On an effect of inhomogeneous constraints for a maximizing problem of the Sobolev embedding associated with the space of bounded variation

2019/05/19 by Ishiwata, Michinori, Wadade, Hidemitsu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.07759

Abstract

In this paper, we consider a maximizing problem associated with the Sobolev type embedding on the space of bounded variation. We show that, although the maximizing problem suffers from both of the non-compactness of vanishing and concentrating phenomena, there exists a maximizer for some range of the exponents. Furthermore, we show that any maximizer must be given by a characteristic function on a ball.

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• Optimal limiting embeddings for Δ-reduced Sobolev spaces in L¹
• Defect of compactness in spaces of bounded variation