Mathematical Sciences: Partial Differential Equations and Harmonic Analysis

9401355 Jerison这项数学研究的主要目的是获得椭圆型偏微分方程解的存在性、正则性以及解的大小和形状的估计。这些解是变分解，即能量的最小化或更高临界点或一些其他成本函数。所采用的方法有傅立叶分析形式的对称性和不变性、重标度和迭代、曲率和凸性、极大值原理和比较函数、格林公式和更精细的分部积分公式、渐近分析和扰动以及能量估计的试函数。第一个问题是估计定义在凸域或正曲面上的特征函数的特征值。好的估计需要定位节线，也就是特征函数消失的圆弧。节点线定位问题被看作是一个两相自由边界问题。此外，还将对两相自由边界问题的演化到平衡进行研究。第二个要解决的问题是蒙格在18世纪首次提出的经典分配问题。这个问题可以看作是一个线性规划问题。它可以被表述为决定如何以最小的成本再培训一批工人，以适应新的工作。所使用的方法是几何的，与N-空间的保体映射的研究和Monge Ampere方程的理论有关，Monge Ampere方程是一个完全非线性的偏微分方程。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

9401355 Jerison The primary goal of this mathematical research is to obtain existence, regularity and estimation on the size and shape of solutions of elliptic partial differential equations. The solutions are variational solutions, that is, minimizers or higher critical points of energy or some other cost function. The methods employed are symmetry and invariance in the form of Fourier analysis, rescaling and iteration, curvature and convexity, maximum principle and comparison functions, Green's formula and more elaborate integration by parts formulas, asymptotic analysis and perturbations and trial functions for energy estimation. The first problem is to estimate eigenvalues of eigenfunctions defined on convex domains or on positively curved surfaces. Good estimation necessitates locating the nodal line, that is, the arc where the eigenfunction vanishes. The problem of locating the nodal line is viewed as a two-phase free boundary problem. In addition, it work will be done on the evolution to equilibrium for two-phase free boundary problems. A second problem to be addressed is a classical allocation problem first proposed by Monge in the 18th century. The problem can be viewed as a linear programming problem. It can be formulated as one of deciding how to retrain a group of workers to fit new jobs at minimal cost. The approach to be used is geometric and related to the study of volume-preserving maps of N-space and the theory of the Monge Ampere equation, a fully nonlinear partial differential equation. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***