【摘要】：Hessian方程的障礙問題在微分幾何中有著重要應(yīng)用,該問題起源于研究歐公式空間上一定條件下具有上(下)障礙的超曲面問題。本文針對黎曼流形上一類Hessian方程的障礙問題,研究障礙問題的解的存在性與正則性。本文利用引入一類懲罰函數(shù),將Hessian方程的障礙問題轉(zhuǎn)化為奇異置換方程;通過對奇異置換方程的容許解的研究,得到Hessian方程的障礙問題的解的存在性與正則性;對于奇異置換方程,先驗(yàn)估計(jì)可以保證容許解的存在性及正則性,因此Hessian方程的障礙問題容許解的先驗(yàn)估計(jì)就變得十分重要。首先在n維帶邊緊致黎曼流形M上運(yùn)用最大值原理得到容許解的0C先驗(yàn)估計(jì)。其次,借助0C先驗(yàn)估計(jì),由最大值原理得到容許解在黎曼流形邊界上的1C先驗(yàn)估計(jì);通過選取適當(dāng)?shù)脑囼?yàn)函數(shù),借助U的定義得到了容許解在黎曼流形內(nèi)部的1C先驗(yàn)估計(jì);至此,則得到了黎曼流形上這類Hessian方程障礙問題的1C先驗(yàn)估計(jì)。最后,利用到邊界的距離函數(shù)構(gòu)造適當(dāng)?shù)拈l函數(shù),證明了容許解在黎曼流形邊界上的2C先驗(yàn)估計(jì);對于容許解在流形內(nèi)部的2C先驗(yàn)估計(jì),通過選取適當(dāng)?shù)脑囼?yàn)函數(shù),并利用最大值原理及1C先驗(yàn)估計(jì)即可得到;至此,則得到了黎曼流形上Hessian方程障礙問題的1C先驗(yàn)估計(jì)。
[Abstract]:The obstacle problem of Hessian equation has an important application in differential geometry. The problem originates from the study of hypersurface with upper (lower) obstacle under certain conditions in the space of Euclidean formula. In this paper, we study the existence and regularity of solutions for a class of Hessian equations on Riemannian manifolds. In this paper, by introducing a class of penalty functions, the barrier problem of Hessian equation is transformed into singular permutation equation, and the existence and regularity of the solution of obstacle problem of Hessian equation are obtained by studying the admissible solution of the singular permutation equation. For singular permutation equations, a priori estimate can guarantee the existence and regularity of admissible solutions, so a priori estimate of admissible solutions for Hessian equations becomes very important. First, the 0-C priori estimate of admissible solutions is obtained by using the maximum principle on the n-dimensional edge-compact Riemannian manifold M. Secondly, the 1C priori estimate of admissible solution on the boundary of Riemannian manifold is obtained by means of the 0C priori estimate and the 1C priori estimate of admissible solution in the interior of Riemannian manifold is obtained by means of the definition of U by selecting appropriate test function. At this point, we obtain the 1C priori estimate for the obstacle problem of Hessian equations on Riemannian manifolds. Finally, using the distance function to the boundary to construct the appropriate gate function, we prove the 2C priori estimate of the admissible solution on the Riemannian manifold boundary, and select the appropriate test function for the 2C priori estimate of the admissible solution in the interior of the manifold. By using the maximum principle and the 1C priori estimate, the 1C priori estimate for the obstacle problem of Hessian equation on Riemannian manifold is obtained.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O175;O186.12
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本文編號：2130929