Complutense University Library

Isoperimetric-inequalities in the parabolic obstacle problems

Díaz Díaz, Jesús Ildefonso and Mossino, J. (1992) Isoperimetric-inequalities in the parabolic obstacle problems. Journal de Mathématiques Pures et Appliquées, 71 (3). pp. 233-266. ISSN 0021-7824

[img] PDF
Restricted to Repository staff only until 31 December 2020.

894kB

Official URL: http://cat.inist.fr/?aModele=afficheN&cpsidt=3798149

View download statistics for this eprint

==>>> Export to other formats

Abstract

In this paper, we are concerned with the parabolic obstacle problem (u(t)=partial derivative u/partial derivative t [GRAPHICS] (A is a linear second order elliptic operator in divergence form or a nonlinear "pseudo-Laplacian"). We give an isoperimetric inequality for the concentration of u - psi around its maximum. Various consequences are given. In particular, it is proved that u - psi vanishes after a finite time, under a suitable assumption on psi(t) + A-psi + c- psi - f. Other applications are also given. These results are deduced from the study of the particular case psi=0. In this case, we prove that, among all linear second order elliptic operators A, having ellipticity constant 1, all equimeasurable domains OMEGA, all equimeasurable functions f and u0, the choice giving the "most concentrated" solution around its maximum is: A = -DELTA, OMEGA is a ball OMEGA, f and u0 are radially symmetric and decreasing along thc radii of OMEGA. A crucial point in our proof is a pointwise comparison result for an auxiliary one-dimensional unilateral problem. This is carried out by showing that this new problem is well-posed in L(infinity) in the sense of the accretive operators theory.


Item Type:Article
Uncontrolled Keywords:Stefan problem; rearrangement; parabolic obstacle problems; isoperimetric inequalities; comparison by rearrangement; accretive operators; extinction in finite time
Subjects:Sciences > Mathematics > Differential geometry
Sciences > Mathematics > Differential equations
ID Code:16260
References:

A. ALVINO, P. L. LIONS, G. TROMBETTI,Comparaison des solutions d'équations paraboliques et elliptiques par symétrisation,une méthode nouvelle,C.R.Acad.Sci.Paris,303, Series 1,1986, pp. 975-978.

C. BANDLE, Isoperimetric Inequalities and Applications, Pitman Advanced Publishing Program, Boston,London, Melbourne, 1980.

C. BANDLE, J. MOSSINO, Rearrangement in Variational Inequalities, Annali di Mat.Pura et Applicata,(IV), CXXXVIlI, 1984, pp. 1-14; see also C.R.Acad.Sci.Paris,296, Series J, 1983, pp. 501-504.

C. BANDLE, I. STAKGOLD, Isoperimetric Inequalities for the Effectiveness in Semilinear Parabolic Equations, I.S.N.M, 71, 1984, pp. 289-295.

P. BENILAN, Équations d'évolution dans un espace de Banach quelconque et applications, Doctoral Thesis,Orsay, 1972.

P. BENILAN, M. G. CRANDALL, A. PAZY, Nonlinear Evolution Equations Governed by Accretive Operators (to appear).

P. BENILAN, P. ABORJAILY (to appear).

A. BENSOUSSAN, J. L. LIONS, On the Support of the Solution of some Variational Inequalities of Evolution,J.Math.Soc. Japan,28, No. 1, 1976, pp. 1-17.

H. BREZIS,Problemes unilatéraux,J.Math.Pures Appl.,51,1972, pp. 1-168.

H. BREZIS, Opérateurs Maximaux Monotones et Semigroupes de Contraction dans les Espaces de Hi1bert,North-Holland, Amsterdam, 1973.

H. BREZIS, Monotone Operators, Nonlinear Semigroups and Applications, Proc. Int Congress Math.,Vancouver, 1974.

H. BREZIS, A. FRIEDMAN,Estimates on the Support of Solutions of Parabolic Variationa1 Inequalities III,J. Math.,20,1976, pp. 82-99.

J. I. DÍAZ, Non1inear Partia1 Differentia1 Equations and Free Boundaries, I, Elliptic Equations, Pitman Advanced Publisiling Program, Boston, London, Melboume, 1985.

J. I. DÍAZ, Anu1acion de soluciones para operadores acretivos en espacios de Banach, Rev. Real Acad.Cienc. Exact Fis. Natur. Madrid, 74, 1980, pp. 865-880.

J. I. DÍAZ, J. MOSSINO, Inégalité isopérimétrique dans un probleme d'obstacle parabo1ique, C. R. Acad.Sci. Paris, 305, Series I, 1987, pp. 737-740.

B. GUSTAFSSON, J. MOSSINO, Quelques inégalités isopérimétriques pour le prob1eme de Stefan,C.R.Acad.Sci. Paris,305,Series 1, 1987, pp. 669-672;see also Isoperimetric Inequalities for the Stefan Prob1em,S.IA.M. J. Math. Anal., 20, No. 5, 1989, pp. 1095-1108.

G. S. LADDLE, V. LAKSHMIKANTHAM, A. S. VATSALA, Monotone Iterative Techniques for Non1inear Differential Equations, Monographs Appl. Math., 27, Pitman, 1985.

O. A.LADYZHENSKAYA,N.N.URAL'TSEVA, Linear and Quasilinear Elliptic Equations, Academic Press,New York, 1968.

C. MADERNA, S. SALSA, Some Specia1 Properties of Solutions to Obstacle Problems, Rendiconti del Seminario Matematico dell’Universita di Padova, 71, 1984, pp. 121-129.

J, MOSSINO, Inéga1ités Isopérimétriques et App1ications en Physique, Hermann, Paris, 1984.

J, MOSSINO, J. M. RAKOTOSON, Isoperimetric Inequalities in Parabolic Equations, Ann. Scuo. Norm. Sup.Pisa, Cl. Sci. IV, XIII, No. 1, 1986, pp. 51-73.

J. MOSSINO, R. TEMAM, Directional Derivative of the Increasing Rearrangement Mapping, and Application to a Queer Differential Equation in Plasma Physics, Duke Math. J., 48, 1981, pp. 475-495.

C. PICARD, Opérateurs  -accrétifs et génération de semi-groupes non linéaires, C. R. Acad. Sci. Paris,275, Series A, 1972, pp. 639-641.

G. POLYA, C. SZEGO, Isoperimetric Inequalities in Mathematica1 Physics, Princeton University Press, 1951.

K. SATO, On the Generators of non-Negative Contraction Semi-Groups in Banach Lattices, J. Math. Soc. Japan, 20, 1968, pp. 423-436.

J. L. VÁZQUEZ, Symétrisation pour u,=Δ(u) et applications C. R. Acad. Sci. Paris, 295, Series 1, 1982,pp. 71-74; and 296, 1983, p. 455.

Deposited On:10 Sep 2012 09:08
Last Modified:07 Feb 2014 09:26

Repository Staff Only: item control page