Optimal conditions for anti-maximum principles

• Hans-Christoph Grunau ^[1] ; Guido Sweers ^[2]
  1. [1] University of Bayreuth

     University of Bayreuth

     Kreisfreie Stadt Bayreuth, Alemania

     
  2. [2] Delft University of Technology

     Delft University of Technology

     Países Bajos

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 30, Nº 3-4, 2001, págs. 499-513
• Idioma: inglés
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• Resumen

  • The resolvent for some polyharmonic boundary value problems is positive for h positive and less than the first eigenvalue. It is known that beyond this first eigenvalue a sign-reversing property exists. Such a result is called an anti-maximum principle. Depending on the boundary conditions, the dimension of the domain and the order of the operator, the result is uniform or not. In the non-uniform case the right hand side needs to be in LP(0) with p large enough. Sharp estimates for iterated Green functions are used in order to prove that such restrictions are optimal both for the non-uniform and the uniform anti-maximum principle. We will also use these estimates to give an alternative proof of the (uniform) anti-maximum principle.