For non-experts: Suppose we are looking at the temperature in a room. There are two kinds of questions we can ask. First, we can look at how the temperature is affected by the temperature of the walls, ceiling, and floor (this is called Dirichlet data). For example, exterior walls will probably have a different temperature than interior walls, and heating or cooling vents will have a different temperature than the surrounding walls. This will cause the temperature to vary throughout the room. Secondly, we can look at how the temperature is affected by the heat flow through the walls, ceiling and floor (this is called Neumann data). For example, well-insulated walls will allow very little heat through, but an open door will allow lots of heat through. As before, this will cause the temperature to vary throughout the room. The equations modeling this situation can rarely be solved exactly, so I am interested in the qualitative properties of the solution. For example, I want to know that if the data (the information about the walls, ceiling, and floor) satisfies certain measurements, what measurements will the solution (the temperature distribution throughout the room) satisfy? I'm particularly interested in what happens near the corners of the room, because many of the mathematical tools used for studying problems like this tend to break down near corners. In the example I've described, the solution usually satisfies better measurements than the data, and these measurements are relatively well understood. The problem I look at is a similar problem that involves a complicated mix of Dirichlet and Neumann data, where the measurements of the solution may not be as good as the measurements of the data. My goal is to understand how the geometry of the boundary affects the relationship between these measurements.
For calculus-savvy non-experts: The specific problem I look at comes from studying calculus with complex variables. Because each complex variables corresponds to two real variables, there is roughly twice as much information to consider when looking at limits in complex variables, which means we have a lot more information about complex differentiable functions than we do about the real differentiable functions studied in calculus. In particular, complex differentiable functions have to satisfy a differential equation, and solving this differential equation gives rise to the problem hinted at in the previous paragraph. The measurements mentioned in the previous paragraph involve measuring the size of the derivatives of the data and solution. In the problem I study, it may be possible to differentiate the data as many times as you want, but the solution may not be differentiable (at least near the boundary). Part of my research involved finding cases where the solution can be differentiated as many times at the data, which is good enough for many applications.
For experts: I work in partial differential equations in several complex variables. More specifically, I look at the d-bar-Neumann problem and related operators, including the d-bar_b boundary complex. My areas of interest include bounded plurisubharmonic exhaustion functions, non-smooth domains, and estimates in Sobolev spaces.
Global
Regularity for the
dbar-Neumann Operator and Bounded Plurisubharmonic Exhaustion Functions,
November 2009, revised April 2011, to appear in Adv. Math.
Closed Range for dbar and dbar_b on Bounded Hypersurfaces in Stein
Manifolds (with A. Raich), June 2011, submitted
Defining Functions for Unbounded C^m Domains (with A. Raich), November 2011, submitted
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