Sobolev Embedding Theorem
The Sobolev embedding theorem is a result in functional analysis which proves that certain Sobolev spaces can be embedded in various 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 domain (i.e., a bounded, connected open set) in and let be the intersection of with a hyperplane of dimension in 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
if either or if and . For such , , and , one also has that
for . If instead , then
for . Finally, if and if either or if and , then
for . Note that the above embeddings are all due essentially to Sobolev 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 graph of a Lipschitz function), for example, then the target space of (1) can be replaced with the smaller space . Moreover, if , then
for . If instead , then
with (9) holding also for provided that 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, e.g.,
when satisfies . In the event that and , there is an analogous inequality:
where satisfies . 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 -spaces being embedded above are replaced with the Sobolev spaces (i.e., the Sobolev spaces of functions, the trace of whose -order derivatives vanishes for all ), 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."