Sobolev Space

A Sobolev space is a vector space of functions whose generalized derivatives, together with the functions themselves, satisfy specified integrability conditions. The classical spaces $W^{k,p}(\Omega)$ combine an integer differentiability order $k$ with an integrability exponent $p$. Unlike spaces defined by classical differentiability, they accommodate functions with corners and other limited irregularities. They are fundamental in functional analysis, particularly in the study of partial differential equations. (math.ucdavis.edu)

Weak derivatives

The defining idea is to express differentiation through integration against smooth test functions rather than through pointwise difference quotients.

Let $\Omega$ be an open subset of $\mathbb R^n$, and let $u$ be locally integrable. A locally integrable function $g_i$ is the weak derivative of $u$ in the $i$-th coordinate direction if

Here $C_c^\infty(\Omega)$ denotes the smooth functions with compact support inside $\Omega$. This identity extends integration by parts; for a continuously differentiable function, it reproduces the classical partial derivative. A weak derivative, when it exists, is unique almost everywhere. (math.ucdavis.edu)

For a multi-index $\alpha=(\alpha_1,\ldots,\alpha_n)$, with $|\alpha|=\alpha_1+\cdots+\alpha_n$, higher derivatives are defined by

In distribution theory, the right-hand side always defines a distributional derivative. A weak derivative exists as a function when that distribution is represented by a locally integrable function. (jschoeberl.github.io)

For example, $u(x)=|x|$ has weak derivative $\operatorname{sgn}(x)$, despite its corner at zero. Its second distributional derivative is $2\delta_0$, which is not a locally integrable function. Thus, on $(-1,1)$, it belongs to $W^{1,p}$ for every $1\le p\le\infty$, but not to $W^{2,p}$. A step function likewise has a distributional derivative concentrated at its jump, rather than a weak derivative represented by a function. (jschoeberl.github.io)

Integer-order spaces and their norms

For an integer $k\ge0$ and $1\le p\le\infty$,

where derivatives are weak derivatives. The $L^p$ spaces measure integrability using Lebesgue integration. As in $L^p$, functions equal almost everywhere are identified. In particular, $W^{0,p}(\Omega)=L^p(\Omega)$. (math.ucdavis.edu)

A standard norm is

and

These spaces are Banach spaces: every Cauchy sequence in the Sobolev norm converges within the space. (math.ucdavis.edu)

The notation $H^k(\Omega)=W^{k,2}(\Omega)$ emphasizes the especially important square-integrable case. It is a Hilbert space with inner product

Completeness makes these spaces suitable for passing from smooth approximations to limiting solutions. (finite-element.github.io)

Approximation and boundary values

For $1\le p<\infty$, smooth functions belonging to $W^{k,p}(\Omega)$ are dense in that space. This is the Meyers–Serrin theorem. It allows identities established for smooth functions to be extended by convergence in the Sobolev norm. Such density generally fails in the $W^{k,\infty}$ norm. (people.tamu.edu)

Smooth functions with compact support need not be dense in the entire space. Their closure defines

which encodes homogeneous boundary conditions. For first-order spaces on suitably regular domains, this is the space of functions with zero boundary trace. (jschoeberl.github.io)

Boundary values cannot simply be assigned to arbitrary representatives of an almost-everywhere equivalence class. Instead, a trace operator extends ordinary restriction to the boundary continuously from smooth functions. On a bounded Lipschitz domain, the trace of an $H^1(\Omega)$ function lies in $H^{1/2}(\partial\Omega)$, and

The fractional boundary space expresses the loss of regularity associated with restriction to a lower-dimensional set. (jschoeberl.github.io)

Embedding and compactness

Sobolev embedding theorems convert derivative control into stronger integrability or continuity. For a bounded Lipschitz domain in $\mathbb R^n$, their first-order forms include:

• $1\le p<n$:

  $$W^{1,p}(\Omega)\hookrightarrow L^{p^*}(\Omega), \qquad p^*=\frac{np}{n-p}.$$

• $p=n>1$: embedding into every finite $L^q(\Omega)$, but generally not into $L^\infty(\Omega)$.

• $n<p<\infty$: a representative that is Hölder continuous, with exponent $1-n/p$.

The arrow denotes a continuous inclusion, with the target norm bounded by a constant times the Sobolev norm. These distinctions show why dimension and integrability are inseparable from regularity. (people.tamu.edu)

The Rellich–Kondrachov compactness theorem strengthens certain embeddings: bounded sequences have strongly convergent subsequences in a weaker space. For $1\le p<n$, the inclusion into $L^q(\Omega)$ is compact when $1\le q<p^*$, but generally not at $q=p^*$. Compactness helps control limiting processes in nonlinear equations. (math.ucdavis.edu)

Fractional and negative orders

Sobolev regularity can also have noninteger order. For $0<s<1$ and $1\le p<\infty$, the Sobolev–Slobodeckij space $W^{s,p}(\Omega)$ uses the seminorm

together with the $L^p$ norm. This measures differences across all length scales rather than an integer number of derivatives. (arxiv.org)

On $\mathbb R^n$, the Fourier transform defines $H^s$ for any real $s$ by

For negative $s$, $u$ is understood as a tempered distribution. For $0<s<1$, this definition agrees with $W^{s,2}$, with equivalent norms. For $p\ne2$, Fourier-based Bessel-potential spaces and difference-based fractional spaces generally differ. (arxiv.org)

Negative-order spaces also arise through duality. A common convention is

allowing forcing terms to be continuous linear functionals rather than ordinary square-integrable functions. (jschoeberl.github.io)

Partial differential equations and numerical methods

Consider the Poisson equation with homogeneous Dirichlet boundary conditions,

Its variational formulation seeks $u\in H_0^1(\Omega)$ such that

Only first weak derivatives of $u$ are required, although the original equation contains second derivatives. On a bounded Lipschitz domain, for $f\in H^{-1}(\Omega)$, coercivity and the Lax–Milgram theorem yield a unique weak solution. Higher regularity depends on additional assumptions about the domain and data; it is not automatic from existence in $H^1$. (jschoeberl.github.io)

The finite element method replaces the infinite-dimensional trial space by a finite-dimensional subspace. Continuous piecewise-polynomial functions can belong to $H^1$ even though their gradients jump across element interfaces. Sobolev norms provide the natural measures of approximation error; for coercive variational problems, Céa’s lemma bounds the numerical error by the best approximation error in the chosen subspace. (finite-element.github.io)

Historical development and qualifications

The spaces are named after Sergei L. Sobolev, whose work in the 1930s connected generalized derivatives, function spaces, and existence theory for differential equations. They developed alongside approaches associated with Friedrichs, Leray, and others, and became part of the modern distributional framework of analysis. (mathshistory.st-andrews.ac.uk)

Statements about embeddings, traces, and approximation require their domain and exponent assumptions. Results on smooth or Lipschitz domains need not hold on arbitrary open sets. Likewise, the notation $W^{s,p}$ or $H^s$ at noninteger order must be read with the author’s definition: different constructions are not universally interchangeable. (people.tamu.edu)

References

1. Notes on Partial Differential Equationsmath.ucdavis.edu
2. Chapter 3: Sobolev Spacesmath.ucdavis.edu
3. Generalized derivatives — Interactive Finite Elementsjschoeberl.github.io
4. Chapter 2: Weak derivatives and Sobolev spacespeople.tamu.edu
5. Trace theorems and their applications — Interactive Finite Elementsjschoeberl.github.io
6. Traces spaces — Interactive Finite Elementsjschoeberl.github.io
7. Hitchhiker's guide to the fractional Sobolev spacesarxiv.org
8. The weak formulation of the Poisson equation — Interactive Finite Elementsjschoeberl.github.io
9. Convergence of finite element approximationsfinite-element.github.io
10. Sergei Sobolev (1908–1989) — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk
11. Sobolev Krasovskii — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk