Díaz Díaz, Jesús Ildefonso and Kawohl, B.
(1993)
*On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings.*
Journal of Mathematical Analysis and Applications, 177
(1).
pp. 263-286.
ISSN 0022-247X

PDF
Restricted to Repository staff only until 31 December 2020. 681kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X83712576

## Abstract

"We consider some degenerate parabolic problems on a convex (or starshaped) ring. We prove that if the initial data have convex (or starshaped) level sets, then the solution u(t,⋅) has the same property for any positive t. Similar results are shown for the corresponding stationary problems. Our results imply in particular the convexity (or starshapedness) of certain free boundaries. Other nonlinear parabolic problems are also discussed.''

Item Type: | Article |
---|---|

Uncontrolled Keywords: | porous-medium equation; geometrical properties; obstacle problem; free-boundary; diffusion; stabilization; continuity; regularity; concavity; support |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 16178 |

References: | H W. ALT AND S. LUCKHAUS, Quasilinear elliptic-parabolic differential equations, Mth. Z.183(1983),311-341. R ARIS. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,"Vol. I and II. Clarendon, Oxford, 1975. C. BANDLE AND I. STAKGOLD. The formation of the dead core in parabolic reaction diffusion problems, Trans. Amer. Math. Soc. 286 (1984),275-293. M. BARDl. Asymptotic spherical symmetry of the free boundary in degenerate diffusion equations, Ann. Math. Pura Appl. (4) 148 (1987), 117-130. P. BENILAN, AND J. L. VAZQUEZ, Concavity of solutions of the porous medium equation, Trans. Amer. Math. Soc. 299 (1987), 81--93. M. BERTSCH, "Nonlinear DilTusion Problems: The Large Time Behaviour," Thesis. Leiden, 1983. C. BORELL, Brownian motion in a convex ring and quasiconcavity, "Comm. Math. Phys. 86 (1982), 143-147. H. J. BRASCAMP AND E. H. LlEB, On extensions af the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log-concave functions, and with applications lo the diffusion equation, J. Funct. Anal. 22 (1976), 366-389. H. BREZIS, "Opérateurs maximaux mono tones et semi-groupes de contractions dans les espaces de Hilbert," North-Holland, Amsterdam, 1973. L. A. CAFFARELLl AND J. SPRUCK, Convexity properties of solutions of some classical variational problems, Comm. Partial Differential Equations 7 (1982), 1337-1379. L. A. CAFFARELLI. J. L. VAZQUEZ, AND N. l. WOLANSKl, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J. 36 (1987), 373-401. A. DAMLAMIAN, Some results on the multi-phase Stefan problem, Comm. Partial Differential Equatons 2 (1977),1017-1044. J. I. DÍAZ, Elliptic and parabolic quasilinear equations giving rise to a free boundary: The boundary of the support of the solutions, Proc. Symp. Pure Mth. 45, Part I (1986). 381-393. J. I. DÍAZ, "Nonlinear Partial Differential Equations and Free Boundaries. Vol. I Elliptic Equations," Pitman Rcsearch Notes in Math., Vol. 106, London. Pitman. 1985. J.I.DÍAZ AND J.HERNÁNDEZ, Some results on the existence of free boundarics for parabolic reaction diffusion systems, in"Trends in Theory and Practice of Nonlinear Differential Equations"(Y.Lakshmikantham,Ed.,pp.149-156.Dekker.Ncw York. 1984. J. I. DÍAZ AND J. HERNÁNDEZ,Qualitativc propertics of free boundaries for some nonlinear degenerate parabolic equations, in Nonlinear Parabolic Equations: Qualitative Properties of Solutions"(L.Boccardo and A.Tesei, Eds.), pp. 85--93. Longman. London. 1987. J. I. DÍAZ AND M. A. HERRERO, Estimates of the Support of the solution of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc Edinburg 89 (1981), 249-258. J. I. DÍAZ AND B, KAWOHL, Convexity and starshapedness of level sets for some nonlinear parabolic problems, in "Free Boundary Problems, Theory and Applicalions" (K. Hoffman and J. Sprekels, Eds.), Pitman Res. Notes in Math., Vol. 186, pp. 883-887,Pitman, London, 2990. J. I. DÍAZ AND F. DE THELlN, On a nonlinear parabolic problem arising in some models related to turbulent flows, to appear in SIAM Math. Anal. E. A. DI BENEDETTO, A boundary modulus of continuity for a class of singular parabolic equations, J. Differentia/ Equations 63 (1986), 418-447. E. DI BENEDETTO AND A. FRIEDMAN, Regularity of solutions of non-linear degenerate parabolic systems, J. Reine Angew. Mth. 349 (1984), 83-128. A. EL HACHIMI AND F. DE THELlN, Supersolutions and stabilization of the solutions of the equation ut-pu=f(x, u), Nonlinear Anal. TMA 12 (1988), 1385-1398. A. FRIEDMAN ANO D, KINDERLEHRER, A one phase Stefan problem, Indiana Unv, Math. J. 24 (1975), 1005-1035. R. GABRIEL, A result concerning convex level surfaces of 3-dimensional harmonic functions, J. London Math. Soc. 32 (1957), 286-294. M. E. GURTlN AND R. C. MCCAMY, On the diffusion of biological populations, Math.Biosci. 33 (1977), 35-49. J. HULSHOF AND N. WOLANSKY, Monotone flows in N-dimensional partially saturated porous media, Arch. Rational Mech. Anal. 102 (1988), 287-305. B. KAWOHL, Starshapedness of level sets for the obstacle problem and for the capacitory potential problem, Proc. Amer. Matih. Soc. 89 (1983), 637-640. B. KAWOHL, Geometrical properties of level sets of solutions to elliptic problems, Proc. Symp. Pure Mah. 45, Part 2 (1986), 25-36. S, KAWOHL, "Rearrangements and Convexity of Level Sets in PDE," Springer Lecture Notes in Math., Vol. 1150, Springer, Heidelberg, 1985. B. KAWOHL, Qualitative properties of solutions to semilinear heat equations, Exposition. Math. 4 (1986). 257-270. G. KEADY, The persistence of log-concavity for positive solutions of the one-dimensional hear equation, J. Austral. Math. Soc. Ser. A 48 (1990), 1-16. G. KEADY AND I. STAKGOLD, Some geometric properties of solids in combustion, in "Geometry of Solutions lo Partial Differential Equations" (G. Talenti. Ed.), Symposia Mathcmatica, Vol. XXX, pp. 137-151, Academic Press, London, 1989. A. U. KENNINGTON, Convexity of level curves for an initial value problem, J. Math Anal. Appl.. 133 (1988), 324-330. N. KOREVAAR, Convex solutions to nonlinear elliptic and parabolic boundary problems. Indiana Univ. Math. J. 32 (1983), 603-614. O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, AND N. N. URAL'TSEVA, "Linear and Quasilinear Equations of Parabolic Type," Transl. of' Math, Monographs, Vol. 23, Amer. Math. Soc., Providencc. 1968 M. LANGLAIS. AND D. PHILLIPS. Stabilization of solutionS of non-linear and degenerate evolution equations, Nonlinear Anal. 9 (1985), 321-333. J. L. LEWIS, Capacitary functions in convex rings. Arch. Rational Mech. Anal. 66 (1977).201-224. P. L. LIONS, Two geometrical properties of solutions of semi-linear problems. Appl. Anal. 12 (1981), 267-272. H. MATANO, Nonincrease of the lap number of a solutions for a one-dimensional semilinear parabolic equation. J. Fac. Sei.Univ. Tokio. Sect. IA Math¡. 29 (1982) 401-441. K. NICKEL, Gestaltaussagen über Lösungen parabolischer Differcntíalgleichungen. J. Reine Angew•. Math. 211 (1962). 78-94. G. POLYA, Qualitatives über den Wärmeausgleich. Z . Angew•. Math. Mech. 13 (1933). 125-128. S. SAKAGUCHI, Starshaped coincidence sets in the obstacle problem, Ann. Scuola Norm. Sup. Pisa (4) 11 (1984), 123-128. S. SAKAGUCHI, Coincidence sets in the obstacle problem for the p-harmonic operator. Proc. Amer. Math. Soc. 95 (1985),382-386. J. L. VAZQUEZ, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202. T. I. VOGEL, A free boundary problem arising from a galvanizing process. SIAM J. Math. Anal. 16 (1985), 970-979. M. WlEGNER, On C'-regularity of the gradient of solutions of degenerate parabolic systems, Ann. Math. Pura App/. (4) 145 ( 1986), 385-405. |

Deposited On: | 11 Sep 2012 07:28 |

Last Modified: | 06 Feb 2014 10:39 |

Repository Staff Only: item control page