Nodal Domain Theorem Research Articles

Pleijel nodal domain theorem in non-smooth setting

Pleijel nodal domain theorem in non-smooth setting

On Pleijel's nodal domain theorem for the Robin problem

AbstractWe prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular, we remove the restriction, imposed in previous work, that the Robin parameter be non‐negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter, and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.

Local version of Courant’s nodal domain theorem

Local version of Courant’s nodal domain theorem

A Courant nodal domain theorem for linearized mean field type equations

A Courant nodal domain theorem for linearized mean field type equations

Long Wavelength Coherency in Well Connected Electric Power Networks

We investigate coherent oscillations in large scale transmission power grids, where large groups of generators respond in unison to a distant disturbance. Such long wavelength coherent phenomena are known as inter-area oscillations. Their existence in networks of weakly connected areas is well captured by singular perturbation theory. However, they are also observed in strongly connected networks without time-scale separation, where applying singular perturbation theory is not justified. We show that the occurrence of these oscillations is actually generic. Applying matrix perturbation theory, we show that, because these modes have the lowest oscillation frequencies of the system, they are only moderately sensitive to increased network connectivity between well chosen, initially weakly connected areas, and that their general structure remains the same, regardless of the strength of the inter-area coupling. This is qualitatively understood by bringing together the standard singular perturbation theory and Courant’s nodal domain theorem.

Non-boundedness of the number of super level domains of eigenfunctions

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in this case, the first eigenfunction is constant, and looking at the level sets of $u$ amounts to looking at the nodal sets $\{u-a=0\}$, where $a$ is a real constant. In the first part of the paper, we prove that the Extended Courant property is false for the subequilateral triangle and for regular $N$-gons ($N$ large), with the Neumann boundary condition. More precisely, we prove that there exists a Neumann eigenfunction $u_k$ of the $N$-gon, with labelling $k$, $4 \le k \le 6$, such that the set $\{u_k \not = 1\}$ has $(N+1)$ connected components. In the second part, we prove that there exists a metric $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$), which can be chosen arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $u$ of the associated Laplace-Beltrami operator, such that the set $\{u \not = 1\}$ has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin. As for the positive direction, in Appendix~B, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in $\mathbb{R}^2$.

On Pleijel\u2019s Nodal Domain Theorem for Quantum Graphs

We establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains nu _n of the nth eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with L^1-potentials and a variety of vertex conditions as well as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. Among other things, these results characterise the accumulation points of the sequence (frac{nu _n}{n})_{nin mathbb {N}}, which are shown always to form a finite subset of (0, 1]. This extends the previously known result that nu _nsim ngenerically, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which {nu _n}not sim {n}; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

Spectrum of Signless 1-Laplacian on Simplicial Complexes

We introduce the signless 1-Laplacian and the dual Cheeger constant on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is investigated. Our estimates can be comparable to Hoffman's bounds on Laplacian eigenvalues of simplicial complexes. An interesting inequality involving multiplicity of the largest eigenvalue, independence number and chromatic number is provided, which could be regarded as a variant version of Lovász sandwich theorem. Also, the behavior of 1-Laplacian under the topological operations of wedge and duplication of motifs is studied. The Courant nodal domain theorem in spectral theory is extended to the setting of signless 1-Laplacian on complexes.

Courant-sharp Robin eigenvalues for the square: The case of negative Robin parameter

We consider the cases where there is equality in Courant’s nodal domain theorem for the Laplacian with a Robin boundary condition on the square. In our previous two papers, we treated the cases where the Robin parameter h > 0 is large, small respectively. In this paper we investigate the case where h < 0.

Upper bounds for Courant-sharp Neumann and Robin eigenvalues

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in R n with C 2 boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. MSC classification (2010): 35P15, 49R05, 35P05.

Courant’s nodal domain theorem for positivity preserving forms

We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.

Courant-sharp Robin eigenvalues for the square and other planar domains

This paper is devoted to the determination of the cases where there is equality in Courant’s nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard–Helffer, Helffer–Persson Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter h , we focus on the case where h is large. We consider the case where h is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with piecewise C^{2, \alpha} boundary \alpha > 0 as h large varies.

Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter

This article is the continuation of our first work on the determination of the cases where there is equality in Courant’s Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Berard–Helffer, for the problem with a Dirichlet boundary condition (which corresponds to $$h=+\infty $$). There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer–Persson Sundqvist for the Neumann problem (which corresponds to $$h=0$$) to the case where $$ h>0$$ is small.

Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions

In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary conditions.

Sturm’s theorem on zeros of linear combinations of eigenfunctions

Motivated by recent questions about the extension of Courant’s nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm–Liouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the twentieth-century. Although not specialists in History of Sciences, we have tried to put this theorem into the context of nineteenth century mathematics.

A nodal domain theorem and a higher-order Cheeger inequality for the graph $p$-Laplacian

We consider the nonlinear graph p -Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph p -Laplacian for any p\geq 1 . While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian ( p = 2 ), the behavior changes for p = 1 . We show that the bounds are tight for p\geq 1 as the bounds are attained by the eigenfunctions of the graph p -Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph p -Laplacian for p > 1 . If the eigenfunction associated to the k -th variational eigenvalue of the graph p -Laplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p\to 1 .

Nodal decompositions of graphs

A nodal domain of a function is a maximally connected subset of the domain for which the function does not change sign. Courant's nodal domain theorem gives a bound on the number of nodal domains of eigenfunctions of elliptic operators. In particular, the k-th eigenfunction contains no more than k nodal domains. We prove a generalization of Courant's theorem to discrete graphs. Namely, we show that for the k-th eigenvalue of a generalized Laplacian of a discrete graph, there exists a set of corresponding eigenvectors such that each eigenvector can be decomposed into at most k nodal domains. In addition, we show this set to be of co-dimension zero with respect to the entire eigenspace.

Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions

Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions, Discrete Analysis 2017:13, 20 pp. An important phenomenon that often occurs with linear operators is that the complexity of their eigenfunctions is closely related to the size of their eigenvalues. To give a particularly simple example, the differential operator $f\mapsto f''$, defined on the space of smooth functions on the circle, has trigonometric functions as its eigenfunctions. The eigenvalue associated with a function of the form $A\cos (n\theta)+B\sin (n\theta)$ is $-n^2$, while the function itself oscillates $n$ times as it goes round the circle, so as the size of the eigenvalue increases, the function becomes more complex, in the sense that it oscillates more. A _nodal domain_ of an eigenfunction $f$ defined on a manifold is a connected component where $f$ does not change sign. A fundamental theorem of Courant, his nodal domain theorem, states that the $n$th eigenfunction of a Laplacian (under suitable conditions) has at most $n$ nodal domains. This paper concerns oscillations of the eigenfunctions for a fractional Schrodinger operator on the real line. The authors analyse a related problem concerning functions defined on the upper half plane, where they prove a result similar to the nodal domain theorem. In order to use this to obtain estimates for the number of oscillations of the fractional Schrodinger operator, they use combinatorial arguments in a surprising way. A _noncrossing partition_ is a partition of a totally ordered set $X$ with the property that if $a < b < c < d$, then it is not possible for $a\sim c$ and $b\sim d$, where $\sim$ is the equivalence relation corresponding to the partition. The name is due to the fact that if the points of $X$ are drawn in order on a circle, then a partition is noncrossing if and only if it is possible to decompose the circle into nonoverlapping connected regions such that each region contains the points from one cell of the partition on its boundary. (Equivalently, but slightly less naturally, the convex hulls of the equivalence classes are disjoint.) Noncrossing partitions were introduced by Kreweras in 1972 and have had applications in a wide variety of areas, to which this paper adds another. The one-dimensional Schrodinger operator is the operator $-\frac {d^2}{dx^2} + V(x)$, where $V$ is a suitable potential. A _fractional_ Schrodinger operator is an operator of the form $(-\frac{d^2}{dx^2})^{\alpha/2}+V(x)$, where $0<\alpha<2$ and one makes sense of the fractional Laplacian in the usual way: convert it into a multiplier by taking Fourier transforms, raise the multiplier to the power $\alpha/2$, and take the inverse Fourier transform again. There are many reasons to be interested in the behaviour of its eigenfunctions. The motivation for the authors is to understand better the behaviour of travelling waves for equations where the fractional Laplacian models dispersion. Article image by [Dmitry Belayev](http://people.maths.ox.ac.uk/belyaev/)

Nodal domains of eigenvectors for 1-Laplacian on graphs

The eigenvectors for graph 1-Laplacian possess some sort of localization property: On one hand, the characteristic function on any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques. The Courant nodal domain theorem for graphs is extended to graph 1-Laplacian for strong nodal domains, but for weak nodal domains it is false. The notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the number of independent eigenvectors. A positive answer is given to a question raised in Chang (2016) [3], to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph 1-Laplacian.

Courant-sharp eigenvalues of the three-dimensional square torus

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus ( R / Z ) 3 (\mathbb {R}/\mathbb {Z})^3\, , all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber–Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Pérez, P. Romon, and A. Ros (2004) on the periodic isoperimetric problem.