A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197-220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for...

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Autores principales:	Lederman, Claudia Beatriz, Wolanski, Noemi Irene
Publicado:	2010
Materias:	Combustion Inhomogeneous problems Monotonicity formula Singular perturbation problems
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v189_n1_p25_Lederman http://hdl.handle.net/20.500.12110/paper_03733114_v189_n1_p25_Lederman
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_03733114_v189_n1_p25_Lederman
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spelling	paper:paper_03733114_v189_n1_p25_Lederman2023-06-08T15:37:54Z A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II Lederman, Claudia Beatriz Wolanski, Noemi Irene Combustion Inhomogeneous problems Monotonicity formula Singular perturbation problems In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197-220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions u<sup/> to the singular perturbation problem and for u = lim u<sup/>, assuming that both and u<sup/> and u were defined in an arbitrary domain D in ℝN+1. In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while u<sup/> are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u0 of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u0 in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport. © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2009. Fil:Lederman, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2010 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v189_n1_p25_Lederman http://hdl.handle.net/20.500.12110/paper_03733114_v189_n1_p25_Lederman
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Combustion Inhomogeneous problems Monotonicity formula Singular perturbation problems
spellingShingle	Combustion Inhomogeneous problems Monotonicity formula Singular perturbation problems Lederman, Claudia Beatriz Wolanski, Noemi Irene A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
topic_facet	Combustion Inhomogeneous problems Monotonicity formula Singular perturbation problems
description	In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197-220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions u<sup/> to the singular perturbation problem and for u = lim u<sup/>, assuming that both and u<sup/> and u were defined in an arbitrary domain D in ℝN+1. In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while u<sup/> are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u0 of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u0 in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport. © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2009.
author	Lederman, Claudia Beatriz Wolanski, Noemi Irene
author_facet	Lederman, Claudia Beatriz Wolanski, Noemi Irene
author_sort	Lederman, Claudia Beatriz
title	A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
title_short	A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
title_full	A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
title_fullStr	A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
title_full_unstemmed	A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
title_sort	local monotonicity formula for an inhomogeneous singular perturbation problem and applications: part ii
publishDate	2010
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v189_n1_p25_Lederman http://hdl.handle.net/20.500.12110/paper_03733114_v189_n1_p25_Lederman
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A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

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