Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps. (17th January 2014)

• Record Type: Journal Article
• Authors: Khrabustovskyi, Andrii; Khruslov, Evgeni
• Is Part Of: Mathematical methods in the applied sciences. Volume 38:Number 1(2015:Jan. 15)
• Page Start: 11
• Page End: 26
• Publication Date: 2014-01-17
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3046
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma3046-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="mma3046-para-0001">The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain Ω<sup><italic>ϵ</italic></sup> depending on a small parameter <italic>ϵ</italic> > 0. The domain has the form <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2qt16rz2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:1704214:media:mma3046:mma3046-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>Ω</mml:mi></mml:mrow><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow></mml:msup><mml:mo class="MathClass-rel">=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo class="MathClass-bin">∖</mml:mo><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>, where <italic>S</italic><sup><italic>ϵ</italic></sup> is an <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2qt16s1p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:1704214:media:mma3046:mma3046-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>‐periodic family of trap‐like screens. We prove that, for an arbitrarily large <italic>L</italic>, the spectrum has precisely one gap in [0, <italic>L</italic>] when <italic>ϵ</italic> is small enough; moreover, when <italic>ϵ</italic> → 0, this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of 2D photonic crystals is discussed. Copyright © 2014 John Wiley & Sons, Ltd.</p> </abstract>