On some bounds for frequencies of membranes and plates.

Abstract

The problem of optimizing a domain dependent functional, while keeping a domain’s measure (its volume, perimeter, etc.) fixed, is called an isoperimetric problem. The isoperimetric inequalities have a long history in mathematics dating back to the Greeks and Dido’s problem, when the first classical isoperimetric inequality appeared in the Euclidean geometry. With the introduction of the calculus of variations in the 17th century, the isoperimetric inequalities found their way into mathematical physics. Among the isoperimetric problems, here we propose the investigation of those linking the shape of a membrane to the sequence of its frequencies. The starting point in this research field is the Faber-Krahn inequality, which states that among all fixed membranes of given area, the first frequency is minimal for the circular membrane. As for the second frequency of fixed membranes of given area, we know that it is minimized by the disjoint union of two identical circular membranes (Krahn’s inequality). For other types of membranes several results are known, but a lot of questions remain open. In this thesis we are going to present some classical isoperimetric inequalities, as well as some universal bounds, which are not isoperimetric, but in some cases they represent the best possible bounds obtained at their time. Finally, we will present some new universal bounds we have obtained for the frequencies of clamped and buckled plates.