Sobolev Embedding Theorem

The Sobolev embedding theorem is a result in functional analysis which proves that certain Sobolev spaces can be embedded in various function spaces including , , and for various domains , in and for miscellaneous values of , , , , , , and (usually depending on properties of the domains and ). 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 be a bounded domain (a connected open set) in and let be the intersection of with a translate of a -dimensional linear subspace of for . Let , be integers and let . 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 satisfies a so-called "cone condition" (i.e., if there exists a finite cone such that each is the vertex of a finite cone contained in and congruent to ), then

(1)

if either or if and . Here, consists of functions whose partial derivatives of orders at most are continuous and bounded. For such , , and , one also has that

(2)

and

(3)

for . If instead , then

(4)

and

(5)

for . Finally, if and if either or if and , then

(6)

and

(7)

for . Note that the above embeddings are all due essentially to Sobolev (Adams 1975, §5.5) and depend only on , , , , , , and the dimension of the cone in the cone condition.

Other types of domains also provide a number of embeddings. If satisfies the so-called "strong local Lipschitz condition" (i.e., if each point on the boundary of has a neighborhood whose intersection with is the function graph of a Lipschitz function), for example, then the target function space of (1) can be replaced with the smaller function space . Moreover, if , then

(8)

for . If instead , then

(9)

for , with the endpoint also allowed when and .

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 in L^p for . Indeed, in this context, Minkowski's inequality combined with the Littlewood-Paley decomposition of such a function implies a number of inequalities. For example, if , , and , then

(10)

At the critical exponent , the corresponding homogeneous inequality for is

(11)

These inequalities can be expanded further using fractional differentiation and fractional 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 Sobolev spaces being embedded above are replaced with , the closure of in the Sobolev space , then the resulting embeddings hold for arbitrary domains in . Moreover, it can be shown that the embeddings associated to the above-mentioned cone condition still hold for domains which satisfy only a "weakened cone condition."