Mathematical Sciences: Extremal Problems for Eigenvalues, Heat Kernels and Energies

抽象的劳格森 本文的研究将试图证明：（a）平面单连通区域在Dirichlet边界条件下热核的迹是极小的 当域是圆盘时，假定域在原点具有固定的保角映射半径，以及（B）声学驻波在边界处达到它们的最大位移。 这两个问题的治疗建议是统一的技术变化的度量内的共形类。 接下来，一个开放的能量-最小化问题的金斯堡-朗道能量泛函将被考虑;在这里的解决方案"应该"是一个特定的径向向量场，但困难的是，目前的重组方法的向量是绝对不够的。 请注意，所有这些问题所应用的方法都是数学的，尽管每个问题也有物理意义。 至少从古希腊时代起，人们就对振动和热流问题进行了数千年的研究。 今天，振动和热问题仍然具有挑战性和重要性，因为科学家们在极端规模和极端条件下处理新旧材料。 然而，世纪的科学史表明，在看似实际的问题上取得的进展往往依赖于基础研究中发展起来的方法和见解。 这一建议解决了热流和声振动基本理论中的几个问题，这些问题的"答案"似乎直观地显而易见;这些答案甚至可以在某些情况下通过计算机来证实。难题和挑战在于，没有人能从逻辑上解释为什么这些“答案”是正确的。 找到这样一个合乎逻辑的解释的答案肯定会有益地提高我们的理解和能力的问题的振动和热流。 更具体地说，该提案中的第一个问题涉及（粗略地说）估计在允许热量在一段时间内蒸发后，一个区域中还剩下多少热量。 这里的推测是，一个特定的磁盘失去它的热量 最快 (This也可以用量子力学来重新表述）。 下一个问题涉及声波中最响点的位置，例如汽车内的道路噪音。 据推测，声音在该地区的边缘最大， 而不是在车里 该提案考虑的第三个问题是一个简单的-陈述但仍然-未解决的问题，源于超导理论，这是一个具有巨大实际可能性的领域，我们对它的理论理解很不完整。 最后，对于那些说理论数学在计算机时代已经死了（或者应该死了）的人，我们应该回应说，理论数学 始终如一地提供了计算机不敢涉足的洞察力。 更重要的是，理论数学家的操作成本往往更低。

Abstract Laugesen The research will attempt to establish the conjectures: (a) that the trace of the heat kernel (under Dirichlet boundary conditions) of a simply connected plane domain is minimal when the domain is a disk, given that the domain has a fixed conformal mapping radius at the origin, and (b) that acoustical standing waves attain their maximum displacement at the boundary. The proposed treatment of these two problems is unified by the technique of variation of a metric within a conformal class. Next, an open energy--minimization problem for a Ginzburg--Landau energy functional will be considered; here the solution ``ought'' to be a certain radial vector field, but the difficulty is that current rearrangement methods for vectors are decidedly inadequate. Note that the methods to be applied to all these problems are mathematical, though every problem has physical meaning as well. Problems of vibration and heat flow have been studied for thousands of years, since at least the time of the ancient Greeks. Today, vibration and heat problems remain challenging and important, as scientists grapple with new and old materials on extreme scales and under extreme conditions. The history of 20th century science shows, though, that progress on seemingly practical problems relies very often on the methods and insights developed in basic research. This proposal addresses several questions in the basic theory of heat flow and sound vibration for which the "answers" seem intuitively obvious; these answers can even be confirmed by computer in certain cases. The puzzle, and the challenge, is that no-one can explain logically why these "answers" are correct. Finding such a logical explanation of the answer is sure to profitably enhance our understanding of and capabilities with problems of vibration and heat flow. To be a bit more specific, the first problem in the proposal deals (roughly speaking) with estimating how much heat remains in a region after the heat is allowed to dissipat e for a certain period of time. The conjecture here is that a particular disk loses its heat fastest. (This conjecture can also be re-stated in terms of quantum mechanics.) The next problem concerns the location of the loudest point in a sound wave, such as the road noise inside an automobile. The conjecture is that the sound is loudest at the edge of the region, rather than inside the car, say. The third problem considered by the proposal is a simply--stated but still--unsolved problem arising from the theory of super-conductivity, an area of immense practical possibility of which we have a sadly incomplete theoretical understanding. Finally, to those who say that theoretical mathematics is dead (or should be) in the age of the computer, one should respond that theoretical mathematics consistently provides insight where computers fear to tread. And what is more, theoretical mathematicians are often cheaper to operate.