Improved Poincaré inequalities and solutions of the divergence in weighted norms

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The improved Poincaré inequality ||φ-φΩ||Lp(Ω)≤C||d∇φ||Lp(Ω) Where Ω ⊂ Rn is a bounded domain and d(x) is the distance from x to the boundary of Ω, has many applications. In particular, it can be used to obtain a decomposition of functions with vanishing integral into a sum of locally supported func...

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Autores principales:	Acosta Rodriguez, Gabriel, Duran, Ricardo Guillermo
Publicado:	2017
Materias:	Divergence operator Poincaré inequalities Weights
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1239629X_v42_n1_p211_Acosta http://hdl.handle.net/20.500.12110/paper_1239629X_v42_n1_p211_Acosta
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_1239629X_v42_n1_p211_Acosta
record_format	dspace
spelling	paper:paper_1239629X_v42_n1_p211_Acosta2023-06-08T16:10:12Z Improved Poincaré inequalities and solutions of the divergence in weighted norms Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Divergence operator Poincaré inequalities Weights The improved Poincaré inequality \|\|φ-φΩ\|\|Lp(Ω)≤C\|\|d∇φ\|\|Lp(Ω) Where Ω ⊂ Rn is a bounded domain and d(x) is the distance from x to the boundary of Ω, has many applications. In particular, it can be used to obtain a decomposition of functions with vanishing integral into a sum of locally supported functions with the same property. Consequently, it can be used to go from local to global results, i.e., to extend to very general bounded domains results which are known for cubes. For example, this methodology can be used to prove the existence of solutions of the divergence in Sobolev spaces. The goal of this paper is to analyze the generalization of these results to the case of weighted norms. When the weight is in Ap the arguments used in the un-weighted case can be extended without great difficulty. However, we will show that the improved Poincaré inequality, as well as its above mentioned applications, can be extended to a more general class of weights. Fil:Acosta, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Durán, R.G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1239629X_v42_n1_p211_Acosta http://hdl.handle.net/20.500.12110/paper_1239629X_v42_n1_p211_Acosta
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Divergence operator Poincaré inequalities Weights
spellingShingle	Divergence operator Poincaré inequalities Weights Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo Improved Poincaré inequalities and solutions of the divergence in weighted norms
topic_facet	Divergence operator Poincaré inequalities Weights
description	The improved Poincaré inequality \|\|φ-φΩ\|\|Lp(Ω)≤C\|\|d∇φ\|\|Lp(Ω) Where Ω ⊂ Rn is a bounded domain and d(x) is the distance from x to the boundary of Ω, has many applications. In particular, it can be used to obtain a decomposition of functions with vanishing integral into a sum of locally supported functions with the same property. Consequently, it can be used to go from local to global results, i.e., to extend to very general bounded domains results which are known for cubes. For example, this methodology can be used to prove the existence of solutions of the divergence in Sobolev spaces. The goal of this paper is to analyze the generalization of these results to the case of weighted norms. When the weight is in Ap the arguments used in the un-weighted case can be extended without great difficulty. However, we will show that the improved Poincaré inequality, as well as its above mentioned applications, can be extended to a more general class of weights.
author	Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo
author_facet	Acosta Rodriguez, Gabriel Duran, Ricardo Guillermo
author_sort	Acosta Rodriguez, Gabriel
title	Improved Poincaré inequalities and solutions of the divergence in weighted norms
title_short	Improved Poincaré inequalities and solutions of the divergence in weighted norms
title_full	Improved Poincaré inequalities and solutions of the divergence in weighted norms
title_fullStr	Improved Poincaré inequalities and solutions of the divergence in weighted norms
title_full_unstemmed	Improved Poincaré inequalities and solutions of the divergence in weighted norms
title_sort	improved poincaré inequalities and solutions of the divergence in weighted norms
publishDate	2017
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1239629X_v42_n1_p211_Acosta http://hdl.handle.net/20.500.12110/paper_1239629X_v42_n1_p211_Acosta
work_keys_str_mv	AT acostarodriguezgabriel improvedpoincareinequalitiesandsolutionsofthedivergenceinweightednorms AT duranricardoguillermo improvedpoincareinequalitiesandsolutionsofthedivergenceinweightednorms
_version_	1768544287424774144

Improved Poincaré inequalities and solutions of the divergence in weighted norms

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