|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Dirichlet eigenvalues''' are the [[fundamental mode]]s of [[vibration]] of an idealized drum with a given shape. The problem of whether one can [[hearing the shape of a drum|hear the shape of a drum]] is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function ''u'' ≠ 0 and [[eigenvalue]] λ
| | Sit on Main angling Kayaks: A lay on best canoe is yet another perfectly option to consider if you are a sport fishing canoe. Various lay on very top kayaks are typically entirely created for consultant fisherman and they are built with a few angling add-on. Some of the gadgets put eliminate installed or possibly porch mounted pole members, tackle containers, fish around person, rudder, extra brood and also anglers images. [http://www.youtube.com/watch?v=_-BK5aSNgWs sun dolphin kayak review] [http://www.youtube.com/watch?v=7-kwBrbBq1s dolphin kayak] aruba 10 sit in kayak ([http://www.youtube.com/watch?v=Xx-rUw1-a30 http://www.youtube.com/watch?v=Xx-rUw1-a30]) on peak kayaks usually are known as stable kayaks and are generally an excellent option for starters. <br><br> |
| {{NumBlk|:|<math>\begin{cases}
| |
| \Delta u + \lambda u = 0& \rm{in\ }\Omega\\
| |
| u|_{\partial\Omega} =0.&
| |
| \end{cases}
| |
| </math>|{{EquationRef|1}}}}
| |
| Here Δ is the [[Laplacian]], which is given in ''xy''-coordinates by
| |
| :<math>\Delta u = \frac{\partial^2u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}.</math> | |
| The [[boundary value problem]] ({{EquationNote|1}}) is, of course, the [[Dirichlet problem]] for the [[Helmholtz equation]], and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with [[Neumann eigenvalue]]s: eigenvalues for the corresponding [[Neumann problem]]. The Laplace operator Δ appearing in ({{EquationNote|1}}) is often known as the '''Dirichlet Laplacian''' when it is considered as accepting only functions ''u'' satisfying the Dirichlet boundary condition. More generally, in [[spectral geometry]] one considers ({{EquationNote|1}}) on a [[manifold with boundary]] Ω. Then Δ is taken to be the [[Laplace-Beltrami operator]], also with Dirichlet boundary conditions.
| |
|
| |
|
| It can be shown, using the [[spectral theorem#The spectral theorem for compact self-adjoint operators|spectral theorem for compact self-adjoint operators]] that the eigenspaces are finite-dimensional and that the Dirichlet eigenvalues λ are real, positive, and have no [[limit point]]. Thus they can be arranged in increasing order:
| | The business also carries a variety with boat add-on these include craft kayak add-ons, boat buggies, boat dry baggage, kayak dry containers, boat rudders, kayak center, boat finance, canoe gears, compasses, goblet members, skill plugs and many other. You may also purchase boat maintenance and additionally maintenance merchandise like deck rig packages and also strain connect sets. Present, all of these products are available at marked down selling prices. <br><br> Those sites providing Kayaks discounted have kayaks that can come throughout sorts of capacities after 5ft in order to 12ft, and they are crafted according to the consumption. Cost all these kayaks in addition could differ as per the most important trip standard and shape. Minimal worth of the [http://www.youtube.com/watch?v=7-kwBrbBq1s dolphin kayak] is mostly about $300. |
| :<math>0<\lambda_1\le\lambda_2\le\cdots,\quad \lambda_n\to\infty,</math>
| |
| where each eigenvalue is counted according to its geometric multiplicity. The eigenspaces are orthogonal in the space of [[square-integrable function]]s, and consist of [[smooth function]]s. In fact, the Dirichlet Laplacian has a continuous extension to an operator from the [[Sobolev space]] <math>H^2_0(\Omega)</math> into <math>L^2(\Omega)</math>. This operator is invertible, and its inverse is compact and self-adjoint so that the usual spectral theorem can be applied to obtain the eigenspaces of Δ and the reciprocals 1/λ of its eigenvalues.
| |
| | |
| One of the primary tools in the study of the Dirichlet eigenvalues is the [[Raleigh quotient|max-min principle]]: the first eigenvalue λ<sub>1</sub> minimizes the [[Dirichlet energy]]. To wit,
| |
| :<math>\lambda_1 = \inf_{u\not=0}\frac{\int_\Omega |\nabla u|^2}{\int_\Omega |u|^2},</math>
| |
| the [[infimum]] is taken over all ''u'' of [[compact support]] that do not vanish identically in Ω. By a [[Meyers-Serrin theorem|density argument]], this infimum agrees with that taken over nonzero <math>u\in H_0^1(\Omega)</math>. Moreover, using results from the [[calculus of variations]] analogous to the [[Lax–Milgram theorem]], one can show that a minimizer exists in <math>H_0^1(\Omega)</math>. More generally, one has
| |
| :<math>\lambda_k = \sup\inf \frac{\int_\Omega |\nabla u|^2}{\int_\Omega |u|^2}</math>
| |
| where the [[supremum]] is taken over all (''k''−1)-tuples <math>\phi_1,\dots,\phi_{k-1}\in H^1_0(\Omega)</math> and the infimum over all ''u'' orthogonal to the φ<sub>''i''</sub>.
| |
| | |
| ==Applications==
| |
| [[File:SpiralCladding.png|200px|thumb|right|Fig.1. Spiral-shaped boundary of the domain (blue), its chunk (red), and 3 segments of a ray (green).
| |
| ]]
| |
| The Dirichlet Laplacian may arise from various problems of [[mathematical physics]];
| |
| it may refer to modes of at idealized [[drum]], small waves at the surface of an idealized [[pond|pool]],
| |
| as well as to a mode of an idealized [[optical fiber]] in the [[paraxial approximation]]. | |
| The last application is most practical in connection to the [[double-clad fiber]]s;
| |
| in such fibers, it is important, that most of modes of the fill the domain uniformly,
| |
| or the most of rays cross the core. The poorest shape seems to be the circularly-symmetric domain<ref name=bedo>{{cite journal| author=S. Bedo|coauthors= W. Luthy, and H. P. Weber |
| |
| title=The effective absorption coefficient in double-clad fibers|
| |
| journal=[[Optics Communications]]|
| |
| volume=99| issue=5-6| pages=331–335| year=1993|
| |
| url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-46JGTGD-M5&_user=10&_coverDate=06%2F15%2F1993&_alid=550903253&_rdoc=1&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c8a4c3ecc3d9a4e9ecb84f96cfef0333 |
| |
| doi=10.1016/0030-4018(93)90338-6 | bibcode=1993OptCo..99..331B}}
| |
| </ref><ref name="Doya">{{cite journal|title=Modeling and optimization of double-clad fiber amplifiers using chaotic propagation of pump|
| |
| author= Leproux, P.| coauthors=S. Fevrier, V. Doya, P. Roy, and D. Pagnoux| journal=[[Optical Fiber Technology]]| url=http://www.ingentaconnect.com/content/ap/of/2001/00000007/00000004/art00361| | |
| volume=7 | year=2003 | issue=4 | pages=324–339|doi=10.1006/ofte.2001.0361 | bibcode=2001OptFT...7..324L}}</ref>
| |
| ,.<ref name="Liu">{{cite journal|
| |
| title=The absorption characteristics of circular, offset, and rectangular double-clad fibers|
| |
| author=A. Liu| coauthors= K. Ueda| url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-497C4YV-BW&_user=10&_coverDate=12%2F15%2F1996&_alid=550869877&_rdoc=3&_fmt=summary&_orig=search&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=3&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=688bbca25fdd98e29caadb676b003c1e
| |
| | journal=[[Optics Communications]]| volume=132| year=1996|
| |
| issue=5-6| pages= 511–518|
| |
| doi=10.1016/0030-4018(96)00368-9}}</ref>
| |
| The modes of pump should not avoid the active core used in double-clad [[fiber amplifier]]s.
| |
| The spiral-shaped domain happens to be especially efficient for such an application due to the
| |
| boundary behavior of modes of '''Dirichlet laplacian'''.<ref name="Kouznetsov">{{cite journal
| |
| |title=Boundary behavior of modes of Dirichlet laplacian
| |
| | author= Kouznetsov, D.| coauthors=Moloney, J.V.| journal=[[Journal of Modern Optics]]
| |
| | url=http://www.metapress.com/content/be0lua88cwybywnl/?p=5464d03ba7e7440f9827207df673c804&pi=6
| |
| |volume=51 | year=2004 | issue=13 | pages=1955–1962
| |
| |ref=http://www.ils.uec.ac.jp/~dima/TMOP102136.pdf | doi=10.1080/09500340408232504 | bibcode=2004JMOp...51.1955K}}</ref>
| |
| | |
| The theorem about boundary behavior of the Dirichlet Laplacian if analogy of the property of rays in geometrical optics (Fig.1);
| |
| the angular momentum of a ray (green) increases at each reflection from the spiral part of the boundary (blue), until the ray hits the chunk (red); all rays (except those parallel to the optical axis) unavoidly visit the region in vicinity of the chunk to frop the excess of the
| |
| angular momentum. Similarly, all the modes of the Dirichlet Laplacian have non-zero values in vicinity of the chunk. The normal component of the derivative
| |
| of the mode at the boundary can be interpreted as [[pressure]]; the pressure integrated over the surface gives the [[force]]. As the mode is steady-state
| |
| solution of the propagation equation (with trivial dependence of the longitudinal coordinate), the total force should be zero.
| |
| Similarly, the [[angular momentum]] of the force of pressure should be also zero. However, there exists a formal proof, which
| |
| does not refer to the analogy with the physical system.<ref name="Kouznetsov"/>
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| * {{springer|title=Dirichlet eigenvalues|id=d/d130170|year=2001|first=Rafael D.|last=Benguria}}.
| |
| * {{Cite book|first=Isaac|last=Chavel|title=Eigenvalues in Riemannian geometry|series=Pure Appl. Math.|volume=115|publisher=[[Academic Press]]|year=1984|isbn=0-12-170640-0}}.
| |
| * {{Cite book|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|title=Methods of Mathematical Physics, Volume I|publisher=Wiley-Interscience|year=1962}}.
| |
| | |
| {{DEFAULTSORT:Dirichlet Eigenvalue}}
| |
| [[Category:Differential operators]]
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Spectral theory]]
| |
Sit on Main angling Kayaks: A lay on best canoe is yet another perfectly option to consider if you are a sport fishing canoe. Various lay on very top kayaks are typically entirely created for consultant fisherman and they are built with a few angling add-on. Some of the gadgets put eliminate installed or possibly porch mounted pole members, tackle containers, fish around person, rudder, extra brood and also anglers images. sun dolphin kayak review dolphin kayak aruba 10 sit in kayak (http://www.youtube.com/watch?v=Xx-rUw1-a30) on peak kayaks usually are known as stable kayaks and are generally an excellent option for starters.
The business also carries a variety with boat add-on these include craft kayak add-ons, boat buggies, boat dry baggage, kayak dry containers, boat rudders, kayak center, boat finance, canoe gears, compasses, goblet members, skill plugs and many other. You may also purchase boat maintenance and additionally maintenance merchandise like deck rig packages and also strain connect sets. Present, all of these products are available at marked down selling prices.
Those sites providing Kayaks discounted have kayaks that can come throughout sorts of capacities after 5ft in order to 12ft, and they are crafted according to the consumption. Cost all these kayaks in addition could differ as per the most important trip standard and shape. Minimal worth of the dolphin kayak is mostly about $300.