Showing papers in "Mathematical Inequalities & Applications in 2023"
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TL;DR: Based on an apparently new Lagrange-type identity, a Cauchy-Schwarz-type inequality was proved in this article , which was obtained by using certain macro variables.
Abstract: Based on an apparently new Lagrange-type identity, a Cauchy--Schwarz-type inequality is proved. The mentioned identity is obtained by using certain ``macro'' variables; it is hoped that such a method can be used to prove or produce other identities and inequalities.
TL;DR: In this paper , the authors define the trace on a domain which is definable in an ominimal structure and show that every function vanishing on the boundary in the trace sense satisfies Poincar\'e inequality.
Abstract: We first define the trace on a domain $\Omega$ which is definable in an o-minimal structure. We then show that every function $u\in W^{1,p}(\Omega)$ vanishing on the boundary in the trace sense satisfies Poincar\'e inequality. We finally show, given a definable family of domains $(\Omega_t)_{t\in \mathbb{R}^k}$, that the constant of this inequality remains bounded, if so does the volume of $\Omega_t$.