Qualitative Studies On Eigenvalues And Solutions For Partial Differential Equations

In this thesis, we study eigenvalues and the properties of solution for par-tial differential equations. Partial differential equations involved in a number of issues from the chemistry, physics and mathematical model of the biological field, with a strong practical background. It has become one of the most im-portant fields of study in mathematical, which interacts with many branches of mathematics, such as differential geometry, complex analysis and harmonic analysis. Eigenvalues and the properties of solution are basic problems in partial differential equations. Therefore, the study on eigenvalues and the properties of solution for partial differential equations has scientific significance and potential applications. The main contents are organized as follows:In chapter1, we state the background of eigenvalues and the properties of solution for partial differential equations and the main work of this article.In chapter2, we introduce some mathematical terms and tools as prelimi-naries that will be used in the following chapters.In chapter3, Let Ω (?) Rn(n≧2) be a bounded domain with boundary (?)Ω, ν be the outward unit vector normal to (?)Ω, and0<β<+∞be a parameter. We discuss the following Robin eigenvalue problem The aim of this chapter is twofold. One is an upper bound for the ratio of the first two eigenvalues by Schwarz symmetrization method and rearrange-ment techniques, which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria in [4] and [6], the other is a Chiti type reverse Holder inequality for the first eigenfunction.In chapter4, we concern the following eigenvalue problem of k-Hessian operator where Ω is a smooth bounded convex domain in Rn. First we devote to deduce the first variational formula and some related overdetermined problems for the principle eigenvalue of problem (2), and then prove Serrin type symmetry result for our overdeter-mined problems.