Magnetic Pseudodifferential Operators and Acoustic Black Holes
Translated title
Magnetiske pseudodifferential operatorer og akustiske sorte huller
Author
Sørensen, Kasper Studsgaard
Term
4. term
Education
Publication year
2019
Submitted on
2019-08-09
Pages
33
Abstract
Dette speciale giver en tilgængelig oversigt over to hovedspor inden for differentialligninger: magnetiske pseudodifferential-operatorer og akustiske sorte huller. Først defineres en klasse af magnetiske symboler, der tillader polynomiel vækst i x−x′, og de tilhørende magnetiske pseudodifferential-operatorer repræsenteres via en generaliseret matrix. Denne repræsentation bruges til at analysere spektrale egenskaber: for selvadjungerede symboler er spektret 1/2-Hölder-kontinuert, og ved konstant magnetfelt er spektrummets minimum og maksimum Lipschitz-kontinuerte; tilsvarende er kanterne af et spektralt gap, der ikke lukker, Lipschitz-kontinuerte. Specialet fokuserer særligt på beviserne for Lipschitz-kontinuitet. Dernæst behandles akustiske sorte huller, hvor målet er at minimere refleksionen af en bølge i en plade via formoptimering. Der gives en kort gennemgang af variabelregning og Lagrange-multiplikatorer i ramme af Euler–Bernoulli bjælkteori, og der forsøges en udvidelse til Timoshenko-teori, som viser sig at være mere kompleks og fortsat er under udvikling, med mulige fremtidige numeriske løsninger. Afslutningsvis nævnes kort relateret arbejde om singulære funktioner konstrueret ud fra stationære tidsrækker.
This thesis offers an accessible overview of two main directions in differential equations: magnetic pseudodifferential operators and acoustic black holes. It first introduces a class of magnetic symbols that allow polynomial growth in x−x′, and associates magnetic pseudodifferential operators represented via a generalized matrix. Using this representation, the work analyzes spectral properties: for self-adjoint symbols the spectrum is 1/2-Hölder continuous, and under a constant magnetic field the minimum and maximum of the spectrum are Lipschitz continuous; similarly, the edges of a spectral gap that does not close are Lipschitz continuous. The thesis places particular emphasis on the proofs of Lipschitz continuity. It then turns to acoustic black holes, where the goal is to minimize wave reflection in a plate through shape optimization. A brief review of calculus of variations and Lagrange multipliers is given in the framework of the Euler–Bernoulli beam theory, followed by an attempted extension to Timoshenko theory that proves more complex and remains ongoing, with possible future numerical approaches. Finally, related work on singular functions constructed from stationary time series is noted.
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