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

本研究将试图建立以下猜想：(a)假设单连通平面域在原点处有固定的保角映射半径，单连通平面域为圆盘时，热核的迹线（在Dirichlet边界条件下）最小；(b)声驻波在边界处位移最大。这两个问题的建议处理是统一的技术，一个度规的变化在一个共形类。接下来，将考虑金兹堡-朗道能量泛函的开放能量最小化问题；这里的解“应该”是某个径向矢量场，但困难在于目前矢量的重排方法显然是不充分的。请注意，应用于所有这些问题的方法都是数学的，尽管每个问题也都有物理意义。振动和热流的问题已经被研究了几千年，至少从古希腊时代就开始了。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尺度和极端条件下努力解决新旧材料。然而，20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；在某些情况下，这些答案甚至可以通过计算机来证实。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答案，肯定会有利于提高我们对振动和热流问题的理解和能力。更具体地说，提案中的第一个问题（粗略地说）处理的是在允许热量在一段时间内消散后，估计一个区域内剩余的热量。这里的猜想是某个特定的圆盘损失热量最快。（这个猜想也可以用量子力学重新表述。）下一个问题涉及声波中最响亮点的位置，例如汽车内部的道路噪音。据推测，在这个区域的边缘，声音是最大的，而不是在车内。提案所考虑的第三个问题是由超导理论引起的一个简单但仍未解决的问题，这是一个具有巨大实践可能性的领域，遗憾的是，我们在理论上的认识还不完全。最后，对于那些说理论数学在计算机时代已经死亡（或应该死亡）的人，我们应该回答说，理论数学始终提供计算机不敢涉足的领域。更重要的是，理论数学家的操作成本通常更低。

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.