Analysis of the Spectrum of the Direchlet Laplacian on domains in Euclidean space

Specialeforsvar ved Børge Laursen

Titel: Analysis of the Spectrum of the Dirichlet Laplacian on domains in Euclidean space

Abstract: In this thesis we study the spectrum of the Dirichlet Laplacian on open, bounded domains in Euclidean space. We start out by proving that the spectrum of the Dirichlet Laplacian on open, bounded domains in Euclidean space is purely discrete. Afterwards, we have three main results of different characters. Our first main result is Weyl’s law for open, bounded, Jordan contented domains in Euclidean space which is describes the asymptotic growth of the min-max values. We prove Weyl’s law using the Dirichlet-Neumann bracketing as seen in [1]. Also, in relation to Weyl’s law, we state the Polya conjecture and prove it for tiling domains. Through an application of the Poisson Summation Formula, we determine a three term asymptotic for the heat kernel trace on the square. For general open, bounded, Jordan contented domains we prove a one term asymptotic for the heat kernel trace as a corollary to Weyl’s law. Our next main result is the existence of non-isometric isospectral domains in the plane as seen in Buser et al[2], that is we give an example of two non-isometric open, bounded domains in the plane for which the min-max values of the Dirichlet Laplacians on the two domains agree. Our final main result is the Faber-Krahn inequality which says that the first min-max value of the Dirichlet Laplacian on the ball is less than or equal to the first min-max value of the Dirichlet Laplacian on any open, bounded domain of the same volume (and in the same Euclidean space) as the ball. For the proof we develop a theory of symmetric decreasing rearrangements of functions as seen in [3] and we apply the Riesz rearrangement inequality without proff

Vejleder: Jan Philip Solovej
Censor:   Kim Knudsen, DTU