Mathematical Sciences: Partial Differential Equations and Harmonic Analysis

小行星9401355 数学研究的主要目的是得到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。 解是变分解，即极小值或能量或其他一些成本函数的更高临界点。 所采用的方法是对称性和不变性的形式傅立叶分析，重新缩放和迭代，曲率和凸性，最大值原理和比较功能，绿色的公式和更详细的集成部分公式，渐近分析和扰动和试验功能的能量估计。 第一个问题是定义在凸域或正曲面上的特征函数的特征值估计。 好的估计需要确定节线，也就是本征函数为零的弧。 节点线的定位问题被看作是一个两相自由边界问题。 此外，本文还对两相自由边界问题的平衡态演化进行了研究。 要解决的第二个问题是一个经典的分配问题首先提出的蒙赫在世纪。 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低的成本重新培训一批工人以适应新的工作。 将使用的方法是几何和相关的研究体积保持映射的N-空间和理论的蒙日安培方程，一个完全非线性偏微分方程。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

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. ***