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Elliptiska Partiella Differentialekvationer och Spektralgeometri
KTH, School of Engineering Sciences (SCI).
KTH, School of Engineering Sciences (SCI).
2019 (Swedish)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesisAlternative title
Elliptic Partial Differential Equations and Spectral Geometry (English)
Abstract [sv]

I denna rapport utforskar vi en del teori kring eliptiska partiella differential ekvationer, i vilka problem de uppstår och metoder för att lösa dem. Mer specifikt försöker vi att sätta oss in i och undersöka frågan ställd av Mark Kac 1966: “Can one hear the shape of a drum?”. Genom att använda metoder från en artikel av C. Gordon, D. Webb and S. Wolpert [GWW92] konstruerar vi två plana domäner som är isospektrala under Laplacianen. Detta ger oss svaret till Kac’s fråga: nej, man kan inte höra formen på en trumma. Med numeriska metoder visualiserar vi några egenmoder för dessa isospektrala domäner och jämför deras egenvärden. Fastän man inte kan höra formen på en trumma ger spektrat en del användbar information. Med Weyl’s Lag kan man beräkna arean, eller till och med omkretsen, av domänen vilket vi diskuterar i sista sektionen.

Abstract [en]

In this paper we explore some theory behind elliptic partial differential equations, in what problems they arise and methods of solving them. Specifically we will try to address the question asked by Mark Kac in 1966: “Can one hear the shape of a drum?”. Using the theory from an article by C. Gordon, D. Webb and S. Wolpert [GWW92] we construct two planar domains which are isospectral under the Laplacian. Thus answering Kac’s question negatively that no, one cannot hear the shape of a drum. With numerical methods we visualize some eigenmodes for these isospectral domains and compare their eigenvalues. Even though one can not hear the shape of a drum the spectrum generate some useful information. With Weyl’s Law one can calculate the area, or even the circumference, of the domain which we discuss in the last section.

 

Place, publisher, year, edition, pages
2019.
Series
TRITA-SCI-GRU ; 2019:115
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:kth:diva-254710OAI: oai:DiVA.org:kth-254710DiVA, id: diva2:1334817
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Examiners
Available from: 2019-07-03 Created: 2019-07-03 Last updated: 2019-07-03Bibliographically approved

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