Bounds on resonant frequencies of vibrational systems

The Laplacian is a very important operator with many applications. First we show its importance in the musical theory. Then we define the self-adjoint Dirichlet Laplacian on bounded domains using the quadratic forms and state some of his spectral properties. Applying the min-max principle and using the shrinking or parallel coordinates we obtain two upper bounds for the first eigenvalue of the Dirichlet Laplacian on simply-connected domains. Moreover we introduce our own upper bound for particular, not simply-connected domains. Finally we apply the obtained bounds to some special shapes of simply connected domains, compare them and subsequently we show examples of behavior of our bound on the particular, not simply-connected domains created from the domains introduced before.