Orlicz-type generalizations, trace and extension problems in fractional settings, and optimization problems posed in fractional spaces<sup>[12](https://pijms.com/index.php/pijms/article/view/5)</sup><sup> • </sup><sup>[13](https://arxiv.org/abs/2303.05940)</sup>.

## Applications

The main motivation is that weak solutions of PDEs are far easier to guarantee than classical ones: Sobolev spaces are flexible enough to contain the solutions of boundary value problems while still supporting the functional-analytic machinery of existence, uniqueness, and regularity proofs<sup>[3](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)</sup>. Embedding theorems upgrade weak solutions to classical ones when the regularity is high enough, and compact embeddings convert weak convergence into strong convergence in variational existence arguments<sup>[9](https://www.buttenschoen.ca/MATH725/distributions/embeddings/)</sup>.

In numerical analysis, the scale of Sobolev spaces quantifies the decay rate of the approximation error in finite element methods. The finite element method is a Galerkin approximation scheme in which the finite-dimensional subspaces consist of piecewise polynomial functions; choosing basis functions with small supports yields sparse linear systems, a decisive property for the large systems that continuum-mechanics PDEs produce<sup>[2](https://epubs.siam.org/doi/10.1137/1.9781611973488.ch7)</sup>. In the calculus of variations, minimization of coercive lower semicontinuous functionals on Sobolev spaces admits solutions by the direct method, while without coercivity existence can fail even for convex cost functions<sup>[14](https://epubs.siam.org/doi/10.1137/1.9781611973488.ch15)</sup>.

## History

Sergei L'vovich Sobolev introduced the function spaces that now carry his name in the 1930s, and the study of these spaces immediately became a whole area of functional analysis<sup>[1](https://mathshistory.st-andrews.ac.uk//Biographies/Sobolev/)</sup>. In 1935, working with the \( L_1 \)-space and building on ideas of Günter, he proposed using solutions of the wave equation in a generalized sense in which the solutions need have neither first derivatives nor boundedness; this line of thought led to the revolutionary papers that inaugurated distribution theory, of which he is regarded as a cofounder<sup>[15](https://link.springer.com/article/10.1134/S0037446623050166)</sup>. His notion of generalized function, developed further by [Laurent Schwartz](https://www.edgepedia.org/laurent-schwartz) and [Israel Gelfand](https://www.edgepedia.org/israel-gelfand), became one of the central notions of mathematics<sup>[1](https://mathshistory.st-andrews.ac.uk//Biographies/Sobolev/)</sup>. Sobolev's 1950 monograph *Applications of Functional Analysis in Mathematical Physics* appeared in English translation by the American Mathematical Society in 1963, the same year Schwartz's book on distributions was published<sup>[1](https://mathshistory.st-andrews.ac.uk//Biographies/Sobolev/)</sup>.

## References

1. [Sergei Sobolev (1908–1989), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Sobolev/)
2. [H. Attouch, G. Buttazzo, G. Michaille, Chapter 7: The finite element method, *Variational Analysis in Sobolev and BV Spaces*, SIAM](https://epubs.siam.org/doi/10.1137/1.9781611973488.ch7)
3. [A brief review on function spaces, MA8502, Norwegian University of Science and Technology](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/function_spaces.html)
4. [J.-L. Guermond, Chapter 2: Weak derivatives and Sobolev spaces, Texas A&M lecture notes](https://people.tamu.edu/~guermond/M661_FALL_2019/chap02.pdf)
5. [Appendix: Sobolev spaces, GSM 229, American Mathematical Society](https://www.ams.org/publications/authors/books/postpub/gsm-229-appendix-sobolev.pdf)
6. [P. D. Lamberti, L. Provenzano, On Trace Theorems for Sobolev Spaces, arXiv:1910.03971](https://arxiv.org/abs/1910.03971)
7. [Lecture 05: Sobolev spaces, Reconstructing Mathematics](https://www.youtube.com/watch?v=0TV1MV0582o)
8. [T. Schmidt, Partial Differential Equations II course page, Universität Hamburg](https://www.math.uni-hamburg.de/home/schmidt/lectures/2019_PDG_II.html.en)
9. [Sobolev Embeddings, MATH 725, University of British Columbia Okanagan](https://www.buttenschoen.ca/MATH725/distributions/embeddings/)
10. [Chapter 7: Fourier transform and Sobolev spaces, GSM 116, American Mathematical Society](https://www.ams.org/bookstore/pspdf/gsm-116-prev.pdf)
11. [Trace and extension theorems for Sobolev-type functions in metric spaces, arXiv:1704.06344](https://arxiv.org/abs/1704.06344)
12. [Advances in Sobolev Spaces and Their Applications to Modern Analysis and PDEs, Pi International Journal of Mathematical Sciences](https://pijms.com/index.php/pijms/article/view/5)
13. [A First Course in Fractional Sobolev Spaces, arXiv:2303.05940](https://arxiv.org/abs/2303.05940)
14. [H. Attouch, G. Buttazzo, G. Michaille, Chapter 15: Variational problems with a lack of coercivity, SIAM](https://epubs.siam.org/doi/10.1137/1.9781611973488.ch15)
15. [Sobolev's Worldline and Memes, Siberian Mathematical Journal](https://link.springer.com/article/10.1134/S0037446623050166)

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