Rearrangements and convexity of level sets in PDE.

*(English)*Zbl 0593.35002
Lecture Notes in Mathematics. 1150. Berlin etc.: Springer-Verlag. V, 136 p. DM 21.50 (1985).

The first part of this work is a survey on rearrangement technique and its applications to partial differential equations and the calculus of variations. The author studies in detail the main types of rearrangements for functions of one and several variables and uses the general theory to prove convexity or symmetry properties of solutions of differential equations. The applications include the treatment of some free boundary problems such as the dam problem, the capacitary problem, the theory of jets and cavities.

The second part of the book concerns the applications of the maximum principle to obtain the same type of results. The work contains several original contributions of the author and solves many interesting problems and conjectures on the shape of free boundary of elliptic variational inequalities.

The second part of the book concerns the applications of the maximum principle to obtain the same type of results. The work contains several original contributions of the author and solves many interesting problems and conjectures on the shape of free boundary of elliptic variational inequalities.

Reviewer: V.Barbu

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35R35 | Free boundary problems for PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

26B25 | Convexity of real functions of several variables, generalizations |

26D10 | Inequalities involving derivatives and differential and integral operators |

35A15 | Variational methods applied to PDEs |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35B50 | Maximum principles in context of PDEs |

35J20 | Variational methods for second-order elliptic equations |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

49R50 | Variational methods for eigenvalues of operators (MSC2000) |