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

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 arisen. 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.

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The work of R.B. has been supported by Fondecyt (Chile) Projects # 116–0856 and #114-1155. The work of C.V. has been supported by a “Beca Presidente de la República” (Chile) fellowship and by a “Beca Padre Hurtado” (PUC) fellowship. The work of H. VDB. has been partially supported by CONICYT (Chile) (PCI) project REDI170157 and partially by Millennium Nucleus “Center for Analysis of PDE” NC130017.

Received 5 February 2018

Accepted 19 June 2018

Published 27 November 2019