Gagliardo–Nirenberg–Sobolev inequalities for convex domains in $\mathbb{R}^d$

• DOI: 10.4310/mrl.2019.v26.n5.a3
• Publication Date: 2019-11-27T17:19:06Z
• Authors: Rafael D. Benguria; Cristobal Vallejos; Hanne Van Den Bosch
• Keywords: Mathematics - Analysis of PDEs; FOS: Mathematics; FOS: Physical sciences; Mathematical Physics (math-ph); 0101 mathematics; 01 natural sciences; Mathematical Physics; Analysis of PDEs (math.AP)

A special type of Gagliardo-Nirenberg-Sobolev (GNS) inequalities in $\mathbb{R}^d$ has played a key role in several proofs of Lieb-Thirring inequalities. Recently, a need for GNS inequalities in convex domains of $\mathbb{R}^d$, in particular for cubes, has arised. The purpose of this manuscript is two-fold. First we prove a GNS inequality for convex domains, with explicit constants which depend on the geometry of the domain. Later, using the discrete version of Rumin's method, we prove GNS inequalities on cubes with improved constants.<br/>15 pages, v2: typo in eq(9), p.3 fixed<br/>