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.
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.
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
Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms.
A typical example is measuring the energy of a temperature or velocity distribution by an
It is therefore important to develop a tool for differentiating Lebesgue space functions.
is a natural number, and for all infinitely differentiable functions with compact support
and we are using the notation: The left-hand side of this equation still makes sense if we only assume
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
can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely, where
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
The transition to multiple dimensions brings more difficulties, starting from the very definition.
does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.
For example, as can be easily checked using spherical polar coordinates for the function
then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in
is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in
to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient
The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see (Maz'ya 2011, §1.1.3).
defined to be the closure of the infinitely differentiable functions compactly supported in
consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).
Sobolev spaces are often considered when investigating partial differential equations.
Roughly speaking, this theorem extends the restriction operator to the Sobolev space
Note that the trace operator T is in general not surjective, but for 1 < p < ∞ it maps continuously onto the Sobolev–Slobodeckij space
For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers[3][4]) that the space
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
Furthermore, In the case of the Sobolev space W1,p(Ω) for 1 ≤ p ≤ ∞, extending a function u by zero will not necessarily yield an element of
This idea is generalized and made precise in the Sobolev embedding theorem.
for the Sobolev space of some compact Riemannian manifold of dimension n. Here k can be any real number, and 1 ≤ p ≤ ∞.
There are similar variations of the embedding theorem for non-compact manifolds such as