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.
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Acknowledgments. The first and the third author wish to thank Berardo Ruffini for discussions on his paper . The first author has been partially supported by the research project FIR (Futuro in Ricerca) 2013 ‘Geometrical and qualitative aspects of PDE’s’. The third author acknowledges financial support from the research project ‘Singular perturbation problems for differential operators’ Pro-getto di Ateneo of the University of Padova and from the research project ‘IN-dAM GNAMPA Project 2015 – Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione’. The first and the third author are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
- Biharmonic operator
- Neumann boundary conditions
- Quantitative isoperimetric inequality
- Steklov boundary conditions