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Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues [article]

A. H. Barnett, Andrew Hassell
<span title="2010-06-18">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain Ω⊂^n with piecewise smooth boundary.  ...  The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.  ...  For star-shaped domains we strengthened the inclusion bounds (Theorem 5.1), achieving a sharp power of E and sharp constant, in the limit of small tension, when tension is weighted by a special geometric  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1006.3592v1">arXiv:1006.3592v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sykwejspvbgedjypyef2yejfhi">fatcat:sykwejspvbgedjypyef2yejfhi</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1006.3592/1006.3592.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3f/3b/3f3bce8100f66e499cc38caf09386be098700d1c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1006.3592v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues

A. H. Barnett, A. Hassell
<span title="">2011</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cf6hwtzc6baajcjfclnyxmzxgy" style="color: black;">SIAM Journal on Numerical Analysis</a> </i> &nbsp;
We study eigenfunctions φ j and eigenvalues E j of the Dirichlet Laplacian on a bounded domain Ω ⊂ R n with piecewise smooth boundary.  ...  In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp.  ...  The authors are grateful for discussions with Dana Williams, Timo Betcke, and Chen Hua.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/100796637">doi:10.1137/100796637</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fxe3okl2lzbdnpccuebanjlodm">fatcat:fxe3okl2lzbdnpccuebanjlodm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170830001316/https://math.dartmouth.edu/~ahb/papers/b_resub.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/52/5a/525af24c607ae841993d245746ca6b2bbd997708.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/100796637"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Estimates on Neumann eigenfunctions at the boundary, and the "Method of Particular Solutions" for computing them [article]

A. H. Barnett, Andrew Hassell
<span title="2011-07-12">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We consider the "Method of particular solutions" for numerically computing eigenvalues and eigenfunctions of the Laplacian Δ on a smooth, bounded domain Omega in RR^n with either Dirichlet or Neumann boundary  ...  This is advantageous for the accurate computation of large eigenvalues. The Dirichlet case can be treated using elementary arguments and has appeared in SIAM J. Num.  ...  of 'quasi-orthogonality' of ψ i and ψ j , when |λ i − λ j | is small.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1107.2172v1">arXiv:1107.2172v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7kobdeqhebecrd76ywxs2ks62i">fatcat:7kobdeqhebecrd76ywxs2ks62i</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1107.2172/1107.2172.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1107.2172v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Mini-Workshop: Boundary Value Problems and Spectral Geometry

Jussi Behrndt, Konstantin Pankrashkin, Olaf Post
<span title="">2012</span> <i title="European Mathematical Publishing House"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mnv35em3uvgxdkcbx2cvykhugu" style="color: black;">Oberwolfach Reports</a> </i> &nbsp;
Some recent contributions in the field of boundary value problems and spectral geometry concern, e.g., construction of isospectral manifolds with boundary, eigenvalue and resonance distribution for large  ...  Boundary value problems and spectral geometry is an attractive and rapidly developing area in modern mathematical analysis.  ...  The rigorous bound of our method allows us to prove an eigenvalue inclusion here.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4171/owr/2012/02">doi:10.4171/owr/2012/02</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wvepbssjbrehdclkjeuqmj4lti">fatcat:wvepbssjbrehdclkjeuqmj4lti</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20130409140327/http://w3.bretagne.ens-cachan.fr/math/people/virginie.bonnaillie/articles/Bo12Ober.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3c/b3/3cb35af30492c973c811f83b05ae33d7d58dac6c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4171/owr/2012/02"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues [article]

Alex Barnett, Andrew Hassell, Melissa Tacy
<span title="2015-12-14">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For smooth bounded domains in R, we prove upper and lower L^2 bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window.  ...  This 'inclusion bound' improves over previously known bounds by a factor of E^5/6. It is analogous to a recently improved inclusion bound in the Dirichlet case, due to the first two authors.  ...  Estimates (1.10), (1.11) are analogous to the quasi-orthogonality results for normal derivatives of Dirichlet eigenfunctions proved in [4, 7] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.04165v1">arXiv:1512.04165v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bg3radf3lbbuplhy6lrnrj2c7q">fatcat:bg3radf3lbbuplhy6lrnrj2c7q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826050134/https://arxiv.org/pdf/1512.04165v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8d/2e/8d2e540dd0027895e874f4ab9ba1d83041b36c1a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.04165v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The Hodge laplacian on manifolds with boundary

Pierre Guerini, Alessandro Savo
<span title="">2003</span> <i title="Cellule MathDoc/CEDRAM"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5equ4iokbfhj3ftldxh7i7phe4" style="color: black;">Séminaire de théorie spectrale et géométrie</a> </i> &nbsp;
The Hodge laplacian on manifolds with boundary Séminaire de Théorie spectrale et géométrie, tome 21 (2002-2003), p. 125-146 © Séminaire de Théorie spectrale et géométrie (Grenoble), 2002-2003, tous droits  ...  Then, in the case of the Dirichlet problem, the first eigenvalue of the Laplacian cannot be small if Vol (D.) is not large.  ...  the first (positive) eigenvalue for the Dirichlet conditions. 3.1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5802/tsg.338">doi:10.5802/tsg.338</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/znrncgmb5jg27lswr6cd7y4a5q">fatcat:znrncgmb5jg27lswr6cd7y4a5q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220116220219/https://tsg.centre-mersenne.org/article/TSG_2002-2003__21__125_0.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/87/4d/874d59f2f9b99e6a574993efe3fe725675f51751.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5802/tsg.338"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Asymptotic spherical shapes in some spectral optimization problems [article]

Dario Mazzoleni, Benedetta Pellacci, Gianmaria Verzini
<span title="2019-09-25">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of the mixed Dirichlet-Neumann Laplacian.  ...  , with center at points of ∂Ω having large mean curvature.  ...  As before, if Ω ⊂ R N is a bounded Lipschitz domain and D ⊂ Ω is quasi-open, with |D| < |Ω|, the inclusion H 1 0 (D, Ω) ֒→ L 2 (Ω) is compact.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1811.01623v3">arXiv:1811.01623v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cytubb2s2rea5d52bg3hcxzvoa">fatcat:cytubb2s2rea5d52bg3hcxzvoa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200906153405/https://arxiv.org/pdf/1811.01623v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/92/db/92db38a09327c68d78f04d920031e376573ded9f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1811.01623v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Geometrical structure of Laplacian eigenfunctions [article]

Denis S. Grebenkov, Binh-Thanh Nguyen
<span title="2013-02-08">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition.  ...  The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.  ...  Sapoval for many valuable discussions and for his passion to localization that strongly motivated our work.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1206.1278v2">arXiv:1206.1278v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pmeiikacdrhjxjqjpoh44tvaii">fatcat:pmeiikacdrhjxjqjpoh44tvaii</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200915101638/https://arxiv.org/pdf/1206.1278v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/df/9c/df9c7db0af1ff35eeeadb937fcf36be2105ea265.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1206.1278v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Any three eigenvalues do not determine a triangle [article]

Javier Gómez-Serrano, Gerard Orriols
<span title="2020-01-21">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Despite the moduli space of triangles being three dimensional, we prove the existence of two triangles which are not isometric to each other for which the first, second and fourth Dirichlet eigenvalues  ...  To do so, we develop new tools to rigorously enclose eigenvalues to a very high precision, as well as their position in the spectrum.  ...  Acknowledgements We are grateful to Princeton University, where this research was performed, and to the VSRC Program and the Department of Mathematics for partially supporting the second author's stay.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.06758v2">arXiv:1911.06758v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p2b362qlvjb3nfx3rk2xage5ii">fatcat:p2b362qlvjb3nfx3rk2xage5ii</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200924230751/https://arxiv.org/pdf/1911.06758v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f3/e6/f3e6a06677bd4e8876104b1be6e0f647f76394ad.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.06758v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

A Survey on the Krein–von Neumann Extension, the Corresponding Abstract Buckling Problem, and Weyl-type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains [chapter]

Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, Roman Shterenberg, Gerald Teschl
<span title="">2013</span> <i title="Springer Basel"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cz3im356gvg2jat63ae7tnorvq" style="color: black;">Operator Theory: Advances and Applications</a> </i> &nbsp;
on C^∞_0(Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂R^n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C^  ...  In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed  ...  for many helpful discussions and very valuable correspondence on various topics discussed in this survey.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-0348-0591-9_1">doi:10.1007/978-3-0348-0591-9_1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dpoiiqmnanch5gyulph26cd4wm">fatcat:dpoiiqmnanch5gyulph26cd4wm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706081115/http://www.mat.univie.ac.at/~gerald/ftp/articles/BucklingKreinWeyl.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/7a/3d/7a3d76eb62ceb404d5a9be05681a88acd4fa16bd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-0348-0591-9_1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea, Gerald Teschl
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/37jjomjmvfhrzf3di2gjnrrfuu" style="color: black;">Advances in Mathematics</a> </i> &nbsp;
We study spectral properties for H_K,Ω, the Krein--von Neumann extension of the perturbed Laplacian -Δ+V defined on C^∞_0(Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂R^n  ...  Our work builds on that of Grubb in the early 1980's, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining  ...  Eduard Tsekanovskii for many helpful discussions and very valuable correspondence on various topics of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aim.2009.10.006">doi:10.1016/j.aim.2009.10.006</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/f57epsu6yjeu5efannjz4xamyq">fatcat:f57epsu6yjeu5efannjz4xamyq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190412214358/https://core.ac.uk/download/pdf/81216104.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9c/e9/9ce95035b5f77bdfa7e14f9cf92baba41d71a5ff.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aim.2009.10.006"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Optimal unions of scaled copies of domains and Pólya's conjecture [article]

Pedro Freitas, Jean Lagacé, Jordan Payette
<span title="2019-08-22">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that this sequence encodes information yielding conditions for Ω to satisfy Pólya's conjecture with either Dirichlet or Neumann boundary conditions.  ...  This approach allows us to recover a stronger version of Pólya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing  ...  We shall also think of the tiling F as a mere quasi-inclusion and we will not use F in our notations. Since V is bounded, there exists R > 0 such that V ⊂ rV for all r ≥ R.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1908.08441v1">arXiv:1908.08441v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ub6toefkcjh43nk2gaua2zliza">fatcat:ub6toefkcjh43nk2gaua2zliza</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200901020533/https://arxiv.org/pdf/1908.08441v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a5/0c/a50cf6761b0b8a67fa5886bc4f6804991fd36c20.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1908.08441v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Eigenvalue and gap estimates for the Laplacian acting on $p$-forms

Pierre Guerini, Alessandro Savo
<span title="2003-08-25">2003</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
The other usual boundary problem for differential forms is given by the relative boundary conditions: J ω = J δω = 0. It generalizes the Dirichlet boundary problem for functions.  ...  If ∂M = ∅, we consider the following eigenvalue problem defined by the absolute boundary conditions: ∆ω = µω, where J is the inclusion ∂M → M andν is the inward unit normal vector at each point of ∂M .  ...  The following corollary shows that Theorem 3.1 implies some well-known sharp lower bounds for the first Dirichlet eigenvalue λ 1,0 (M ). λ 1,0 (M ) ≥ 1 4 (n − 1) 2 H 2 + π 2 4R 2 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-03-03336-1">doi:10.1090/s0002-9947-03-03336-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j2hzeziwdndvfcoqukbkglkl6a">fatcat:j2hzeziwdndvfcoqukbkglkl6a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200813025500/https://www.ams.org/journals/tran/2004-356-01/S0002-9947-03-03336-1/S0002-9947-03-03336-1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/48/81/4881205cf53b06f32d83e7d307324c9f3347a3e8.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-03-03336-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs [article]

Tonći Antunović, Ivan Veselić
<span title="2008-11-27">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups.  ...  The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem  ...  Nakić and F. Sobieczky for stimulating discussions related to the topics of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0707.4292v2">arXiv:0707.4292v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tzjrcdsldzaj7ikoyezlh5bxxa">fatcat:tzjrcdsldzaj7ikoyezlh5bxxa</a> </span>
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Minimization problems for eigenvalues of the Laplacian [chapter]

Antoine Henrot
<span title="">2003</span> <i title="Birkhäuser Basel"> Nonlinear Evolution Equations and Related Topics </i> &nbsp;
For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.  ...  After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic.  ...  λ 1 with Dirichlet boundary condition on the outer boundary Γ and Neumann boundary condition on the boundary of the holes. • Several holes, Neumann boundary condition on the outer boundary Γ, Dirichlet  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-0348-7924-8_24">doi:10.1007/978-3-0348-7924-8_24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eeswi3lj4rcrxnicrn3mjm3jli">fatcat:eeswi3lj4rcrxnicrn3mjm3jli</a> </span>
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