Optimization For Eigenvalues Of Boundary Value Problems

In this thesis, several important extremal problems on eigenvalues of ordinary differential equations(ODE) will be solved. For the second-order ODE, we will show how the Pontryagin’s Maximum Principle(PMP) can be applied to the construction of nondegenerate potentials and the extremal problems of eigenvalues of the p-Laplacian with potentials. For the fourth-order beam equations, a typical extremal problem on eigenvalues will lead to the beam equations with measures, not with potentials. In order to give a complete treatment for fourth-order ODE, we will first introduce the fourth-order linear measure differential equations(MDE). It will be shown that the solutions of initial value problems of MDE are well-defined. As a crucial result, we will show that solutions of MDE depend on measures in a very strong way, i.e. solutions are continuously dependent on measures when the weak*topology for measures is considered. Then we will use the minimization of the Rayleigh form to show that the first eigenvalue of the MDE is well-defined. Moreover, it will be shown that the first eigenvalue of MDE, as a nonlinear functional of measures, is also strong continuous in measures. Finally, after establishing a relation between the the minimization problems of the first eigenvalues of ODEs and MDEs, the minimization of the first eigenvalues of the beam equations will be completely solved by finding the solution of the corresponding problem for MDE.The thesis is organized as follows.Firstly, we will give a background introduction to extremal problems of eigenvalues.Some important approaches to these problems of second-order ODE and important results will be briefly commented.Secondly, by using the Pontryagin’s Maximum Principle(PMP), we will construct a new, optimal class of non-degenerate potentials for second-order ODE. Inspired also by PMP, the corresponding extremal problems on eigenvalues for such a construction of non-degenerate potentials will be solved for the p-Laplacian with potentials.Thirdly, we will establish some basic results on the fourth-order linear MDE. Given a measure μ(t) on the unit interval, it will be shown that solutions of the initial value problems of the MDEdy(3)(t) + y(t)dμ(t) = 0are uniquely defined on the whole interval. As a new result, we will show that solutions have a strong continuous dependence on measures. For the corresponding Rayleigh form for the beam equation with a measure, it will be shown that the minimum does define the first eigenvalue λ1(μ) for MDE dy(3)(t) + y(t)dμ(t) = λy(t)dt,with the typical boundary conditions. As mentioned before, a crucial result is that λ1(μ)is strongly continuous in μ when the weak*topology is considered for measures μ.Finally, we will first show that the minimization of the first eigenvalue of the forthorder MDE in balls of measures will be realized by the Dirac measures. Then, by using the strong continuity of the first eigenvalues and approximation of Dirac measures by potentials, it will be shown that the minimization of the first eigenvalues of ODE in the balls of potentials are reduced to the problems for the ODE.