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The concentration-compactness principle in the calculus of variations. The limit case. I. (English) Zbl 0704.49005
Summary: After the study made in the locally compact case for variational problems with some translation invariance, we investigate variational problems (with constraints), for example in \({\mathbb{R}}^ N\), where the invariance of \({\mathbb{R}}^ N\) by the group of dilatations creates some possible loss of compactness. This is, for example, the case for all the problems associated with the determination of extremal functions in functional inequalities (such as, for example, the Sobolev inequalities). We show how the concentration-compactness principle has to be modified in order to be able to treat this class of problems, and we present applications to functional analysis, mathematical physics, differential geometry and harmonic analysis. [For part II see the author, ibid. No.2, 45-121 (1985; Zbl 0704.49006).]

MSC:
49J10 Existence theories for free problems in two or more independent variables
49J27 Existence theories for problems in abstract spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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