arXiv research

Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.

Proves Payne conjecture for buckling and membrane eigenvalues.

In this paper, we investigate universal estimates for eigenvalues of a buckling problem. For a bounded domain in a Euclidean space, we give a positive contribution for obtaining a sharp universal inequality for eigenvalues of the buckling problem. For a domain in the unit sphere, we give an important improvement on the…

This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimat…

We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere $(-Δ)^p u=Λ(-Δ) u$ with order $p(\geq2)$. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th eigenvalues independent of the domains. In particular, for $p=2$, our result is shar…

We obtain universal inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains in $\mathbb{M}\times\mathbb{R}$.

The study classifies biharmonic submanifolds in a sphere using specific eigenmaps.

In this paper we study the eigenvalues of buckling problem on domains in a unit sphere. By introducing a new parameter and using Cauchy inequality, we optimize the inequality obtained by Wang and Xia in [12].

In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate on the second eigenvalue in terms of the first eigenvalue, which improves one recent result obtained by Wang-Xia in [7].

We investigate the eigenvalues of the buckling problem of arbitrary order on compact domains in Euclidean spaces and spheres. We obtain universal bounds for the $k$th eigenvalue in terms of the lower eigenvalues independently of the particular geometry of the domain.

Liu's paper contains an error regarding eigenvalues.

For curves of prescribed length embedded into the unit disc in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds $2π$ and in the large length limit. In the small excess length case, we prove convergence to a fourth order obstacle type problem with integral constraint on…

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key…

We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with $g_0$ the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, D…

Extends plate problems to differential forms on manifolds.

In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.

Frustration causes buckling-like behavior in tubular foldable mechanisms.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

A generalization of the Euler-Plateau problem to account for the energy contribution due to twisting of the bounding loop is proposed. Euler-Lagrange equations are derived in a parameterized setting and a bifurcation analysis is performed. A pair of dimensionless parameters govern bifurcations from a flat, circular gro…

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensiona…

The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…

The paper explores inequalities between eigenvalues on Riemannian manifolds.

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues,…

Let p an integer. We define a family of idempotents (and nilpotents) in the Temperley - Lieb algebras at 4p-th roots of unity which generalize the usual Jones-Wenzl idempotents. These new idempotents correspond to finite dimentional simple and projective indecomposable representations of the restricted quantum group Uq…

The paper studies biharmonic functions and bi-eigenfunctions on spheres and model spaces.

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

Sharp bounds derived for eigenvalues on specific geometric spaces.

Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.

Study finds knots with ideal length need not have smallest volume.

The study investigates how branch points affect the shape and mechanics of hyperbolic surfaces.

Develops discrete geometry for non-constant curvature surfaces.

Local GP approach improves simulation efficiency for large datasets.

Study develops efficient algorithm for probabilistic penetration response of composite plates.

The paper proposes a method to model non-smooth functions using clustering, classification, and Gaussian process modeling.

We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equa…