Mathematical Sciences: Elliptic and Parbolic Partial Differential Equations

该项目集中在数学的三个领域 偏微分方程的研究 第一个问题涉及 退化椭圆Bellman方程的一般理论 基本拟线性算子 这些方程的出现是因为 控制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，财富 椭圆方程的理论不再适用。 的 这项工作的目标是开发新的技术， 解的先验估计，非线性问题的研究 与真实的和复Monge-Ampere方程有关的方程 研究具有给定曲率的曲面方程。 一 第二个研究领域集中在粘度解决方案， 非线性椭圆型方程 这个概念的使用， 类似于控制理论中的价值函数， 可以用来研究非线性方程的解 而不需要假设它们是可微的。 一旦解决方案 已知存在粘度意义上的，这是一个重要的问题， 出现数值或其它近似。 工作将 我已经开发出估算收敛速度的方法 近似于真实解 构建 好的近似值还必须做进一步的研究 粘度解决方案的平滑性。 第三 这项研究的内容涉及的解决方案的性质， 带白色噪声强迫项的线性抛物方程。 这些 是随机性是一种自然成分的方程。 它们出现在过滤问题，遗传学，量子力学 和磁动力学 工作做得很少 来近似求解这些方程。 这项研究将 集中讨论那些定义在有界域上的方程， 边界条件引入不寻常的问题， 在确定性方程中遇到的 偏微分方程是数学的基础， 物理世界的建模。 数学的作用 分析不是为了建立方程，而是为了 提供有关的定性和定量信息 解决方案 这可能包括以下问题的答案 独特性、平滑性和成长性。 此外，分析往往 发展的解决方案和估计的近似方法 这些近似值的准确性。

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i.e. the governing matrix in the equation becomes singular, and the wealth of theory on elliptic equations is no longer applicable.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.