Mathematical Sciences: Elliptic and Parbolic Partial Differential Equations

本项目集中在偏微分方程的三个数学研究领域。第一部分涉及具有拟线性算子的退化椭圆型Bellman方程的一般理论。这些方程是受控扩散的动态规划方程。在许多重要的情况下，方程变得退化，即方程中的控制矩阵变得奇异，丰富的椭圆方程理论不再适用。这项工作的目标是开发新的技术来证明解的先验估计，研究与实和复蒙日-安培方程有关的非线性方程，并研究具有规定曲率的曲面方程。第二个研究领域集中在非线性椭圆方程的粘性解上。这个概念的使用，类似于在控制论中寻找值函数，允许人们研究非线性方程的解，而不假设它们是可微的。一旦一个解已知存在于粘度意义上，一个重要的数值或其他近似问题就出现了。我们将研究估计逼近真解的收敛速度的方法。为了构造良好的近似，还必须进一步研究粘度解的平滑性。本研究的第三个部分涉及带白噪声强迫项的线性抛物方程解的性质。在这些方程中，随机性是一个自然因素。它们出现在滤波问题、遗传学、量子力学和磁动力学中。对这类方程的近似解所做的工作很少。本研究将集中于那些定义在有界域中的方程，其中边界条件引入了确定性方程中没有遇到的异常问题，偏微分方程构成了物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

This project concentrates on three areas of mathematical research in partial differential equations. The first concerns the general theory of degenerate elliptic Bellman equations with underlying quasilinear operators. These equations arise as the dynamic programming equations for controlled diffusions. In many important cases the equations become degenerate, i.e. the governing matrix in the equation becomes singular, and the wealth of theory on elliptic equations is no longer applicable. The goals of this work are to develop new techniques for proving a prior estimates on solutions, investigation of nonlinear equations related to the real and complex Monge-Ampere equations and to study equations of surfaces with prescribed curvature. A second area of investigation focuses on viscosity solutions of nonlinear elliptic equations. The use of this concept, which is similar to that of finding the value function in control theory, allows one to investigate solutions of nonlinear equations without assuming that they are differentiable. Once a solution is known to exist in the viscosity sense, an important problem of numerical or other approximation arises. Work will be done developing methods for estimating the speed of convergence of approximations to the true solution. To construct good approximations one must also make further investigations into the smoothness properties of viscosity solutions. The third element of this research concerns properties of solutions of linear parabolic equations with white noise forcing terms. These are equations in which randomness is a natural ingredient. They arise in filtering problems, genetics, quantum mechanics and magneto-dynamics. Very little work has been done in approximating solutions to such equations. This research will concentrate on those equations defined on bounded domains in which the boundary conditions introduce unusual problems not encountered in the deterministic equations 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.