Nontrivial Curve Research Articles

On the Fučík spectrum of the [formula omitted]-Laplacian with no-flux boundary condition

In this paper, we study the quasilinear elliptic problem −Δpu=au+p−1−bu−p−1inΩ,u=constanton∂Ω,0=∫∂Ω|∇u|p−2∇u⋅νdσ,where the operator is the p-Laplacian and the boundary condition is of type no-flux. In particular, we consider the Fučík spectrum of the p-Laplacian with no-flux boundary condition which is defined as the set Πp of all pairs (a,b)∈R2 such that the problem above has a nontrivial solution. It turns out that this spectrum has a first nontrivial curve C being Lipschitz continuous, decreasing and with a certain asymptotic behavior. Since (λ2,λ2) lies on this curve C, with λ2 being the second eigenvalue of the corresponding no-flux eigenvalue problem for the p-Laplacian, we get a variational characterization of λ2. This paper extends corresponding works for Dirichlet, Neumann, Steklov and Robin problems.

Monodromy bootstrap for SU(2|2) quantum spectral curves: from Hubbard model to AdS3/CFT2

We propose a procedure to derive quantum spectral curves of AdS/CFT type by requiring that a specially designed analytic continuation around the branch point results in an automorphism of the underlying algebraic structure. In this way we derive four new curves. Two are based on SU(2|2) symmetry, and we show that one of them, under the assumption of square root branch points, describes Hubbard model. Two more are based on SU(2|2) × SU(2|2). In the special subcase of zero central charge, they both reduce to the unique nontrivial curve which furthermore has analytic properties compatible with PSU(1, 1|2) × PSU(1, 1|2) real form. A natural conjecture follows that this is the quantum spectral curve of AdS/CFT integrable system with AdS3 × S3 × T4 background supported by RR-flux. We support the conjecture by verifying its consistency with the massive sector of asymptotic Bethe equations in the large volume regime. For this spectral curve, it is compulsory that branch points are not of the square root type which qualitatively distinguishes it from the previously known cases.

Homogenization of Steklov eigenvalues with rapidly oscillating weights

In this article we study the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions. The results are obtained via a careful study of oscillating functions on the boundary and a precise estimate of the \(L^\infty \) bound of eigenfunctions. As an application we provide some estimates on the first nontrivial curve of the Dancer–Fučík spectrum.

Dancer–Fuc̆ik spectrum for fractional Schrödinger operators with a steep potential well on [formula omitted

In this paper, we study Dancer–Fuc̆ik spectrum of the fractional Schrödinger operators which is defined as the set of (α,β)∈R2 such that (−Δ)su+Vλ(x)u=αu++βu−in RNhas a nontrivial solution u, where the potential Vλ has a steep potential well for sufficiently large parameter λ>0. It is allowed that (−Δ)s+Vλ has essential spectrum with finitely many eigenvalues below the infimum of σess(−Δ)s+Vλ. Many difficulties are caused by general nonlocal operators, we develop new techniques to overcome them to construct the first nontrivial curve of Dancer–Fuc̆ik point spectrum by minimax methods, to show some qualitative properties of the curve, and to prove that the corresponding eigenfunctions are foliated Schwartz symmetric. As applications we obtain the existence of nontrivial solutions for nonlinear Schrödinger equations with nonresonant nonlinearity.

Fučik spectrum for the Kirchhoff-type problem and applications

In this study, we focus on the Fučik spectrum for the Kirchhoff-type problem, which is defined as a set Σ comprising those (α,β)∈R2 such that (0.1)−∫Ω|∇u|2Δu=α(u+)3+β(u−)3,inΩ,u=0,on∂Ωhas a nontrivial solution, where Ω is an open ball in RN for N=1,2,3; or Ω⊂R2 is symmetric in x and y, and convex in the x and y directions, u+=max{u,0}, u−=min{u,0}, and u=u++u−. First, we prove that the curves {μ1}×R, R×{μ1}, and C≔{(s+c(s),c(s)):s∈R} belong to Σ, where c(s)=min{β:(s+β,β)∈Σ0} and Σ0 comprises those (α,β)∈R2 such that (0.1) has a sign changing solution. We refer to {μ1}×R and R×{μ1} as trivial curves in Σ in the sense that any solution of (0.1) with (α,β)∈{μ1}×R or R×{μ1} is signed. We denote C as the first nontrivial curve in Σ in the sense that any solution of (0.1) with (α,β)∈C is sign changing and for each s∈R, we consider the line that passes through (s,0) with a slope of 1 in the αOβ plane R2, then the first point on this line that intersects with Σ0 is simply (s+c(s),c(s))∈C. Second, we investigate some properties of the function c and the curve C. In particular, c is Lipschitz continuous, decreasing on R and c(s)→μ1 as s→∞, and C is asymptotic to the broken line ℒ2≔{μ1}×[μ1,∞)∪[μ1,∞)×{μ1}. Furthermore, we show that the point (α,β) corresponding to the signed solution of (0.1) is from ℒ≔({μ1}×R)∪(R×{μ1}), the point (α,β) corresponding to the sign changing solution of (0.1) is on the upper right of ℒ2, and no nontrivial solution of (0.1) exists when (α,β) is between ℒ2 and C. Finally, as an application, we establish the multiplicity of solutions to the following Kirchhoff-type problem: −1+∫Ω|∇u|2Δu=f(x,u),inΩ,u=0,on∂Ω,where the nonlinearity f is asymptotically linear at zero and asymptotically 3-linear at infinity. To the best of our knowledge, this is the first study to consider that the nonlinearity has an extension property at both the zero and infinity points.

Waist size for cusps in hyperbolic 3-manifolds II

The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In this paper, it is proved that the next two smallest waist sizes are realized uniquely for the cusps in the $$5_2$$ knot complement and the manifold obtained by (2,1)-surgery on the Whitehead link. One application is an improvement on the universal upper bound for the length of an unknotting tunnel in a 2-cusped hyperbolic 3-manifold.

On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

Let B 1 B_1 be a ball in R N \mathbb {R}^N centred at the origin and let B 0 B_0 be a smaller ball compactly contained in B 1 B_1 . For p ∈ ( 1 , ∞ ) p\in (1, \infty ) , using the shape derivative method, we show that the first eigenvalue of the p p -Laplacian in annulus B 1 ∖ B 0 ¯ B_1\setminus \overline {B_0} strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 p \to 1 and p → ∞ p \to \infty are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p p -Laplacian on bounded radial domains.

A representation formula of incompressible liquid crystal flow and its applications

In this paper, we give a representation formula for the solutions of incompressible liquid crystal flow in arbitrary dimensions. As a direct application, one can obtain a strong global solution in two dimensions, as long as the image of the initial orientation field of the liquid crystal lies in a plane, or is a nontrivial curve on the unit sphere. The results also hold for inhomogeneous incompressible liquid crystals. This generalizes the results in Gong et al (2015 Nonlinearity 28 3677–94) and Dong and Lei (2012 (arXiv:org/abs/1205.3697)).

On the eigenvalues and Fučik spectrum of p-fractional Hardy-Sobolev operator with weight function

In this article, we study the nonlinear eigenvalue problem of fractional Hardy–Sobolev operator where is a bounded domain in with Lipschitz boundary containing 0, , , , and the weight function V, having nontrivial positive part, belongs to suitable integrable class and may change sign. We investigate some properties of the first eigenvalue such as simplicity and isolation. Moreover, we also study the Fučik spectrum of fractional Hardy-Sobolev operator, which is defined as the set such that has a non-trivial solution u. We show the existence of a first nontrivial curve of this spectrum and also we prove some properties of this curve. At the end, we study a nonresonance problem with respect to the weighted Fučik spectrum.

The Fucík spectrum of the p-Laplacian and jumping nonlinear problems

We prove the existence of a nontrivial curve in the Fucík spectrum of the p-Laplacian using the sequence of minimax eigenvalues constructed by cohomological index. Furthermore, we study the solvability of jumping nonlinear problems:{−Δpu=α(u+)p−1−β(u−)p−1+f,x∈Ω,u=0,x∈∂Ω.

On the first curve of the Fučík spectrum for elliptic operators

In this Note we present a new variational characterization of the first nontrivial curve of the Fučík spectrum for elliptic operators with Dirichlet or Neumann boundary conditions. Moreover, we describe the asymptotic behaviour and some properties of this curve and of the corresponding eigenfunctions. In particular, this new characterization allows us to compare the first curve of the Fučík spectrum with the infinitely many curves we obtained in previous works (see [8, 9]): for example, we show that these curves are all asymptotic to the same lines as the first curve, but they are all distinct from such a curve.

Convergence Rates in a Weighted Fučik Problem

Abstract In this work we consider the Fučik problem for a family of weights depending on ε with Dirichlet and Neumann boundary conditions. We study the homogenization of the spectrum. We also deal with the special case of periodic homogenization and we obtain the rate of convergence of the first non-trivial curve of the spectrum.

On the Fučik spectrum of non-local elliptic operators

In this article, we study the Fu\v{c}ik spectrum of fractional Laplace operator which is defined as the set of all $(\al,\ba)\in \mb R^2$ such that \begin{equation*} \quad \left. \begin{array}{lr} \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\ \end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$, where $\Om$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $n>2s$, $s\in(0,1)$. The existence of a first nontrivial curve $\mc C$ of this spectrum, some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to Fu\v{c}ik spectrum.

The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues

It is well-known that the second eigenvalue λ2 of the Dirichlet Laplacian on the ball is not radial. Recently, Bartsch, Weth and Willem proved that the same conclusion holds true for the so-called nontrivial (sign changing) Fucik eigenvalues on the first curve of the Fucik spectrum which are close to the point (λ2, λ2). We show that the same conclusion is true in dimensions 2 and 3 without the last restriction.

On the Fu c ˘ ik spectrum for the [formula omitted]-Laplacian with Robin boundary condition

The aim of this paper is to study the Fuc˘ik spectrum of the p -Laplacian with Robin boundary condition given by − Δ p u = a ( u + ) p − 1 − b ( u − ) p − 1 in Ω , | ∇ u | p − 2 ∂ u ∂ ν = − β | u | p − 2 u on ∂ Ω , where β ≥ 0 . If β = 0 , it reduces to the Fuc˘ik spectrum of the negative Neumann p -Laplacian. The existence of a first nontrivial curve C of this spectrum is shown and we prove some properties of this curve, e.g., C is Lipschitz continuous, decreasing and has a certain asymptotic behavior. A variational characterization of the second eigenvalue λ 2 of the Robin eigenvalue problem involving the p -Laplacian is also obtained.

A-twisted Landau–Ginzburg models

In this paper we discuss correlation functions in certain A-twisted Landau–Ginzburg models. Although B-twisted Landau–Ginzburg models have been discussed extensively in the literature, virtually no work has been done on A-twisted theories. In particular, we study examples of Landau–Ginzburg models over topologically nontrivial spaces–not just vector spaces–away from large-radius limits, so that one expects nontrivial curve corrections. By studying examples of Landau–Ginzburg models in the same universality class as nonlinear sigma models on nontrivial Calabi–Yaus, we obtain nontrivial tests of our methods as well as a physical realization of some simple examples of virtual fundamental class computations.

On the Fučik point spectrum for Schrödinger operators on $${\mathbb{R}}^{N}$$

We investigate the Fucik point spectrum of the Schrodinger operator \(S_{\lambda} = - \Delta + V_{\lambda}\, {\rm in}\, L^{2}({\mathbb{R}}^{N})\) when the potential Vλ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that Sλ has essential spectrum with finitely many eigenvalues below the infimum of \(\sigma_{\rm ess}(S_\lambda)\). We construct the first nontrivial curve in the Fucik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schrodinger equations with jumping nonlinearity.

An asymmetric Neumann problem with weights

We prove the existence of a first nonprincipal eigenvalue for an asymmetric Neumann problem with weights involving the p-Laplacian (cf. (1.2) below). As an application we obtain a first nontrivial curve in the corresponding Fučik spectrum (cf. (1.4) below). The case where one of the weights has meanvalue zero requires some special attention in connexion with the (PS) condition and with the mountain pass geometry.

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let \Omega\subset\R^N be a ball or an annulus and f:\R\to\R absolutely continuous, superlinear, subcritical, and such that f(0)=0 . We prove that the least energy nodal solution of -\Delta u= f(u) , u\in H^1_0(\Omega) , is not radial. We also prove that Fučik eigenfunctions, i.e. solutions u\in H^1_0(\Omega) of -\Delta u=\lambda u^+-\mu u^- , with eigenvalue (\lambda,\mu) on the first nontrivial curve of the Fučik spectrum, are not radial. A related result holds for asymmetric weighted eigenvalue problems. An essential ingredient is a quadratic form generalizing the Hessian of the energy functional J\in C^1(H^1_0(\Omega)) at a solution. We give new estimates on the Morse index of this form at a radial solution. These estimates are of independent interest.

Lipshitz maps from surfaces

We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, then the Riemannian 2-sphere has Uryson 1-width less than 12 and contains a closed geodesic of length less than 160. Similarly, if a closed oriented Riemannian surface does not admit a degree 1 map to the unit 2-sphere with Lipshitz constant 1, then it contains a closed homologically non-trivial curve of length less than 4π. On the other hand, we give examples of high genus surfaces with arbitrarily large Uryson 1-width which do not admit a map of non-zero degree to the unit sphere with Lipshitz constant 1.