Díaz Díaz, Jesús Ildefonso and Padial Molina, Juan Francisco and Rakotoson, Jean Michel Theresien
(2007)
*On some Bernoulli free boundary type problems for general elliptic operators.*
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 137
(5).
pp. 895-911.
ISSN 0308-2105

PDF
Restringido a Repository staff only hasta 31 December 2020. 198kB |

Official URL: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1199064

## Abstract

We consider some Bernoulli free boundary type problems for a general class of quasilinear elliptic (pseudomonotone) operators involving measures depending on unknown solutions. We treat those problems by applying the Ambrosetti-Rabinowitz minimax theorem to a sequence of approximate nonsingular problems and passing to the limit by some a priori estimates. We show, by means of some capacity results, that sometimes the measures are regular. Finally, we give some qualitative properties of the solutions and, for a special case, we construct a continuum of solutions.

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

Uncontrolled Keywords: | t-set; equations; existence; confinement; stellarator; regularity; datum. |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 15288 |

References: | H. W. Alt and L. A. Caffarelli. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981), 105–144. H. W. Alt, L. A. Caffarelli and A. Friedman. Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282 (1984), 431–461. H. Amann and P. Quitner. Semilinear parabolic equations involving measures and low regularity data. Trans. Am. Math. Soc. 356 (2004), 1045–1119. A. Ambrosetti and H. Rabinowitz. Dual variational methods in critical point theory and applications. J. Funct. Analysis 14 (1973), 349–381. Ph. Bénilan and H. Brezis. Nonlinear problems related to the Thomas-Fermi equation. Dedicated to Philippe Bénilan. J. Evol. Eqns 3 (2003), 673–770. Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez. An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa 22 (1995), 241–273. L. Boccardo and T. Gallouët. Nonlinear elliptic and parabolic equation involving measure as data. J. Funct. Analysis 87 (1989), 149–169. L. Boccardo, T. Gallouët and L. Orsina. Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Analysis Math. 73 (1997), 203–223. A. Beurling. On free-boundary problems for the Laplace equation. Seminars on Analytic Functions, vol. 1, pp. 248–263 (Princeton, NJ: Institute for Advanced Study, 1958). H. Brézis. Opérateurs maximaux monotones (Amsterdam: North-Holland, 1973). O. P. Bruno and P. Laurence. Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure. Commun. Pure Appl. Math. 49 (1996), 717–764. P. Cardaliaguet and R. Tahraoui. Some uniqueness results for Bernoulli interior freeboundary problems in convex domains. Electron. J. Diff. Eqns 2002 (2002), 1–16. J. I. Díaz. Nonlinear partial differential equations and free boundaries, Research Notes in Mathematics, vol. 106 (London: Pitman, 1985). J. I. Díaz and J. Hernández. Global bifurcation and continua of nonegative solutions for a quasilinear elliptic problem. C. R. Acad. Sci. Paris Sér. I 329 (1999), 587–592. J. I. Díaz and J. M. Rakotoson. On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a Stellarator geometry. Arch. Ration. Mech. Analysis 134 (1996), 53–95. J. I. Díaz, F. Padial and J. M. Rakotoson. Mathematical treatment of the magnetic confinement in a current carrying Stellarator. Nonlin. Analysis 34 (1998), 857–887. J. I. Díaz, J. F. Padial and J. M. Rakotoson. On some Bernoulli free boundary type problems without compactness conditions on the diffusion. (In preparation.) M. Flucher and M. Rumpf. Bernoulli's free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486 (1997), 165–204. A. Friedman and Y. Liu. A free boundary problem arising in magnetohydrodynamic system. Ann. Scuola Norm. Sup. Pisa IV 22 (1995), 375–448. A. Henrot. Subsolutions and supersolutions in a free boundary problem. Ark. Mat. 32 (1994), 79–90. J. L. Lions. Quelques méthodes de résolution de problèmes aux limites nonlinéaires (Paris: Dunod, 1969). J. M. Rakotoson. Properties of solutions of quasilinear equations in a T-set when the datum is a Radon measure. Indiana Univ. Math. J. 46 (1997), 247–297. J. M. Rakotoson. Propriétés qualitatives de solutions d'équation à donnée mesure dans un T-ensemble. C. R. Acad. Sci. Paris Sér. I 323 (1996), 335–340. J. M. Rakotoson. Generalized solution in a new type sets for problems with measure as data. Diff. Integ. Eqns 6 (1993), 27–36. W. Ziemer. Weakly differentiable functions (Springer, 1989). |

Deposited On: | 21 May 2012 10:00 |

Last Modified: | 06 Feb 2014 10:20 |

Repository Staff Only: item control page