Universal Inequality And Upper Bounds Of Eigenvalues For Fractional Laplacian

This paper mainly study the universal inequality and upper bounds of eigenvalues for a class of fractional Laplacian.We consider the fractional Laplacian(-?)s|?,which is defined as the pseudo-differential operator restricted to ?,where ?(?)Rn is a bounded open domain and s>0.By using the Fourier transform,we can conveniently define the operator as(-?)s|?u(x)=X?F-1(|?|2su(?)).Due to the theory of operator,there is a sequence of discrete eigenvalues for this operator.In this paper,we will prove the universal inequality and upper bounds of eigenvalues for the fractional Laplacian when s is a positive rational number.The thesis is concerned about the lower and upper bounds as well as the uni-versal inequality of eigenvalues for classical and extended Laplacian including some non-degenerate elliptic operators such as bi-Laplacian,poly-Laplacian and fractional Laplacian.In addition,after the pioneering work of Hormander,as one of the gen-eralized eigenvalue problems of classical Laplacian,there are many interesting known results of the eigenvalue problems of degenerate elliptic operators and we shall give a brief introduction to these results in the paper.The details are divided into the following three parts:In Chapter 1,we will first introduce the well-known results of the lower bounds of eigenvalues for the Dirichlet Laplacian and its generalized eigenvalue problems on a bounded domain ? C Rn and then the universal inequality and upper bounds of these eigenvalue problems.Besides,we are going to give a comprehensive introduction to the known results of fractional Laplacian and its eigenvalue problems which we will mainly study in this paper.At last,we shall give the main results of this paper.In Chapter 2,we provide the proof of the main results.Firstly,we introduce two propositions which will play an important role in our proof.Secondly,we give a proof of our main theorem,which leads to the universal inequality of eigenvalues for the fractional Laplacian(-?)s|?(s>0 is a rational number).Finally,if s?1,then by applying the universal inequality of this paper and the variant of Chebyshev sum inequality,we will prove the so-called Yang type inequality for corresponding eigenvalue problem which is the extension to the case of poly-Laplacian operators.Moreover,we can derive the upper bounds of corresponding eigenvalues from the Yang type inequality.In Chapter 3,in spite of the complicated process,a separated proof of a vital proposition in the paper will be given,which is the key point to the way of estimating the upper bounds of eigenvalues.