Short Time Heat Content and the Heat Kernel Asymptotics

摘要奖：DMS-9820904主要研究者：Peter B。Gilkey将研究由热方程控制的时间相关过程以及由此产生的短时间渐近性。第一个项目涉及热含量渐近，从静态设置移动到数据（度量，内部热源，边界条件）与时间相关的设置。以往的工作都假定边界条件是光滑的，而本课题的边界条件是不连续的.第二个项目是研究热传导方程基本解的短时展开的渐近性。 以往的研究大多涉及静态几何，但研究含时几何的渐近展开式，对于相当一般的拉普拉斯型算子和相当一般的齐次边界条件是很自然的。本研究将涉及扩展通常的伪微分算子的微积分以建立所需的短时渐近性的存在性，以及确定Neumann边界条件的度量和耦合常数的随时间变化如何影响热核的短时渐近性。例如，一个漂浮在冰水中的物体，浸没在水中的部分满足狄利克雷边界条件，其余部分满足诺依曼边界条件（假设作为第一近似，没有热量从空气传递到物体）;边界条件沿水/空气界面是不连续的沿着。理解来自水/空气界面的额外边界贡献可能具有很大的物理重要性，并将该情况的微分几何与该主题的物理基础联系起来。也有许多情况下，几何不是静态的;举一个例子，宇宙正在膨胀。物理环境中的边界条件可能随时间而变化;例如，建筑物外的温度随一天中的时间以及季节而变化;因此，相关的热流不能很好地用无定向设置来模拟。了解这种情况下的热流有明显的物理应用。 热方程的渐近性和热含量的渐近性已被证明是理论物理中的重要问题。 例如，这些渐近性在弯曲空间中场论的重整化以及弦和膜理论中起着重要的作用，并且它们与zeta函数重整化有关。

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.