Bonnesen型对称混合不等式
发布时间：2018-09-05 11:50
【摘要】：数学中最经典的几何不等式是等周不等式,它刻画了欧氏平面R2中的由简单闭曲线所围成域的面积与周长间的关系.Bonnesen型不等式是加强的等周不等式,经Chern,Bonnesen,Hadwiger,Osserman,Santal′o,任德麟,周家足,张高勇等人的发展,Bonnesen型不等式与Laplacian算子的第一特征值,Wullf流,Sobolev不等式等密切联系.反向的Bonnesen型不等式,即逆Bonnesen型不等式也逐渐被关注.等周不等式的推广之一是关于平面两凸域的对称混合等周不等式,加强的对称混合等周不等式是关于平面两凸域的Bonnesen型对称混合不等式.本文主要研究关于平面两凸域的Bonnesen型对称混合不等式及逆Bonnesen型对称混合不等式.在第3章中,我们首先研究关于平面两凸域的Bonnesen型对称混合不等式,利用积分几何中的Poincar′e运动公式和Blaschke运动公式估计关于平面两凸域K0和K1的对称混合等周亏格?2(K0,K1),得到一些Bonnesen型对称混合不等式,并且证明了其等号成立的条件.这些不等式推广了Bonnesen和Kotlyar等人的结果.然后我们研究关于平面两凸域的逆Bonnesen型对称混合不等式,由Poincar′e运动公式,Blaschke运动公式及Blaschke滚动定理,我们得到一些新的对平面卵形域成立的逆Bonnesen型对称混合不等式.此外我们还得到对任意平面凸域均成立的逆Bonnesen型对称混合不等式,其条件比著名的Bottema不等式的条件弱.最后我们推广平面上的Bol-Fujiwara定理,即得到关于平面两卵形域的广义Bol-Fujiwara定理.我们还进一步介绍了关于平面两凸域的Bonnesen型对称混合不等式在估计第二类完全椭圆积分方面的应用.第4章讨论常曲率曲面中两凸域的对称混合等周不等式及Bonnesen型对称混合不等式.
[Abstract]:The most classical geometric inequality in mathematics is the isoperimetric inequality, which describes the relationship between the area and the perimeter of a domain enclosed by a simple closed curve in the Euclidean plane R2. Bonnesen-type inequality is a strengthened isoperimetric inequality, developed by Chern, Bonnesen, Hadwiger, Osserman, Santal'o, Ren Delin, Zhou Jiazu, Zhang Gaoyong, etc. Inequalities are closely related to the first eigenvalues of Laplacian operators, Wullf flows, Sobolev inequalities, etc. Reverse Bonnesen-type inequalities, i.e. inverse Bonnesen-type inequalities, are also gradually concerned. Bonnesen-type symmetric mixed inequalities for biconvex domains. In this paper, we mainly study the Bonnesen-type symmetric mixed inequalities for planar biconvex domains and the inverse Bonnesen-type symmetric mixed inequalities for planar biconvex domains. In chapter 3, we first study the Bonnesen-type symmetric mixed inequalities for planar biconvex domains. Ashke's motion formula estimates the symmetric mixed isoperimetric genus? 2 (K0, K1) with respect to planar biconvex domains K0 and K1. Some Bonnesen-type symmetric mixed inequalities are obtained and the conditions under which their symbols hold are proved. These inequalities generalize the results of Bonnesen and Kotlyar et al. Then we study the inverse Bonnesen-type symmetric mixing with respect to planar biconvex domains. By using Poincar'e motion formula, Blaschke motion formula and Blaschke rolling theorem, we obtain some new inverse Bonnesen-type symmetric mixed inequalities for planar oval domains. In addition, we obtain inverse Bonnesen-type symmetric mixed inequalities for arbitrary planar convex domains, whose conditions are more than those of the famous Bottema inequality. Finally, we generalize the Bol-Fujiwara theorem on the plane, that is, we obtain the generalized Bol-Fujiwara theorem on two oval domains in the plane. We further introduce the application of Bonnesen type symmetric mixed inequalities for two convex domains in the plane to estimate the second kind of complete elliptic integrals. Chapter 4 discusses two convex domains on a surface with constant curvature. Symmetric mixed isoperimetric inequalities and Bonnesen type symmetric mixed inequalities.
【学位授予单位】：西南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O178
本文编号：2224175
[Abstract]:The most classical geometric inequality in mathematics is the isoperimetric inequality, which describes the relationship between the area and the perimeter of a domain enclosed by a simple closed curve in the Euclidean plane R2. Bonnesen-type inequality is a strengthened isoperimetric inequality, developed by Chern, Bonnesen, Hadwiger, Osserman, Santal'o, Ren Delin, Zhou Jiazu, Zhang Gaoyong, etc. Inequalities are closely related to the first eigenvalues of Laplacian operators, Wullf flows, Sobolev inequalities, etc. Reverse Bonnesen-type inequalities, i.e. inverse Bonnesen-type inequalities, are also gradually concerned. Bonnesen-type symmetric mixed inequalities for biconvex domains. In this paper, we mainly study the Bonnesen-type symmetric mixed inequalities for planar biconvex domains and the inverse Bonnesen-type symmetric mixed inequalities for planar biconvex domains. In chapter 3, we first study the Bonnesen-type symmetric mixed inequalities for planar biconvex domains. Ashke's motion formula estimates the symmetric mixed isoperimetric genus? 2 (K0, K1) with respect to planar biconvex domains K0 and K1. Some Bonnesen-type symmetric mixed inequalities are obtained and the conditions under which their symbols hold are proved. These inequalities generalize the results of Bonnesen and Kotlyar et al. Then we study the inverse Bonnesen-type symmetric mixing with respect to planar biconvex domains. By using Poincar'e motion formula, Blaschke motion formula and Blaschke rolling theorem, we obtain some new inverse Bonnesen-type symmetric mixed inequalities for planar oval domains. In addition, we obtain inverse Bonnesen-type symmetric mixed inequalities for arbitrary planar convex domains, whose conditions are more than those of the famous Bottema inequality. Finally, we generalize the Bol-Fujiwara theorem on the plane, that is, we obtain the generalized Bol-Fujiwara theorem on two oval domains in the plane. We further introduce the application of Bonnesen type symmetric mixed inequalities for two convex domains in the plane to estimate the second kind of complete elliptic integrals. Chapter 4 discusses two convex domains on a surface with constant curvature. Symmetric mixed isoperimetric inequalities and Bonnesen type symmetric mixed inequalities.
【学位授予单位】：西南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O178
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