Some Problems Related To Asymptotic Behavior Of The Laplacian Spectrum And DtN Spectrum

This dissertation focuses on some problems related to the spectrum asymptotics of Lapacian and DtN operator.In the past century,many mathematicians had worked on the Weyl's asymptotic and the heat trace asymptotic expansion.They improved the remainder term in the Weyl's law,and found some new spectral invariants from the heat trace.By using these spectral invariants,many valuable theorems in the inverse spectral problems were proved,too.Since V.Lazutkin and D.Terman had constructed convex planar domains whose remainder term of Weyl's asymptotic were as worse as possible,we can not find an improved order of remainder term for general planar domain.But this improvement still can be done for some special domains which are "completely integrable",such as the planar disk.Y.Colin de Verdiere was able to convert this into remainder term estimation of a lattice point problem,and by the standard rmethod in the lattice problem he proved that the order is no more than 2/3.Combining the Weyl-van der Corput inequality(the A-process)and the Poisson summation formula followed by the method of stationary phase(the B-process),we can improve the order of the remainder term for planar disk in chapter 2.To find the optimal domain such that the specific eigenvalue is maximal(or min-imal)is a famous problem in spectral geometry.After the work of P.Antunes and P.Freitas in 2012,there are more and more research works on finding the optimal domain for the eigenvalues in the asymptotic sense.These problems are always equivalent to finding the optimal stretched parameter such that the domain enclosed most(or least)lattice points in the asymptotic sense.We shall study these problems on some model domains of finite type.While people always handled the domain whose boundary cur-vature is non-vanishing,there are some points with vanish curvature on the boundary of these model domains.We will prove that the volumes of the intersections of the optimal domain with each coordinate hyperplane are equal.Finally,based on some previous results,we will study the heat trace asymptotic ex-pansion of DtN operator with potential.We will compute the relative heat invariant a4 will be computed by Seeley's method,and then prove the general form of other relative heat invariants follows by an induction argument.By using these spectral invariants,we can prove that the integral of the potential functions in the isospectral set along the boundary should be determined by spectrum.Then,by the property of general trace operator and the Sobolev embedding theorem,we obtain some lower bound for certain Sobolev norms of these potential.Moreover,if the potential in the isospectral set satis-fies some additional conditions,such as they are radially symmetry,then their Cauchy data are determined by the spectrum.