Sharp Sobolev inequalities involving boundary terms revisited

Tang, Zhongwei; Xiong, Jingang; Zhou, Ning

doi:10.1007/s00526-021-02036-z

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2022

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“…Let (M, g) be a (n + 1)-dimensional smooth compact Riemannian manifold with boundary ∂M, n ≥ 2. There has been much work on the sharp Sobolev-type inequalities and sharp Sobolev-type trace inequalities on (M, g) and their applications, see, for example, [21], [22], [16], [17], [26], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds

Tang¹,

Zhou²

2022

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Given (M, g) a smooth compact (n + 1)-dimensional Riemannian manifold with boundary ∂M . Let ρ be a defining function of M and σ ∈ (0, 1). In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of W 1,2 (ρ 1−2σ , M ) ֒→ L p (∂M ), where p = 2n n−2σ . More precisely, under some assumptions on the manifold, we prove that there exists a constant B > 0 such that, for all u ∈ W 1,2 (ρ 1−2σ , M ),. This inequality is sharp in the sense that µ −1 cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, µ −1 does not depend on n and σ only, but depends on the manifold.

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