TOPICS
Search

Sobolev Embedding Theorem


The Sobolev embedding theorem is a result in functional analysis which proves that certain Sobolev spaces W^(k,p)(Omega) can be embedded in various spaces including W^(l,q)(Omega^'), L^r(Omega^'), and C^(j,lambda)(Omega^_^') for various domains Omega, Omega^' in R^n and for miscellaneous values of j, k, l, p, q, r, and lambda (usually depending on properties of the domains Omega and Omega^'). Because numerous such embeddings are possible, many individual results may be termed "the" Sobolev embedding theorem, whereas in actuality the phrase "Sobolev embedding theorem" is best thought of as an umbrella term encompassing all such results.

To proceed, let Omega be a domain (i.e., a bounded, connected open set) in R^n and let Omega_k be the intersection of Omega with a hyperplane of dimension k in R^n for 1<=k<=n. Let j>=0, m>=1 be integers and let 1<=p<infty. Under these constructions, one has a number of function space embeddings, the collection of which will be referred to as the Sobolev embedding theorem.

For example, if Omega satisfies a so-called "cone condition" (i.e., if there exists a finite cone C such that each x in Omega is the vertex of a finite cone C_x contained in Omega and congruent to C), then

 W^(j+m,p)(Omega)↪C_B^j(Omega)
(1)

if either mp>n or if m=n and p=1. For such m, n, and p, one also has that

 W^(j+m,p)(Omega)↪W^(j,q)(Omega_k)
(2)

and

 W^(m,p)(Omega)↪L^q(Omega)
(3)

for p<=q<=infty. If instead mp=n, then

 W^(j+m,p)(Omega)↪W^(j,q)(Omega_k)
(4)

and

 W^(m,p)(Omega)↪L^q(Omega)
(5)

for p<=q<infty. Finally, if mp<n and if either n-mp<k<=n or if p=1 and n-m<=k<=n, then

 W^(j+m,p)(Omega)↪W^(j,q)(Omega_k)
(6)

and

 W^(m,p)(Omega)↪L^q(Omega)
(7)

for p<=q<=kp/(n-mp). Note that the above embeddings are all due essentially to Sobolev and depend only on n, m, p, q, j, k, and the dimension of the cone C in the cone condition.

Other types of domains also provide a number of embeddings. If Omega satisfies the so-called "strong local Lipschitz condition" (i.e., if each point x on the boundary partialOmega of Omega has a neighborhood U_x whose intersection with partialOmega is the graph of a Lipschitz function), for example, then the target space C_B^j(Omega) of (1) can be replaced with the smaller space C^j(Omega^_). Moreover, if mp>n>(m-1)p, then

 W^(j+m,p)(Omega)↪C^(j,lambda)(Omega^_)
(8)

for 0<lambda<=m-(n/p). If instead n=(m-1)p, then

 W^(j+m,p)(Omega)↪C^(j,lambda)(Omega^_)
(9)

with (9) holding also for lambda=1 provided that n=m-1 and p=1.

A number of the above-stated results can be proved either entirely or almost so by way of the so-called Sobolev embedding inequalities. These inequalities follow from the Littlewood-Paley decomposition of a function f in L-p for 1<=p<infty. Indeed, in this context, Minkowski's inequality combined with the Littlewood-Paley decomposition of such a function f implies a number of inequalities, e.g.,

 ||f||_(L^q(R^n))<~||f||_(L^p(R^n))+||del f||_(L^p(R^n))
(10)

when 1<=p<q<=infty satisfies 1/p-1/n>1/q. In the event that p!=1 and q!=infty, there is an analogous inequality:

 ||f||_(L^q(R^n))<~||del ||_(L^p(R^n))
(11)

where 1<p<q<infty satisfies 1/p-1/n=1/q. Inequalities (10) and (11) are true whenever the respective right-hand sides are finite and can be expanded even further using fractional differentiation and integration operators to yield many of the embedding results stated previously.

The above results can be further altered to allow for even more general embeddings. For example, if the W^(m,p)-spaces being embedded above are replaced with the W_0^(m,p) Sobolev spaces (i.e., the Sobolev spaces of functions, the trace of whose k-order derivatives vanishes for all k<m), then the resulting embeddings hold for arbitrary domains Omega in R^n. Moreover, it can be shown that the embeddings associated to the above-mentioned cone condition still hold for domains Omega which satisfy only a "weakened cone condition."


See also

Fractional Derivative, Fractional Integral, Littlewood-Paley Decomposition, L-p-Space, Sobolev Space

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha

References

Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011.

Cite this as:

Stover, Christopher. "Sobolev Embedding Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SobolevEmbeddingTheorem.html

Subject classifications