偏微分方程中的等周不等式及其相关问题的研究

项目名称： 偏微分方程中的等周不等式及其相关问题的研究

项目编号： No.11271120

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 戴求亿

作者单位： 湖南师范大学

项目金额： 65万元

中文摘要： 等周问题在数学的发展中具有很重要的地位。该问题最早起源于几何，目前已散见于数学的许多分支。本项目研究偏微分方程中的等周问题，特别关注k-Hessian 算子Dirichlet第一特征值上、下界的等周估计和第一特征函数积分模的等周估计等等。同时，还关注在研究上述估计时派生出来的一些重要问题，如：第一特征值的Brunn-Minkowski不等式、最优形状的光滑性以及超定问题的对称性等等。因为在处理线性和拟线性问题中行之有效的Schwarz对称化方法不能应用于完全非线性问题,我们采用变分方法来估计k-Hessian 算子特征值的下界。这一方法要求对最优形状的光滑性和超定问题的对称性进行深入的研究。同样，为了克服Schwarz对称化方法不能应用的困难，我们将运用第一特征值的Brunn-Minkowski不等式来导出特征值的上界估计。这一方法是本项目的独创，可用于发现偏微分方程中更多的等周估计。

中文关键词： 等周不等式；曲率测度；k-曲率算子；双曲空间；椭圆方程

英文摘要： Isoperimetric problem is very important in the development of mathematics. This problem is arised in geometry at first and so far can be seen in many mathematical branches. In this program, we will study isoperimetric problem in partial differential equations. Emphasis is posed on the isoperimetric estimate of lower and upper bound for the first eigenvalue of k-Hessian operator and isoperimetric estimate of integral norm for the first eigenfunction of k-Hessian operator. Emphasis is also posed on important problems, such as Brunn-Minkowski inequality for the first eignvalue, regularity of the optimal shape and symmetry of overdetermind problem, arised in the study of isoperimetric estimate for the first eigenvalue and for the integral norm of the first eigenfunction. Since the Schwarz symmetrization method which works well in the study of linear and quasilinear problem is not applicable in the study of fully nonlinear problem, we use variational method to estimate the lower bound of the first eigenvlue of k-Hessian operator. This approach needs a deep study of the regularity of the optimal shape and the symmetry of an overdetermind problem. Also, to overcome the difficult that the Schwarz symmetrization method is not applicable, we estimate the upper bound of the first eigenvlue of k-Hessian operator by making u

英文关键词： Isoperimetric inequality；Curvature measure；k-curvature operator；Hyperbolic space；elliptic equations