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

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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.