Fundamental Gap Problem Of Eigenvalues Of Partial Differential Operators

Eigenvalue problems have a wide range of applications in partial differential equations and geometric analysis,and also play an important role in the discipline of physical theory.In particular,the minimum fundamental gap problem of the first two eigenvalues has important applications in quantum mechanics and statistical mechanics.And it has corresponding mathematical meanings.Currently,the research on the minimum fundamental gap problem of second order elliptic operators is the most extensive,but it mainly focuses on the minimum fundamental gap problem of Schrodinger operators.This dissertation focuses on the minimum fundamental gap problem of one dimensional vibrating string equation and a class of second order elliptic operators degenerated at the boundary,it is mainly divided into the following points:First of all,the eigenvalue minimum fundamental gap problem for onedimensional vibrating string equation which density functions constrained by the L∞norm or L1 norm is studied.The existence of the optimal density function is proved using standard compactness theory.The characteristics of the optimal density function are described using spectral perturbation theory combined with optimization techniques.In particular,the corresponding"switching principle" is derived.The characterization of the eigenfunction with respect to the optimal density function is depicted by calculation and variational analysis,and its optimal geometric shape is given.Secondly,the eigenvalue minimum fundamental gap problem for a class of one-dimensional degenerate elliptic equations which potential functions constrained by the L∞norm is discussed.By defining a weighted Sobolev space,a corresponding compact embedding theorem is established,and a wellposedness study on weak solutions of equations is given.In addition,the properties of eigenvalues and corresponding eigenfunctions are described in detail.On the basis of these works,the corresponding spectral perturbation theorem is proved from the perspective of partial differential equations.Combining the corresponding compactness theory,the existence of the optimal potential function is derived,and the characteristics of the optimal potential function are characterized using direct optimization techniques.Thirdly,the minimum fundamental gap problem for the first two eigenvalues of a class of two-dimensional Grushin type elliptic operators which potential functions constrained by the L∞ norm is investigated.By defining a suitable weighted Sobolev space,using the properties of A∞ weighted functions,the corresponding compact embedding theorem and Poincare inequality are established,and the existence and uniqueness of weak solutions of the equation are discussed.The existence and characteristic theorems of the optimal potential function are proved by using the compact embedding theorem and the corresponding spectral perturbation theory.Finally,the maximum problem of the first eigenvalue and the minimum fundamental gap problem of the first two eigenvalues for a class of two-dimensional degenerate elliptic operators which potential functions constrained by the Lp norm are studied,where 1<p<∞ and 2<p<∞,respectively.By defining a suitable weighted Sobolev space,the corresponding compact embedding theorem and Poincare inequality are established,and the well-posedness for weak solutions of the equation is studied.Moreover,the local boundedness and regularity of weak solutions are proved,where the potential function are restricted by the norm of Lp.Combining with Harnack’s inequality,the existence and uniqueness of the optimal potential function for the maximum problem of the first eigenvalue are proved,and the corresponding estimates are given.In addition,the existence of the minimum fundamental gap of the eigenvalue is proved by using the compact embedding theorem,and the characterization of the optimal potential function of the minimum fundamental gap is given by utilizing the spectral perturbation theory.