Short Time Heat Content and the Heat Kernel Asymptotics

摘要奖：DMS-9820904首席研究员：Peter B.Gilkey Gilkey将研究受热方程控制的依赖时间的过程以及由此产生的短时渐近性。第一个项目涉及热含量的渐近性，从静态设置转移到数据(公制、内部热源、边界条件)是时间相关的设置。以往的工作总是假定边界条件是光滑的，而在本工程中，边界条件是不连续的。第二个项目涉及研究热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问题，但对于一般的拉普拉斯算子和相当一般的齐次边界条件，研究含时几何的渐近展开式是很自然的。这项研究将涉及扩展通常的伪微分算子演算，以确定所需的短时渐近性的存在，以及确定Neumann边界条件的度规常数和耦合常数的随时间的变化如何影响热核的短时渐近性。例如，漂浮在冰水中的物体在浸入水中的部分满足Dirichlet边界条件，其余部分满足Neumann边界条件(作为第一近似值，假设空气不向物体传递热量)；边界条件在水/空气界面上是不连续的。了解来自水/空气界面的额外边界贡献可能具有非常重要的物理意义，并将情况的微分几何与对象的物理基础联系起来。在许多情况下，几何学也不是静态的；举个例子，宇宙正在膨胀。物理环境中的边界条件可能会随时间而变化；例如，建筑物外的温度随一天中的时间和季节而变化；因此，相关的热流不能很好地用无定常环境来模拟。了解这一环境中的热流具有明显的物理应用。热方程的渐近性和热含量的渐近性已被证明在理论物理中具有中心重要性。例如，这些渐近性在曲线空间中场理论的重正化以及弦和膜理论中起着重要的作用，并且它们与Zeta函数重正化有关。

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.