RellichKondrachov

RellichKondrachov is a term that refers to a pair of related concepts in the field of partial differential equations, specifically concerning the regularity of solutions. The Rellich theorem and the Kondrachov embedding theorem are fundamental results in Sobolev spaces, which are function spaces equipped with norms that measure the integrability of the function and its derivatives.

The Rellich theorem establishes a compactness criterion for embedding certain Sobolev spaces into L^p spaces. It states that if a Sobolev space W^{k,p}(Omega) is compactly embedded into L^q(Omega) for some p, q, and domain Omega, then certain conditions on k, p, q, and the domain must be met. This compactness property is crucial for proving the existence of solutions to various differential equations, as it allows one to pass from convergence of sequences of approximate solutions in the Sobolev space to pointwise convergence or convergence in the L^q space.

The Kondrachov embedding theorem provides more precise conditions for these embeddings and establishes the conditions under which the embedding is compact. It essentially refines the Rellich theorem by giving explicit characterizations of the parameters for which compact embeddings hold. These theorems are particularly important in the study of elliptic and parabolic partial differential equations, where they are used to establish the existence and regularity of solutions through compactness arguments. The interplay between these two theorems underpins much of the modern analysis of PDEs.