Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

2016/09/16 by Makoto Mizuguchi, Kazuaki Tanaka, Mizuguchi, Makoto +5 · 3 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1609.05184

openalex publication_date 2016/09/16 · openalex created_date 2016/09/30 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with an explicit value of the embedding constant from W^1,q(Ω) to L^p(Ω) for a bounded domain Ω⊂ℝ^N~(N∈ℕ), where 1≤ q≤ p≤ ∞. To obtain this value, we previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein's extension operator, in the article (K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and Applications, Vol. 389, pp. 1-23, 2015). This formula is also applicable to a domain that can be divided into Lipschitz domains. However, the values computed by the previous formula are very large. In this paper, we propose several sharper estimations of the embedding constant on a bounded domain that can be divided into convex domains.

Citations

• An optimal Poincaré inequality for convex domains

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