發(fā)布時(shí)間：2018-09-05 11:50

【摘要】：數(shù)學(xué)中最經(jīng)典的幾何不等式是等周不等式,它刻畫了歐氏平面R2中的由簡(jiǎn)單閉曲線所圍成域的面積與周長(zhǎng)間的關(guān)系.Bonnesen型不等式是加強(qiáng)的等周不等式,經(jīng)Chern,Bonnesen,Hadwiger,Osserman,Santal′o,任德麟,周家足,張高勇等人的發(fā)展,Bonnesen型不等式與Laplacian算子的第一特征值,Wullf流,Sobolev不等式等密切聯(lián)系.反向的Bonnesen型不等式,即逆Bonnesen型不等式也逐漸被關(guān)注.等周不等式的推廣之一是關(guān)于平面兩凸域的對(duì)稱混合等周不等式,加強(qiáng)的對(duì)稱混合等周不等式是關(guān)于平面兩凸域的Bonnesen型對(duì)稱混合不等式.本文主要研究關(guān)于平面兩凸域的Bonnesen型對(duì)稱混合不等式及逆Bonnesen型對(duì)稱混合不等式.在第3章中,我們首先研究關(guān)于平面兩凸域的Bonnesen型對(duì)稱混合不等式,利用積分幾何中的Poincar′e運(yùn)動(dòng)公式和Blaschke運(yùn)動(dòng)公式估計(jì)關(guān)于平面兩凸域K0和K1的對(duì)稱混合等周虧格?2(K0,K1),得到一些Bonnesen型對(duì)稱混合不等式,并且證明了其等號(hào)成立的條件.這些不等式推廣了Bonnesen和Kotlyar等人的結(jié)果.然后我們研究關(guān)于平面兩凸域的逆Bonnesen型對(duì)稱混合不等式,由Poincar′e運(yùn)動(dòng)公式,Blaschke運(yùn)動(dòng)公式及Blaschke滾動(dòng)定理,我們得到一些新的對(duì)平面卵形域成立的逆Bonnesen型對(duì)稱混合不等式.此外我們還得到對(duì)任意平面凸域均成立的逆Bonnesen型對(duì)稱混合不等式,其條件比著名的Bottema不等式的條件弱.最后我們推廣平面上的Bol-Fujiwara定理,即得到關(guān)于平面兩卵形域的廣義Bol-Fujiwara定理.我們還進(jìn)一步介紹了關(guān)于平面兩凸域的Bonnesen型對(duì)稱混合不等式在估計(jì)第二類完全橢圓積分方面的應(yīng)用.第4章討論常曲率曲面中兩凸域的對(duì)稱混合等周不等式及Bonnesen型對(duì)稱混合不等式.
[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.
【學(xué)位授予單位】：西南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O178

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