On inequalities of Friedrichs and Babuška-Aziz in dimension three
type: Item
instance of: scholarly article
title: On inequalities of Friedrichs and Babuška-Aziz in dimension three
zbMATH Open document ID: 0957.35121
DOI: 10.4171/ZAA/854
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2001-03-25T00:00:00Z
review text: The author extends Friedrichs inequality for conjugate harmonic functions to functions of three independent variables replacing conjugate harmonic functions by pairs of functions \(p\) and vector-valued functions \(q= (q_1,q_2,q_3)\) satisfying the equations \[ \text{rot }q= -\nabla p\quad\text{and}\quad \text{div }q= 0.\tag{1} \] It is shown that for a bounded and simply-connected open domain \(\Omega\) with boundary in the class \(C^2\) there exist constants \(\Gamma\) and \(\widetilde\Gamma\) depending only on the shape of \(\Omega\) such that for any pairs of functions \(p\), \(q\) satisfying (1) in the distribution sense the following inequalities hold: \[ \int_\Omega p^2dV\leq \Gamma \int_\Omega|q|^2dV\quad\text{provided that }\int_\Omega p dV= 0, \] and \[ \int_\Omega|q|^2dV\leq \widetilde\Gamma \int_\Omega p^2dV\quad\text{provided that }q\cdot n= 0\quad\text{on }\partial\Omega. \] The optimal constants \(\Gamma\) and \(\widetilde\Gamma\) are related by \[ \Gamma= C-1\quad\text{and }\widetilde\Gamma= \widetilde C- 1 \] to the optimal constants \(C\) and \(\widetilde C\) in the inequalities \[ D(w)\leq Cd(w)\quad\text{for all }w\in W\oplus \text{Ker rot} \] and \[ D(w)\leq \widetilde Cr(w)\quad\text{for all }w\in W\oplus \text{Ker div}, \] where \[ D(w,\widetilde w)= d(w,\widetilde w)+ r(w,\widetilde w), \] \[ d(w,\widetilde w)= \int_\Omega \text{div }w\text{ div }\widetilde w dV\quad\text{and}\quad r(w,\widetilde w)= \int_\Omega \text{rot }w\cdot\text{rot }\widetilde w dV, \] and \(W\) is the subspace in the decomposition \[ H^1_0(\Omega)^3= \text{Ker div}\oplus \text{Ker rot}\oplus W. \] It is also shown how these optimal constants \(\Gamma\), \(\widetilde\Gamma\), \(C\) and \(\widetilde C\) are connected with the ones in related eigenvalue problems and Stokes equations.
reviewed by: Hiroki Tanabe
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 35P99
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 74B05
zbMATH DE Number: 1270165
zbMATH Keywords: functional inequalities
zbMATH Keywords: linear elasticity
zbMATH Keywords: Babushka-Azis inequality
zbMATH Keywords: Friedrichs inequality
zbMATH Keywords: optimal constants
zbMATH Keywords: eigenvalue problems
zbMATH Keywords: Stokes equations
author: Waldemar Velte
MaRDI profile type: Publication
cites work: Q3993273
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cites work: On inequalities of Friedrichs and Babuška-Aziz
description: scientific article; zbMATH DE number 1270165