Resumen de Optimal conditions for anti-maximum principles

Hans-Christoph Grunau, Guido Sweers

• 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.