L1 Compactness of Bounded BV Sets

2002/12/15 by Isidore Fleischer, Fleischer, Isidore
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Nonlinear Partial Differential Equations #math.FA

paper · pdf · doi:10.48550/arxiv.math/0212198

3 pages

arxiv created 2002/12/15 · arxiv updated 2009/11/30

Abstract

Functions, uniformly bounded in BV norm in some bounded open set U in Rⁿ, are compact in L₁(U). This result is known when U has Lipschitz boundary [EG Th. 4 p. 176], [G 1.19 Th. p. 17], [Z 5.34 Cor. p. 227]; the proof for general U here, after identifying the operator theoretic definition of bounded BV norm with that of the Tonelli variation, appeals to the standard compactness criterion in L₁ [DS 21 TH. p. 301] [Y, p. 275] (For completeness, these two auxiliary results are also presented).

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