Mathematical Sciences: Absolute Continuity of Parabolic Measure and Regularity of PDE's

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Abstract Lewis 9531642 This research is concerned with mutual absolute continuity of parabolic and Lebesgue measure as well as related Dirichlet-Neumann problems. In certain time varying domains Lewis and Murray have shown that parabolic measure for the heat equation is mutually absolutely continuous with respect to a certain projective Lebesgue measure. This investigation will begin with the study of a model pde whose prototype is the pullback pde obtained from the heat equation by way of a certain mapping onto the above time varying domain. Since the pullback pde has a parabolic measure that is mutually absolutely continuous with respect to Lebesgue measure, the object of the investigation will be to determine what properties of the model pde are actually needed to guarantee mutual absolute continuity of the above measures. As for the Dirichlet and Neumann problems they are now well understood for square integrable functions defined on the boundary of the above time varying domains. The next step is to consider the Neumann problem for p th power integrable functions when p is between 1 and 2. Again the above pullback pde needs to be analyzed closely. Many phyical processes can be analyzed using partial differential equations. The most famous classical partial differential equations are Laplace's equation, the heat equation, and the wave equation each of which originated in the 18 th and 19 th centuries and found uses in the study of gravity, electricity, fluid flow, electromagnetic waves, to mention only a few topics. My research concerns problems of the following type : Given the temperature on the walls of a room, find the temperature at any place in the room at any later time ? This problem is called the Dirichlet problem for the heat equation. The Neumann problem can be stated similarly in terms of the rate at which heat is flowing out of the walls of the room. Mathematically if the temperature on the walls of the room is fixed and nice enough (cont inuous), then the Dirichlet problem can be shown to have a unique solution. Part of my research has been concerned with whether this problem has a unique solution when the walls of the room and temperature on the walls is allowed to vary. This problem has now been essentially completely solved and the corresponding Neumann problem is being studied. Possible applications of this research are to free boundary problems where the size and temperature of an object are constantly changing (ice melting, gases expanding, nuclear waste solidifying).