This paper is concerned with an explicit value of the embedding constant from$W^{1,q}(\Omega)$ to $L^{p}(\Omega)$ for a bounded domain$\Omega\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$.To obtain this value, we previously proposed a formula for estimating theembedding constant on bounded and unbounded Lipschitz domains by estimating thenorm of Stein's extension operator, in the article (K. Tanaka, K. Sekine, M.Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant ondomains with minimally smooth boundary using extension operator, Journal ofInequalities and Applications, Vol. 389, pp. 1-23, 2015). This formula is alsoapplicable to a domain that can be divided into Lipschitz domains. However, thevalues computed by the previous formula are very large. In this paper, wepropose several sharper estimations of the embedding constant on a boundeddomain that can be divided into convex domains.
展开▼
机译：本文涉及有界域$ \ Omega \ subset \ mathbb {$ W ^ {1，q}（\ Omega）$到$ L ^ {p}（\ Omega）$的嵌入常数的显式值R} ^ N〜（N \ in \ mathbb {N}）$，其中$ 1 \ leq q \ leq p \ leq \ infty $。为获得该值，我们先前提出了一个公式，用于估计有界和无界Lipschitz上的嵌入常数文章（K.Tanaka，K.Sekine，M.Mizuguchi和S.Oishi，使用扩展算子对Sobolev型嵌入常数ondomain的估计，使用扩展算子，估计Stein扩展算子的范数），不等式与应用学报，第389卷，第1-23页，2015年）。该公式也适用于可以分为Lipschitz域的域。但是，由先前公式计算出的值非常大。在本文中，我们对有界域上的嵌入常数提出了一些更清晰的估计，该界域可以分为凸域。
展开▼
