Minimal Smoothness Questions for Inverse Problems and Boundary Value Problems

罗素·布朗将在两个领域进行研究： 偏微分方程与边值正则性 非光滑域中的问题。 自然界的许多基本定律都是以局部的形式来表达的 微分方程在逆问题中，人们试图恢复 信息的偏微分方程中的系数 关于方程的解这提供了一个数学模型 用于确定物体的物理性质（例如其 电导率）从测量（如电压和电流在 边界）。在我将要研究的问题中， 问题，我们考虑的正是上述的例子：确定一个 从电流测量的空间不均匀电导率， 在边界处产生的电压。我研究的创新之处在于 试图确定限制性最小的假设， 可以做出决定。 研究的第二个领域与边值问题有关 非光滑域上的各种偏微分方程。在 与上面考虑的逆问题相反，我们在这里 考虑到更直接的问题， 在边界处给出信息的偏微分方程。这是一个数学模型的问题（为 例如，从内部找到电压电势， 了解边界的潜力。在我的创新 研究是建立最小的（和更现实的）假设， 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.