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An embedding theorem. (English) Zbl 0702.46020

Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 919-924 (1989).
[For the entire collection see Zbl 0671.00007.]
Suppose A is a Young function, i.e. A maps \([0,\infty [\) into \([0,\infty [\) is convex and \(A(0)=0.\)
The author proves the following theorem: If \(\int^{\infty}_{0}[r/A(r)]^{(1/n-1)}dr<\infty\) and \(\int_{{\mathbb{R}}^ n}A(| \text{grad} u(x)|)dx<\infty\) then \(ess \sup | u| <\infty,\) where the support of u is bounded.
This theorem includes embedding theorems for Sobolev spaces [see R. A. Adams, Sobolev spaces, Academic Press (1975; Zbl 0314.46030)].
Reviewer: V.V.Kovrizkin

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems