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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2010
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| 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 |
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| Sumario: | 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. |
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