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Abstract:

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 to the singular perturbation problem and for u = lim u, assuming that both and u 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 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.

Registro:

Documento: Artículo
Título:A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II
Autor:Lederman, C.; Wolanski, N.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Combustion; Inhomogeneous problems; Monotonicity formula; Singular perturbation problems
Año:2010
Volumen:189
Número:1
Página de inicio:25
Página de fin:46
DOI: http://dx.doi.org/10.1007/s10231-009-0099-4
Título revista:Annali di Matematica Pura ed Applicata
Título revista abreviado:Ann. Mat. Pura Appl.
ISSN:03733114
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v189_n1_p25_Lederman

Referencias:

  • Berestycki, H., Caffarelli, L.A., Nirenberg, L., Uniform estimates for regularization of free boundary problems (1990) Analysis and Partial Differential Equations. Lecture Notes in Pure and Applied Mathematics, 122, pp. 567-619. , In: Sadosky, C. (ed.), Marcel Dekker, New York
  • Buckmaster, J.D., Ludford, G.S.S., (1982) Theory of Laminar Flames, , Cambridge: Cambridge University Press
  • Caffarelli, L.A., A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz (1989) Comm. Pure Appl. Math., 42, pp. 55-78
  • Caffarelli, L.A., Lederman, C., Wolanski, N., Uniform estimates and limits for a two phase parabolic singular perturbation problem (1997) Indiana Univ. Math. J., 46 (2), pp. 453-490
  • Evans, L., Gariepy, R., (1992) Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, , Boca Raton: CRC Press
  • Lederman, C., Wolanski, N., Singular perturbation in a nonlocal diffusion model (2006) Commun. PDE, 31 (2), pp. 195-241
  • Lederman, C., Wolanski, N., A local monotonicity formula for an inhomogeneous singular perturbation problem and applications (2008) Ann. Math. Pura Appl., 187 (2), pp. 197-220
  • Lederman, C., Wolanski, N., A two phase elliptic singular perturbation problem with a forcing term (2006) J. Math. Pures Appl., 86, pp. 552-589
  • Vazquez, J.L., The free boundary problem for the heat equation with fixed gradient condition (1996) Free Boundary Problems, Theory and Applications (Zakopane, 1995), Pitman Res. Notes Math. Ser., 363, pp. 277-302. , In: Niezgódka, M., Strzelecki, P. (eds.), Longman, Harlow
  • Weiss, G.S., A singular limit arising in combustion theory: Fine properties of the free boundary (2003) Calc. Var. Partial Differ. Equ., 17 (3), pp. 311-340

Citas:

---------- APA ----------
Lederman, C. & Wolanski, N. (2010) . A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II. Annali di Matematica Pura ed Applicata, 189(1), 25-46.
http://dx.doi.org/10.1007/s10231-009-0099-4
---------- CHICAGO ----------
Lederman, C., Wolanski, N. "A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II" . Annali di Matematica Pura ed Applicata 189, no. 1 (2010) : 25-46.
http://dx.doi.org/10.1007/s10231-009-0099-4
---------- MLA ----------
Lederman, C., Wolanski, N. "A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II" . Annali di Matematica Pura ed Applicata, vol. 189, no. 1, 2010, pp. 25-46.
http://dx.doi.org/10.1007/s10231-009-0099-4
---------- VANCOUVER ----------
Lederman, C., Wolanski, N. A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II. Ann. Mat. Pura Appl. 2010;189(1):25-46.
http://dx.doi.org/10.1007/s10231-009-0099-4