Nodal Domains Research Articles

Existence of ground state sign-changing solutions for a class of quasilinear scalar field equations originating from nonlinear optics

This paper is mainly concerned with the existence of ground state sign-changing solutions for a class of second order quasilinear elliptic equations in bounded domains, which derived from nonlinear optics models. Combining a non-Nehari manifold method due to Tang and Cheng [J. Differ. Equations 261, 2384–2402 (2016)] and a quantitative deformation lemma with Miranda theorem, we obtain that the problem has at least one ground state solution and one ground state sign-changing solution with two precise nodal domains and we present the concrete lower bounds of ground state solution and ground state energy. We also obtain that any weak solution of the problem has C1,σ-regularity for some σ ∈ (0, 1). With the help of Maximum Principle, we reach the conclusion that the energy of the ground state sign-changing solutions is strictly larger than twice that of the ground state solutions.

Solvation modeling and optical properties of CdSO4-Doped L-Valine crystals

The utilization of computational solvation models for crystals proves effective in determining the amalgamation of molecular structures within the crystal medium, offering valuable insights into optimized crystal properties. This study focuses on identifying the arrangement of molecular strips within the crystal, crucial for recognizing active planes and the presence of non-linear optical (NLO) properties. To delve into the crystal's intricacies, a computational model was developed, and conformational analysis was conducted to ensure NLO property. The synthesis of the CdSO4-doped L-Valine metallo-organic complex crystal was achieved through the melt-freezing technique. The computational model was constructed to elucidate the formation of sub-planes within a confirmed orthorhombic crystal lattice. Mapping the molecular charge distribution with the dispersion of charge gradient facilitated the identification of chemical potential inhibition and its equipotential nodal domains. The estimation of electron density potential over critical points of the core carbons provided further insights. Chemical dynamics on the electron content of molecules for NLO properties were investigated by screening control parameters. The inelastic scattering ability of heteronuclear bonds for enhancing boosting potential was observed. Interactive frontier orbitals of degenerate energy states were illustrated, and the chemical potential of molecular zones was analyzed in-depth.

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

$2$-nodal domain theorems for higher-dimensional circle bundles

We prove that the real parts of equivariant (but non-invariant) eigenfunctions of generic bundle metrics on non-trivial principal S^{1} bundles over manifolds of any dimension have connected nodal sets and exactly 2 nodal domains. This generalizes earlier results of the authors in the 3 -dimensional case. The failure of the results on for non-free S^{1} actions is illustrated on even-dimensional spheres by one-parameter subgroups of rotations whose fixed point set consists of two antipodal points.

Coarse nodal count and topological persistence

Courant’s theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant’s theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the Bézout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.

Pleijel nodal domain theorem in non-smooth setting

Pleijel nodal domain theorem in non-smooth setting

The impact of two different rituximab-based strategies in cryoglobulinaemic vasculitis secondary to Sjögren's disease: a monocentric cohort study.

To compare two different rituximab (RTX)-based therapeutic approaches on vasculitic and lymphoproliferative-related disease activity and on non-Hodgkin lymphoma (NHL) development in a cohort of patients affected by cryoglobulinaemic vasculitis secondary to Sjögren's disease (Sjögren-CryoVasc). Three Sjögren-CryoVasc treatment groups were identified: 1) early RTX induction followed by maintenance; 2) late RTX induction with possible on-demand retreatment; 3) no RTX treatment. The following outcomes were evaluated: a) changes in cumulative ESSDAI, considering vasculitic-related and lymphoproliferative-related domains and changes in ESSDAI specific to each single vasculitic-related and lymphoproliferative-related domain; b) development of NHL; c) occurrence of persistent hypogammaglobulinemia associated with serious infections. 13 Sjögren-CryoVasc patients were identified: 1) 5/13 treated earlier with RTX with subsequent maintenance; 2) 5/13 treated late with RTX with possible on-demand retreatment; 3) 3/13 not treated with RTX. The two RTX groups showed a decrease in the ESSDAI score with group 1 showing the most substantial reduction (p=0.028). Patients receiving RTX exhibited significant improvement in cutaneous, PNS, and articular vasculitic-related ESSDAI domains (p=0.007; p=0.006; p=0.03, respectively). By contrast RTX did not greatly affect the lymphoproliferative-related ESSDAI domains, even if an improvement was noted in the glandular and nodal domains for group 1 (p=0.03; p=0.03, respectively). No differences in NHL occurrence or safety concerns were observed among the groups. RTX is an effective and safe treatment to control Sjögren-CryoVasc disease activity with a greater impact when administered earlier with a maintenance regimen. RTX alone cannot, however, affect the possible evolution of Sjögren-CryoVasc into an overt NHL.

Sign-changing solutions for critical Choquard equation on bounded domain

We consider the existence of sign-changing solutions for the following nonlinear Choquard equation{−Δu=(Iα⁎|u|p)|u|p−2u+λu,x∈Ω,u∈H01(Ω), where N≥3,α∈(0,N),λ∈R, p∈(1,N+αN−2] and Ω⊂RN is a smooth bounded domain. Iα:RN→R is the Riesz potential. In critical case p=N+αN−2, if Ω is symmetric about the axis x1, we firstly develop the limit profiles for the symmetric Palais-Smale sequence by the concentration compactness principle. Then we conclude that the problem admits an odd solution with exactly two nodal domains for λ∈(0,λ1),N≥4andα∈(2,N). In contrast, the local Brezis-Nirenberg type problem −Δu=|u|4N−2u+λu,u∈H01(Ω) can not permit such type of odd symmetry solution.

Zigzag phase transition of electrons confined within a thin annulus region

We consider a semiclassical approach to study the possible stabilization of a zigzag phase for a system of interacting electrons confined within a thin annulus region with infinite inner/outer walls. The electrons are considered spinless and interact via the standard Coulomb interaction potential. The classical minimum energy configuration of the system due to the Coulomb repulsion between electrons is accurately determined by using the simulated annealing method. As the number of electrons increases we see the appearance and stabilization of a two-ring structure with zigzag patterns. Further increase of the number of electrons appears to eventually lead to the collapse of the two-ring zigzag structure. A simple quantum treatment of the nodal features of the free many-particle wave function shows the appearance of nodal domain patterns that are consistent with the zigzag features that were observed in the classical model.

Distinguishing Bronchial Carcinoid from Benign Bronchocele using 68Ga-DOTATOC PET/CT Imaging.

Bronchial carcinoids are low-grade neuroendocrine tumors with slow growth rates and the potential to spread to nearby lymph nodes. Here we present a challenging case of bronchial carcinoid visualized alongside an adjacent benign bronchocele. Chest computed tomography (CT) identified the endobronchial mass with unclear morphological and diagnostic insights. A differential diagnosis of several benign and malignant etiologies was made. Subsequently, an endobronchial biopsy confirmed lung carcinoid. For better evaluation, a 68Ga-labeled 1,4,7,10-tetraazacyclododecane-N,N',N'',N'''-tetraacetic acid-d-Phe1-Tyr3-octreotide (68Ga-DOTATOC) positron emission tomography/CT scan was performed. The scan revealed a locally confined endobronchial mass with intense 68Ga-DOTATOC expression. Adjacent benign bronchocele was visualized with insignificant 68Ga-DOTATOC expression. Histopathological examination of the resected upper lobe confirmed these findings. This case highlights the importance of somatostatin receptor imaging in accurately identifying the extent of carcinoid tumors in the primary, nodal, and metastatic domains.

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Equivariant Solutions to the Optimal Partition Problem for the Prescribed Q-Curvature Equation

We study the optimal partition problem for the prescribed constant Q-curvature equation induced by the higher-order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed Q-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant Q-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of GJMS operators on Ricci solitons, compute the Q-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with a prescribed number of nodal domains on the Koiso–Cao Ricci soliton.

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.

Nodal Decompositions of a Symmetric Matrix

Abstract Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is $\Omega (n/\log n)$ and $o(n)$.

Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains

In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p,q)-Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at least γ(Ωλ∖{0}) pairs (±u) of odd weak solutions with precisely two nodal domains, where γ stands for the genus.

Modelling and non-linear analysis of beam-elements in reinforced concrete Vierendeel sandwich plates considering nodal stiffness domain effects

Modelling and non-linear analysis of beam-elements in reinforced concrete Vierendeel sandwich plates considering nodal stiffness domain effects

Local version of Courant’s nodal domain theorem

Local version of Courant’s nodal domain theorem

Nodal domain theorems for $p$-Laplacians on signed graphs

We establish various nodal domain theorems for p -Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph p -Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of p -Laplacians and Atay–Liu's multi-way Cheeger constants on signed graphs. In the particular case of p=1 , this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph 1 -Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of p -Laplacians with p>1 on connected bipartite graphs.

Nehari manifold approach for superlinear double phase problems with variable exponents

In this paper we consider quasilinear elliptic equations driven by the variable exponent double phase operator with superlinear right-hand sides. Under very general assumptions on the nonlinearity, we prove a multiplicity result for such problems whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains.

Exact Morse index of radial solutions for semilinear elliptic equations with critical exponent on annuli

Let Nge 3, R>rho >0 and A_{rho }:={xin mathbb {R}^N; rho<|x|<R}. Let U^{pm }_{n,rho }, nge 1, be a radial solution with n nodal domains of ΔU+|x|α|U|p-1U=0inAρ,U=0on∂Aρ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\Delta U+|x|^{\\alpha }|U|^{p-1}U=0 &{} \ ext {in}\\ A_{\\rho },\\\\ U=0 &{} \ ext {on}\\ \\partial A_{\\rho }. \\end{array}\\right. } \\end{aligned}$$\\end{document}We show that if p=frac{N+2+2alpha }{N-2}, alpha >-2 and Nge 3, then U^{pm }_{n,rho } is nondegenerate for small rho >0 and the Morse index {{textsf{m}}}(U^{pm }_{n,rho }) satisfies m(Un,ρ±)=n(N+2ℓ-1)(N+ℓ-1)!(N-1)!ℓ!for smallρ>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {{\ extsf{m}}}(U^{\\pm }_{n,\\rho }) =n\\frac{(N+2\\ell -1)(N+\\ell -1)!}{(N-1)!\\ell !} \\quad \ ext {for small}\\ \\rho >0, \\end{aligned}$$\\end{document}where ell =[frac{alpha }{2}]+1. Using Jacobi elliptic functions, we show that if (p,alpha )=(3,N-4) and Nge 3, then the Morse index of a positive and negative solutions {{textsf{m}}}(U^{pm }_{1,rho }) is completely determined by the ratio rho /Rin (0,1). Upper and lower bounds for {{textsf{m}}}(U^{pm }_{n,rho }), nge 1, are also obtained when (p,alpha )=(3,N-4) and Nge 3.