【作者】 叶运华；

【导师】 麻希南；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2010， 博士

【摘要】 本文讨论三维调和函数、三维极小图凸水平集的最小主曲率的估计，浸入极小曲面凸水平集的高斯曲率估计以及Hessian型方程允许解的对数梯度估计全文共分六章，第一章回顾椭圆方程解的水平集的凸性的历史；第二章引入刻画函数水平集主曲率的曲率矩阵（？）；第三、四章利用该曲率矩阵证明了如下一些定理：定理0.1设Ω为Rⁿ中的区域，3≤n≤5且u为Ω上的p-调和函数，即设在（？），（？）,u的水平集关于梯度方向（？）为严格凸的，则u的水平集的主曲率不能在Ω的内部达到最小值，除非u是常数利用凸环上p-调和函数的水平集是严格凸的，可以得到下面的推论推论0.2设u为下面边值问题的解，其中Ω₀,Ω₁为Rⁿ中的有界凸集，满足若3<n<5且，则解u的水平集的主曲率在上达到最小值对于极小图情形有定理0.3设为的区域，u为Ω上的极小图，即u满足极小曲面方程假设在Ω上。若u水平集关于梯度方向（?）为严格凸的，则u的水平集的主曲率不能在Ω内部达到极小，除非u是常数利用凸环上的极小图的水平集是严格凸的，同样可以得到下面的推论推论0.4设u满足其中Ω₀和Ω₁为。中满足的有界凸区域。则的水平集的主曲率在的边界上达到极小值第五章利用活动标架计算得到了浸入极小曲面凸水平集高斯曲率的估计定理0.5设为浸入极小曲面，是关于方向∈的高度函数，并且没有临界点，如果u的所有水平集关于方向为严格凸的，那么对于或，函数不能在的内部取得最小值，除非它为常数第六章考虑Hessian型方程允许光滑解的对数梯度估计定理0.6设为下面Hessian型方程的允许光滑解更多还原

【Abstract】 In this dissertation, we discuss the smallest principal curvature estimates of convex level sets of three-dimensional harmonic function and minimal graph, Gaussian curvature estimates of convex level sets of minimal surface. Logarithmic gradient estimates of admissible solution to Hessian type equation arising in geometric optics is also considered. The dissertation consists of six chapters.In chapter 1, we briefly recall the history of convex level sets of solutions to elliptic equations. In chapter 2, we introduce the symmetric curvature matrix {aij} to describe the principal curvatures of level sets of a function. In chapter 3 and 4, we use this curvature matrix to prove the following results:Theorem 1 Let be a bounded smooth domain, 3 < n < 5 and u be a p-harmonic function inΩ, i.e.Assume in < p < 3 and the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ，unless it is constant.Using the fact that the level sets of p-harmonic function over convex ring are strictly convex, we haveCorollary 0.1 Let u be the solution of the following boundary value problem,where and are two bounded convex sets in satisfying C If 3 < n < 5 and，then the principal curvature of the level sets of u attains its minimum on . For the case of the three-dimensional minimal graph, we haveTheorem 2 Let be a domain in and u be a minimal graph over，i.e., u satisfy the minimal surface equationAssume in . If the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ, unless it is constant.Using the fact that the level sets of the minimal graph over convex ring are strictly convex , we get the followingCorollary 0.2 Let u satisfywhere and are bounded convex domains in R3; satisfying Then the principal curvature of the level sets of u attains its minimum value on d .In chapter 5, we discuss the Gaussian curvature estimates of convex level sets of immersed minimal surface using moving frame method and obtain the followingTheorem 3 Let be an immersed minimal surface. Let u be the height function of Mn corresponding to the direction .Ifu has no critical point, and let the level sets of u are all strictly convex with respect to direction Vu. Denote the Gaussian curvature by K and let . Then the function can not attain its minimum at an interior point of Mn, unless it is constant. In chapter 6, we give logarithmic gradient estimate of admissible solution to Hessian type equation and obtain the followingTheorem 4 be a smooth admissible solution to the following equationwhereLet f be a positive function. Then we have更多还原