A local monotonicity formula for an inhomogenous singular perturbation problem and applications
In this paper we prove a local monotonicity formula for solutions to an inhomogeneous singularly perturbed diffusion problem of interest in combustion. This type of monotonicity formula has proved to be very useful for the study of the regularity of limits u of solutions of the singular perturbation...
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2008
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| 100 | 1 | |a Lederman, C. | |
| 245 | 1 | 2 | |a A local monotonicity formula for an inhomogenous singular perturbation problem and applications |
| 260 | |c 2008 | ||
| 270 | 1 | 0 | |m Lederman, C.; Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina; email: clederma@dm.uba.ar |
| 506 | |2 openaire |e Política editorial | ||
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| 504 | |a Buckmaster, J.D., Ludford, G.S.S., (1982) Theory of Laminar Flames, , Cambridge University Press Cambridge | ||
| 504 | |a Caffarelli, L.A., A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C 1,α (1987) Rev. Matem. Iberoamericana, 3, pp. 139-162. , 2 | ||
| 504 | |a Caffarelli, L.A., A Harnack inequality approach to the regularity of free boundaries (1989) Part II: Flat Free Boundaries Are Lipschitz. Comm. Pure Appl. Math., 42, pp. 55-78 | ||
| 504 | |a Caffarelli, L.A., Uniform Lipschitz regularity of a singular perturbation problem (1995) Diff. Int. Eqs., 8, pp. 1585-1590. , 7 | ||
| 504 | |a Caffarelli, L.A., Jerison, D., Kenig, C.E., Global energy minimizers for free boundary problems and full regularity in three dimensions. Noncompact problems at the intersection of geometry, analysis, and topology (2004) Contemp. Math., 350, pp. 83-97. , Amer. Math. Soc., Providence | ||
| 504 | |a Caffarelli, L.A., Kenig, C., Gradient estimates for variable coefficient parabolic equations and singular perturbation problems (1998) Amer. J. Math., 120, pp. 391-439. , 2 | ||
| 504 | |a 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, pp. 453-490. , 2 | ||
| 504 | |a Caffarelli, L.A., Petrosyan, A., Shahgholian, H., Regularity of a free boundary in parabolic potential theory (2004) J. Am. Math. Soc., 17, pp. 827-869. , 4 | ||
| 504 | |a Caffarelli, L.A., Vazquez, J.L., A free boundary problem for the heat equation arising in flame propagation (1995) Trans. Amer. Math. Soc., 347, pp. 411-441 | ||
| 504 | |a Fife, P., Some Nonclassical Trends in Parabolic and Parabolic-Like Evolutions (2003) Trends in Nonlinear Analysis, pp. 153-191. , Springer, Berlin | ||
| 504 | |a Lederman, C., Wolanski, N., Singular perturbation in a nonlocal diffusion problem (2006) Commun. PDE, 31, pp. 195-241. , 2 | ||
| 504 | |a Lederman, C., Wolanski, N., A two phase elliptic singular perturbation problem with a forcing term (2006) J. Math. Pures. Appl., 86, pp. 552-589. , 6 | ||
| 504 | |a Vázquez, J.L., The free boundary problem for the heat equation with fixed gradient condition (1995) Proceedings International Conference on Free Boundary Problems and Applications, , Zakopane, Poland | ||
| 504 | |a Weiss, G.S., Partial regularity for weak solutions of an elliptic free boundary problem (1988) Comm. Partial Diff. Eqs., 23, pp. 439-455. , 3,4 | ||
| 504 | |a Weiss, G.S., A homogeneity property improvement approach to the obstacle problem (1999) Invent. Math., 138, pp. 23-50 | ||
| 504 | |a Weiss, G.S., A singular limit arising in combustion theory: Fine properties of the free boundary (2003) Calc. Variations Partial Diff. Eqs., 17, pp. 311-340. , 3 | ||
| 520 | 3 | |a In this paper we prove a local monotonicity formula for solutions to an inhomogeneous singularly perturbed diffusion problem of interest in combustion. This type of monotonicity formula has proved to be very useful for the study of the regularity of limits u of solutions of the singular perturbation problem and of {u > 0}, in the global homogeneous case. As a consequence of this formula we prove that u has an asymptotic development at every point in {u > 0} where there is a nonhorizontal tangent ball. These kind of developments have been essential for the proof of the regularity of {u > 0} for Bernoulli and Stefan free boundary problems. We also present applications of our results to the study of the regularity of {u > 0} in the stationary case including, in particular, its regularity in the case of energy minimizers. We present as well a regularity result for traveling waves of a combustion model that relies on our monotonicity formula and its consequences.The fact that our results hold for the inhomogeneous problem allows a very wide applicability. Indeed, they may be applied to problems with nonlocal diffusion and/or transport. © 2007 Springer-Verlag. |l eng | |
| 593 | |a Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina | ||
| 650 | 1 | 7 | |2 spines |a COMBUSTION |
| 690 | 1 | 0 | |a INHOMOGENEOUS PROBLEMS |
| 690 | 1 | 0 | |a MONOTONICITY FORMULA |
| 690 | 1 | 0 | |a SINGULAR PERTURBATION PROBLEMS |
| 700 | 1 | |a Wolanski, N. | |
| 773 | 0 | |d 2008 |g v. 187 |h pp. 197-220 |k n. 2 |p Ann. Mat. Pura Appl. |x 03733114 |w (AR-BaUEN)CENRE-1530 |t Annali di Matematica Pura ed Applicata | |
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