On the stability of some isoperimetric inequalities for the fundamental   tones of free plates
  We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger's argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem. 
Davide Buoso, L. Mercredi Chasman, Luigi Provenzano
math.SP math.OC