Searcharxiv⌕ Search

arXiv subjects

Peng-Fei Yao

Publications and source records attributed to Peng-Fei Yao.

12 recordsLinked to original sources
arXiv2026

Rigidity of Strain Tensors for Surfaces With Mixed Type and Applications To Shell Theory

This paper investigates the rigidity of strain tensors on surfaces with sign-changing Gaussian curvature (mixed-type surfaces) and applies the results to determine the optimal thickness exponent in the first Korn inequality for thin shells. Using tools from Riemannian geometry and generalized tensor analysis, we derive an infinitesimal rigidity lemma for strain tensors, which establishes \(L^2\) regularity estimates for displacements decomposed into tangential and normal components. Specifically, we show that for a mixed-type shell with a middle surface \(S = S^+ \cup Γ_0 \cup S^-\) (where \(S^+\), \(S^-\) have positive and negative curvature, respectively, and \(Γ_0\) is a parabolic interface), the optimal constant in Korn's inequality scales as \(h^{4/3}\), matching the behavior previously established for hyperbolic shells. This result is obtained via a combination of geometric decomposition, Fredholm theory for linear operators, and compactness arguments to handle the curvature transition across \(Γ_0\). The findings bridge the gap between elliptic and hyperbolic shell theories, providing a unified framework for understanding rigidity in complex geometries with mixed curvature. The derived estimates are shown to be sharp, offering critical insights for the mechanical design of thin-walled structures with non-uniform curvature.

math.AP↗
arXiv2026

Rigidity Estimate for Hyperbolic Shells and its Application in $Γ$-limit Theory

This paper establishes novel rigidity estimates for hyperbolic shells (surfaces with negative Gaussian curvature) and applies them to derive the \(Γ\)-limit of thin elastic shells. We prove a nonlinear rigidity estimate for \(H^1\) deformations on the mid-surface, and a nonlinear rigidity estimate for hyperbolic shells with clamped lateral boundary. The latter yields the optimal exponent \(h^{-4/3}\). As the main application, we characterize the \(Γ\)-limit of the nonlinear elastic energy for clamped hyperbolic shells across all scaling regimes \(β\in [0,2) \cup (8/3,\infty)\).

math.AP↗
arXiv2025

On Exponents of Thickness in Geometry Rigidity Inequality for Shells

We study exponents of thickness in Frieseck-James-Müller's inequalities for shells. We derive the following results: (a) the exponent of thickness $μ(S)\leq15/8$ if the middle surface $S$ is parabolic; (b) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a minimal surface with negative curvature; (c) the exponent of thickness $μ(S)\leq11/6$ if the middle surface $S$ is a ruled surface with negative curvature. The exponents of thickness in Frieseck-James-Müller's inequalities for thin shells represent the relationship between rigidity and thickness $h$ of a shell when the large deformations take place, i. e., the rigidity of the shell related to the thickness $h$ is $$Ch^{μ(S)}.$$ Thus the above results of $μ(S)<2$ show that those shells are strictly more rigid than plates since $μ(S)=2$ for plates. Moreover, we present another result which shows that when $μ(S)<2,$ any $W^{2,2}$ isometry of the middle surface is rigid.

math.AP↗
arXiv2023

Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations

In this paper, we consider the inverse problem of recovering a time-dependent nonlinearity for a third order nonlinear acoustic equation, which is known as the Jordan-Moore-Gibson-Thompson equation (J-M-G-T equation for short). This third order in time equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained for the small initial and boundary data. By the higher order linearization to the nonlinear equation, and construction of complex geometric optics (CGO for short) solutions for the linearized equation, we derive the uniqueness of recovering the nonlinearity.

math.AP↗
arXiv2021

Strain Tensors and Matching Property on Surfaces with the Gauss curvature changing sign

We prove the regularity of solutions to the strain tensor equation on a region $S$ with the Gauss curvature changing sign. Furthermore, we obtain the density property that smooth infinitesimal isometries are dense in the $W^{2,2}(S,\mathbb{R}^3)$ infinitesimal isometries. Finally, the matching property is established. Those results are important tools in obtaining recovery sequences ($Γ$-lim sup inequality) for dimensionally-reduced shell theories in elasticity.

math.AP↗
arXiv2021

Strain Tensors and Matching Property on Degenerated Hyperbolic Surfaces

We prove the regularity of solutions to the strain tensor equation on degenerated hyperbolic surfaces $S$ where the Gauss curvature is zero on a part of boundary. Furthermore, we obtain the density property that smooth infinitesimal isometries are dense in the $W^{2,2}({S},\R^3)$ infinitesimal isometries. Finally, the matching property is established. Those results are important tools in obtaining recovery sequences ($\Ga$-lim sup inequality) for dimensionally-reduced shell theories in elasticity.

math.AP↗
arXiv2018

Linear Strain Tensors on Hyperbolic Surfaces and Asymptotic Theories for Thin Shells

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces. We prove that if the surface is a smooth noncharacteristic region, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. The implications of this result for the elasticity of thin hyperbolic shells are discussed.

math-ph↗
arXiv2014

Radial Deformations and Cavitation in Riemannian Manifolds with Applications to Membrane Shells

This study is a geometric version of Ball's work, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557-611. Radial deformations in Riemannian manifolds are singular solutions to some nonlinear equations given by constitutive functions and radial curvatures. A geodesic spherical cavity forms at the center of a geodesic ball in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the constitutive functions and the radial curvatures. Some close relationships are shown among radial curvature, the constitutive functions, and the behavior of bifurcation of a singular solution from a trivial solution. In the incompressible case the bifurcation depends on the local properties of the radial curvature near the geodesic ball center but the bifurcation in compressible case is determined by the global properties of the radial curvatures. A cavity forms at the center of a membrane shell of isotropic material placed in tension by means of given boundary tractions or displacements when the Riemannian manifold under question is a surface of $\R^3$ with the induced metric. In addition, cavitation at the center of ellipsoids of $\R^n$ is also described if the Riemannian manifold under question is $(\R^n g)$ where $g(x)$ are symmetric, positive matrices for $x\in\R^n.$

math.AP↗
arXiv2013

Space of Infinitesimal Isometries and Bending of Shells

We discuss infinitesimal isometries of the middle surfaces and present some characteristic conditions for a function to be the normal component of an infinitesimal isometry. Our results show that those characteristic conditions depend on the Gaussian curvature of the middle surfaces: Normal components of infinitesimal isometries satisfy an elliptic problem, or a parabolic one, or a hyperbolic one according to the middle surface being elliptic, or parabolic, or hyperbolic, respectively. In those cases, a problem of determining an infinitesimal isometry is changed into that of 1-dimension. Then we apply those results to the energy functionals of bending of shells which has been obtained as two-dimensional problems by the limit theory of Gamma-convergence from the three-dimensional nonlinear elasticity. Therefore the limit theory of Gamma-convergence reduces to be a one-dimensional problem in the those cases.

math.AP↗
arXiv2006

Boundary controllability for the quasilinear wave equation

We study the boundary exact controllability for the quasilinear wave equation in the higher-dimensional case. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical condition. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.

math.AP↗