Short Time Heat Content and the Heat Kernel Asymptotics

点击翻译按钮获取中文摘要

AbstractAward: DMS-9820904Principal Investigator: Peter B. GilkeyGilkey will study time dependent processes which are controlledby the heat equation and the short time asymptotics which arisethereby. The first project involves the heat content asymptotics,moving from the static setting to the setting where the data(metric, internal heat sources, boundary conditions) are timedependent. Previous work always assumed smooth boundaryconditions; in this project the boundary conditions arediscontinuous. The second project involves studying theasymptotics of the short time expansion of the fundamentalsolution of the heat equation. Most previous studies haveinvolved static geometries, but it is natural to study theseasymptotic expansions for time dependent geometries for quitegeneral operators of Laplace type and for quite generalhomogeneous boundary conditions. This investigation will involveexpanding the usual calculus of pseudo differential operators toestablish the existence of the required short time asymptotics aswell as determining how the time dependent variation of themetric and the coupling constants for Neumann boundary conditionsinfluences the short time asymptotics of the heat kernel.There are many physical settings where boundary conditions arediscontinuous. For example, a body floating in ice watersatisfies Dirichlet boundary conditions on the part immersed inthe water and Neumann boundary conditions on the remainder(assuming as a first approximation that there is no heat transferfrom the air to the body); the boundary conditions arediscontinuous along the water/air interface. Understandingadditional boundary contributions coming from the water/airinterface are likely to be of great physical importance, andlinks the differential geometry of the situation to the physicalunderpinnings of the subject. There are also many conditionswhere the geometry is not static; the Universe is expanding tocite one example. Boundary conditions in physical settings mayvary with time; for example, the temperature outside a buildingvaries with the time of day as well as with the season;consequently the associated heat flow is not well modeled by astatic setting. Understanding the heat flow in this setting hasobvious physical applications. The heat equation asymptotics andheat content asymptotics have proven to be of central importancein theoretical physics. For example, these asymptotics play animportant role in the renormalization of field theory in curvedspace and also in string and membrane theory, and they arerelated to zeta function renormalization.