Minimal Smoothness Questions for Inverse Problems and Boundary Value Problems

罗素·布朗将在两个领域进行研究：偏微分方程逆问题和非光滑区域中边值问题的正则性。自然界的许多基本定律都用偏微分方程式表示。在反问题中，人们试图从偏微分方程解的信息中恢复该方程中的系数。这提供了一个数学模型，用于根据测量结果(如边界上的电压和电流)确定对象的物理属性(如其传导性)。在我将要研究的问题--电导率逆问题中，我们恰好考虑了上面描述的例子：通过测量边界上的电流和电压来确定空间上不均匀的电导率。我的研究的创新之处在于试图确定限制最少的假设，在这些假设下可以做出这一决定。第二个研究领域涉及非光滑区域上的各种偏微分方程边值问题。与上面讨论的反问题不同，这里我们考虑的是更直接的问题，即在边界给定信息的情况下获得偏微分方程解。这是一个数学模型，用于(例如)根据边界上的电势知识求出内部的电势。我的研究的创新之处在于建立了最小的(更现实的)假设，在这些假设下，解决方案可以被证明是存在的。

Russell Brown will pursue research in two areas: inverse problems for partial differential equations and regularity for boundary value problems in nonsmooth domains. Many of the fundamental laws of nature are expressed as partial differential equations. In an inverse problem, one attempts to recover a coefficient in a partial differential equation from information about solutions of the equation. This provides a mathematical model for determining physical properties of an object (such as its conductivity) from measurements (such as voltage and current at the boundary). In the problem I will study, the inverse conductivity problem, we consider exactly the example described above: determine a spatially inhomogeneous conductivity from measurements of current and voltage made at the boundary. The innovation in my research is to attempt to determine the least restrictive hypotheses under which this determination can be made. The second area of investigation is related to boundary value problems for various partial differential equations in nonsmooth domains. In contrast to the inverse problem considered above, here we are considering the more straightforward problem of obtaining solutions to a partial differential equation given information at the boundary. This is a mathematical model for the problem of (for example) of finding the voltage potential in the interior from knowledge of the potential at the boundary. The innovation in my research is to establish minimal (and more realistic) hypotheses under which the solution can be shown to exist.