黎曼流形上一类Hessian方程的障碍问题
发布时间：2018-07-17 21:04
【摘要】：Hessian方程的障碍问题在微分几何中有着重要应用,该问题起源于研究欧公式空间上一定条件下具有上(下)障碍的超曲面问题。本文针对黎曼流形上一类Hessian方程的障碍问题,研究障碍问题的解的存在性与正则性。本文利用引入一类惩罚函数,将Hessian方程的障碍问题转化为奇异置换方程;通过对奇异置换方程的容许解的研究,得到Hessian方程的障碍问题的解的存在性与正则性;对于奇异置换方程,先验估计可以保证容许解的存在性及正则性,因此Hessian方程的障碍问题容许解的先验估计就变得十分重要。首先在n维带边紧致黎曼流形M上运用最大值原理得到容许解的0C先验估计。其次,借助0C先验估计,由最大值原理得到容许解在黎曼流形边界上的1C先验估计;通过选取适当的试验函数,借助U的定义得到了容许解在黎曼流形内部的1C先验估计;至此,则得到了黎曼流形上这类Hessian方程障碍问题的1C先验估计。最后,利用到边界的距离函数构造适当的闸函数,证明了容许解在黎曼流形边界上的2C先验估计;对于容许解在流形内部的2C先验估计,通过选取适当的试验函数,并利用最大值原理及1C先验估计即可得到;至此,则得到了黎曼流形上Hessian方程障碍问题的1C先验估计。
[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.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175;O186.12
，
本文编号：2130929
[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.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175;O186.12
，
本文编号：2130929
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