Nodal Domains Research Articles

Least energy sign-changing solutions for Kirchhoff\u2013Poisson systems

The paper deals with the following Kirchhoff–Poisson systems: \t\t\t0.1{−(1+b∫R3|∇u|2dx)Δu+u+k(x)ϕu+λ|u|p−2u=h(x)|u|q−2u,x∈R3,−Δϕ=k(x)u2,x∈R3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - ( {1+b\\int _{{\\mathbb{R}}^{3}} { \\vert \\nabla u \\vert ^{2}\\,dx} } ) \\Delta u+u+k(x)\\phi u+\\lambda \\vert u \\vert ^{p-2}u=h(x) \\vert u \\vert ^{q-2}u, & x\\in {\\mathbb{R}}^{3}, \\\\ -\\Delta \\phi =k(x)u^{2}, & x\\in {\\mathbb{R}}^{3}, \\end{cases} $$\\end{document} where the functions k and h are nonnegative, 0le lambda , b; 2le ple 4< q<6. Via a constraint variational method combined with a quantitative lemma, some existence results on one least energy sign-changing solution with two nodal domains to the above systems are obtained. Moreover, the convergence property of u_{b} as b searrow 0 is established.

The existence of sign-changing solution for a class of quasilinear Schr\xf6dinger\u2013Poisson systems via perturbation method

This paper is concerned with the existence of a sign-changing solution to a class of quasilinear Schrödinger–Poisson systems. There are some technical difficulties in applying variational methods directly to the problem because the quasilinear term makes it impossible to find a suitable space in which the corresponding functional possesses both smoothness and compactness properties. In order to overcome the difficulties caused by nonlocal term and quasi-linear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. And then, by using the approximation technique, a sign-changing solution with precisely two nodal domains is derived.

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 &gt; 0 as h large varies.

Eigenfunctions with Infinitely Many Isolated Critical Points

AbstractWe construct a Riemannian metric on the 2D torus, such that for infinitely many eigenvalues of the Laplace–Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of our construction implies that each of these eigenfunctions has a level set with infinitely many connected components (i.e., a linear combination of two eigenfunctions may have infinitely many nodal domains).

RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations

We present an open-source software RKPM2D for solving PDEs under the reproducing kernel particle method (RKPM)-based meshfree computational framework. Compared to conventional mesh-based methods, RKPM provides many attractive features, such as arbitrary order of continuity and discontinuity, relaxed tie between the quality of the discretization and the quality of approximation, simple h-adaptive refinement, and ability to embed physics-based enrichment functions, among others, which make RKPM promising for solving challenging engineering problems. The aim of the present software package is to support reproducible research and serve as an efficient test platform for further development of meshfree methods. The RKPM2D software consists of a set of data structures and subroutines for discretizing two-dimensional domains, nodal representative domain creation by Voronoi diagram partitioning, boundary condition specification, reproducing kernel shape function generation, domain integrations with stabilization, a complete meshfree solver, and visualization tools for post-processing. In this paper, a brief overview that covers the key theoretical aspects of RKPM is given, such as the reproducing kernel approximation, weak form using Nitsche’s method for boundary condition enforcement, various domain integration schemes (Gauss quadrature and stabilized nodal integration methods), as well as the fully discrete equations. In addition, the computer implementation aspects employed in RKPM2D are discussed in detail. Benchmark problems solved by RKPM2D are presented to demonstrate the convergence, efficiency, and robustness of the RKPM implementation.

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.

Sex Differences in the Effect of Alcohol Drinking on Myelinated Axons in the Anterior Cingulate Cortex of Adolescent Rats.

Cognitive deficits associated with teenage drinking may be due to disrupted myelination of prefrontal circuits. To better understand how alcohol affects myelination, male and female Wistar rats (n = 7–9/sex/treatment) underwent two weeks of intermittent operant self-administration of sweetened alcohol or sweetened water early in adolescence (postnatal days 28–42) and we tested for macro- and microstructural changes to myelin. We previously reported data from the males of this study showing that alcohol drinking reduced myelinated fiber density in layers II–V of the anterior cingulate division of the medial prefrontal cortex (Cg1); herein, we show that myelinated fiber density was not significantly altered by alcohol in females. Alcohol drinking patterns were similar in both sexes, but males were in a pre-pubertal state for a larger proportion of the alcohol exposure period, which may have contributed to the differential effects on myelinated fiber density. To gain more insight into how alcohol impacts myelinated axons, brain sections from a subset of these animals (n = 6/sex/treatment) were used for microstructural analyses of the nodes of Ranvier. Confocal analysis of nodal domains, flanked by immunofluorescent-labeled contactin-associated protein (Caspr) clusters, indicated that alcohol drinking reduced nodal length-to-width ratios in layers II/III of the Cg1 in both sexes. Despite sex differences in the underlying cause (larger diameter axons after alcohol in males vs. shorter nodal lengths after alcohol in females), reduced nodal ratios could have important implications for the speed and integrity of neural transmission along these axons in both males and females. Alcohol-induced changes to myelinated axonal populations in the Cg1 may contribute to long-lasting changes in prefrontal function associated with early onset drinking.

Implementation of the Nodal Discontinuous Galerkin Method for the Plate Vibration Problem using Linear Elasticity Equations

This work presents a numerical solution of the forced plate vibration problem using the nodal discontinuous Galerkin (DG) method. The plate is modelled as a three-dimensional domain, and its vibration is governed by the linear elasticity equations. The nodal DG method discretises the spatial domain and computes the spatial derivatives of the equations element-wise, while the time integration is conducted using the Runge-Kutta method. This method is in particular of interest as it is very favourable to carry out the computation by a parallel implementation. Several aspects regarding the numerical implementation such as the plate boundary conditions, the point force excitation, and the upwind numerical flux are presented. The numerical results are validated for rectangular concrete plates with different sets of boundary conditions and thicknesses, by a comparison with the exact mobilities that are derived from the classical plate theory (CPT) and the first order shear deformation theory (FSDT). The plate thickness is varied to understand its effect regarding the comparison with the CPT. An excellent agreement between the numerical solution and the FSDT was found. The agreement with the CPT occurs only at the first couple of resonance frequencies, and as the plate is getting thinner. Furthermore, the numerical example is extended to an L-shaped concrete plate. The mobility is then compared with the mobilities obtained by the CPT, FSDT, and linear elasticity equations.

A two-field semi-Lagrangian reproducing kernel model for impact and penetration simulation into geo-materials

In this paper, a displacement–pressure (u–p) semi-Lagrangian reproducing kernel (RK) is introduced to study the penetration depth of various projectile types into dry and fully saturated geo-materials. To describe the poromechanics of saturated materials, Biot theory is incorporated into the semi-Lagrangian formulation and a damage model is embedded into Drucker–Prager constitutive model to simulate the soil behavior and separation during the impact and penetration process. The stabilized nodal domain integration is developed in the two-field semi-Lagrangian RK formulation to ensure numerical stability, and the kernel contact algorithm is implemented to model the interaction between soil and projectile bodies. Several examples are studied to validate and assess the proposed method’s performance in predicting the final penetration depth, and the results are compared to those reported in the literature.

Radial ground state sign-changing solutions for asymptotically cubic or super-cubic fractional Schrödinger-Poisson systems

ABSTRACTIn this paper, we consider the following fractional Schrödinger-Poisson system: where , λ is a positive parameter and f satisfies asymptotically cubic or super-cubic growth that is very different from super-cubic growth in previous literatures. Without assuming the usual Nehari-type monotonic condition on , we establish the existence of one radial ground state sign-changing solution with precisely two nodal domains. Furthermore, we also prove that the energy of any radial sign changing solution is strictly larger than two times the least energy and give a convergence property of as . Our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for s=t=1.

Some remarks on nodal geometry in the smooth setting

We consider a Laplace eigenfunction $$\varphi _\lambda $$ on a smooth closed Riemannian manifold, that is, satisfying $$-\Delta \varphi _\lambda = \lambda \varphi _\lambda $$ . We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $$\varphi _\lambda $$ . This extends previous work of Jakobson and Mangoubi in case (M, g) is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. Ann Math (2) 187(1):241–262, 2018). Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains $$\Omega _\lambda $$ on which $$\varphi _\lambda $$ satisfies exponentially small $$L^\infty $$ bounds, we give some quantitative estimates for radii of inscribed balls.

Saddle solutions for the Choquard equation

We study the Choquard equation $$\begin{aligned} -\Delta u + u= (I_\alpha *|u|^p )|u|^{p-2}u \quad \text { in }\;{\mathbb {R}}^N \end{aligned}$$ where $$N\ge 2$$ , $$I_\alpha $$ is the Riesz potential of order $$\alpha \in (0, N)$$ and $$2\le p<\frac{N+\alpha }{N-2}$$ . For any given integer $$\ell \ge 2$$ , we construct a saddle type nodal solution with its $$2\ell $$ nodal domains meeting at the origin. Moreover, this equation admits nodal solutions of higher dimensional nodal structure with exactly $$2^k$$ nodal domains for each $$1\le k\le N$$ . This is a new phenomenon for the Choquard equation which is nonlocal in nature when compared with its local counterpart the nonlinear Schrodinger equation.

Existence of sign-changing solutions for nonlocal Kirchhoff-Schrödinger-type equations in [formula omitted

In this paper, we investigate the existence of sign-changing solution for a class nonlocal Kirchhoff-Schrödinger-type equation−(a+b∫R3|∇u|2dx)△u+V(x)u=f(x,u),x∈R3, where a and b are positive constants. With the help of the constraint variational method and via a direct approach, we show the existence of sign-changing solution and prove that the sign-changing solution has precisely two nodal domains. This work can be regarded as the complement for some results of the literature.

Local Frequency Interpretation and Non-Local Self-Similarity on Graph for Point Cloud Inpainting.

As 3D scanning devices and depth sensors mature, point clouds have attracted increasing attention as a format for 3D object representation, with applications in various fields such as tele-presence, navigation and heritage reconstruction. However, point clouds usually exhibit holes of missing data, mainly due to the limitation of acquisition techniques and complicated structure. Further, point clouds are defined on irregular non- Euclidean domains, which is challenging to address especially with conventional signal processing tools. Hence, leveraging on recent advances in graph signal processing, we propose an efficient point cloud inpainting method, exploiting both the local smoothness and the non-local self-similarity in point clouds. Specifically, we first propose a frequency interpretation in graph nodal domain, based on which we derive the smoothing and denoising properties of a graph-signal smoothness prior in order to describe the local smoothness of point clouds. Secondly, we explore the characteristics of non-local self-similarity, by globally searching for the most similar area to the missing region. The similarity metric between two areas is defined based on the direct component and the anisotropic graph total variation of normals in each area. Finally, we formulate the hole-filling step as an optimization problem based on the selected most similar area and regularized by the graph-signal smoothness prior. Besides, we propose voxelization and automatic hole detection methods for the point cloud prior to inpainting. Experimental results show that the proposed approach outperforms four competing methods significantly, both in objective and subjective quality.

On the Nodal Structure of Nonlinear Stationary Waves on Star Graphs

We consider stationary waves on nonlinear quantum star graphs, i.e., solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e., the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting, such as the classic Sturm oscillation theorem.

Sup-Norm and Nodal Domains of Dihedral Maass Forms

In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $\phi$ be a dihedral Maass form with spectral parameter $t_\phi$, then we prove that $\|\phi\|_\infty \ll t_\phi^{3/8+\varepsilon} \|\phi\|_2$, which is an improvement over the bound $t_\phi^{5/12+\varepsilon} \|\phi\|_2$ given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindel\"{o}f Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than $t_\phi^{1/8-\varepsilon}$ for any $\varepsilon>0$ for almost all dihedral Maass forms.

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schr\"odinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.

Guided graph spectral embedding: Application to the C. elegans connectome.

Graph spectral analysis can yield meaningful embeddings of graphs by providing insight into distributed features not directly accessible in nodal domain. Recent efforts in graph signal processing have proposed new decompositions—for example, based on wavelets and Slepians—that can be applied to filter signals defined on the graph. In this work, we take inspiration from these constructions to define a new guided spectral embedding that combines maximizing energy concentration with minimizing modified embedded distance for a given importance weighting of the nodes. We show that these optimization goals are intrinsically opposite, leading to a well-defined and stable spectral decomposition. The importance weighting allows us to put the focus on particular nodes and tune the trade-off between global and local effects. Following the derivation of our new optimization criterion, we exemplify the methodology on the C. elegans structural connectome. The results of our analyses confirm known observations on the nematode’s neural network in terms of functionality and importance of cells. Compared with Laplacian embedding, the guided approach, focused on a certain class of cells (sensory neurons, interneurons, or motoneurons), provides more biological insights, such as the distinction between somatic positions of cells, and their involvement in low- or high-order processing functions.

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.

On the Behavior of Clamped Plates under Large Compression

We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.