This work aims to go in-depth in the study of Rayleigh-Faber-Krahn inequality and its proof. This inequality solves the shape optimization problem for the first Dirichlet eigenvalue under a volume constraint. Its proof will be addressed using Rayleigh quotient and Pólya-Szegö inequality, that need from Sobolev spaces to be rigorously understood. Besides, there is an enormous attention to applications of Rayleigh-Faber-Krahn inequality. Specifically, branches such as music, finance, fluids' transport, and quantum mechanics will be studied. Finally, some concrete calculus will be seen: the wave equation will be compared in 2 dimensions and in n dimensions over a generic parallelepiped and a ball through MATLAB. With the several examples and the concrete calculus, we aim to determine which domain has the smallest first Dirichlet eigenvalue among various of the same volume, and hence, motivate the Rayleigh-Faber-Krahn inequality.
