Ellipticity Condition Research Articles

Gradient estimates for unbounded Laplacians with ellipticity condition on graphs

In this article, we prove various gradient estimates for unbounded graph Laplacians which satisfy the ellipticity condition. Unlike common assumptions for unbounded Laplacians, i.e. completeness and non-degenerate measure, the ellipticity condition is purely local that is easy to verify on a graph. First, we establish an equivalent semigroup property, namely the gradient estimate of exponential curvature-dimension inequality, which is a modification of the curvature-dimension inequality and can be viewed as a notion of curvature on graphs. Additionally, we use the semigroup method to prove the Li-Yau inequalities and the Hamilton inequality for unbounded Laplacians on graphs with the ellipticity condition.

Conditions for strong ellipticity of 4th-order elasticity tensors

Conditions for strong ellipticity of 4th-order elasticity tensors

Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

Given a linear elliptic equation \sum a_{ij}u_{ij}=0 in \mathbb{R}^{3} , it is a classical problem to determine if its 1-homogeneous solutions u are linear. The answer is negative in general, by a construction of Martinez-Maure. In contrast, the answer is affirmative in the uniformly elliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a known open problem to determine the degenerate ellipticity condition on (a_{ij}) under which this theorem still holds. In this paper we solve this problem. We prove the linearity of u under the following degenerate ellipticity condition for (a_{ij}) , which is sharp by Martinez-Maure’s example: if \mathcal{K} denotes the ratio between the largest and smallest eigenvalues of (a_{ij}) , we assume \mathcal{K}|_{\mathcal{O}} lies in L_{\rm loc}^{1} for some connected open set \mathcal{O}\subset \mathbb{S}^{2} that intersects any configuration of four disjoint closed geodesic arcs of length \pi in \mathbb{S}^{2} . Our results also give the sharpest possible version under which an old conjecture by Alexandrov, Koutroufiotis and Nirenberg (disproved by Martinez-Maure’s example) holds.

Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)

In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pagès and Panloup (2022) we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in Pagès and Panloup (2022) and Chen et al. (2023). As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularization properties — this is done under an ellipticity condition, using Malliavin calculus for jump processes.

Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier–Stokes Equations: I. Weak Solution Existence

We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution.

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

On the Hm-Regularity for Second Order Elliptic Equations

Our primary objective in this paper is to discuss the Hm-Regularity for second order elliptic equations over Sobolev Spaces. We here consider the cases m = 1 and m≥ 2 separately. We revisit some elementary concepts in Functional Analysis and Abstract Harmonic Analysis before providing a proper definiton to the notion of weak solution of a Dirichlet Problem. While towards the later stages, we shall classify different types of regularity conditions, the main focus lies upon deducing appropriate ellipticity conditions in support of commenting about the existence and uniqueness of the weak solutions to a given problem. Appropriate references are provided in the bibliography section to facilitate further reading for ardent readers and researchers in this field

On the Hm-Regularity for Second Order Elliptic Equations

Our primary objective in this paper is to discuss the Hm-Regularity for second order elliptic equations over Sobolev Spaces. We here consider the cases m = 1 and m≥ 2 separately. We revisit some elementary concepts in Functional Analysis and Abstract Harmonic Analysis before providing a proper definiton to the notion of weak solution of a Dirichlet Problem. While towards the later stages, we shall classify different types of regularity conditions, the main focus lies upon deducing appropriate ellipticity conditions in support of commenting about the existence and uniqueness of the weak solutions to a given problem. Appropriate references are provided in the bibliography section to facilitate further reading for ardent readers and researchers in this field

Multiple solutions of nonlinear Neumann inclusions

We prove new results on the existence of multiple solutions for elliptic inclusions with nonlinear boundary conditions of Neumann type. Our approach is topological and relies on the fixed point index for multivalued map.

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.

On ellipticity of operators with shear mappings

The nonlocal boundary value problems are considered, in which the main operator and the operators in the boundary conditions include the differential operators and twisting operators. The de nition of the trajectory symbols for this class of problems is given. We show that the elliptic problems de ne the Fredholm operators in the corresponding Sobolev spaces. The ellipticity condition of such nonlocal boundary value problem is given.

Machine Learning for Film Thickness Prediction in Elastohydrodynamic Lubricated Elliptical Contacts

This study extends the use of Machine Learning (ML) approaches for lubricant film thickness predictions to the general case of elliptical elastohydrodynamic (EHD) contacts, by considering wide and narrow contacts over a wide range of ellipticity and operating conditions. Finite element (FEM) simulations are used to generate substantial training and testing datasets that are used within the proposed ML framework. The complete dataset entails 915 samples; split into an 823-sample training dataset and a 92-sample testing dataset, corresponding to 90% and 10% of the combined dataset samples, respectively. The proposed ML model consists of a pre-processing stage in which conventional EHD dimensionless groups are used to minimize the number of inputs into the model, reducing them to only three. The core of the model is based on Gaussian Process Regression (GPR), a powerful ML regression tool, well-suited for small-sized datasets, producing output central and minimum film thicknesses, also in dimensionless form. The last stage is a post-processing one, in which the output film thicknesses are retrieved in dimensional from. The results reveal the capabilities and potential of the proposed ML framework, producing quasi-instantaneous predictions that are far more accurate than conventional film thickness analytical formulae. In fact, the produced central and minimum film thickness predictions are on average within 0.3% and 1.0% of the FEM results, respectively.

Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition

Bifurcations and exact solutions in the model of nonlinear elastodynamics of materials with strong ellipticity condition

Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates

We obtain a priori local pointwise second derivative estimates for solutions u to a class of augmented Hessian equations on Riemannian manifolds, in terms of the C1 norm and certain W2,p norms of u. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any W2,p norm can be dropped. Our results are derived using integral estimates.

Hyperbolicity, Mach Lines, and Super-Shear Mode III Steady-State Fracture in Magneto-Flexoelectric Materials, Part II: Crack-Tip Asymptotics

Abstract In our previous study (Part I), the anti-plane steady-state hyperbolic mode III fracture of a magneto-flexoelectric material was solved for the displacement, the polarization, and the magnetic fields. The solution, however, was based on the assumption of the development of strain discontinuities, and the propagation of the crack-tip was related to a critical shear strain. However, in the current study, the asymptotic details of the fields close to the crack-tip were investigated. The asymptotic analysis assumes strain continuity at the crack-tip (discontinuity in the strain gradients) and reveals the existence of a positive dynamic J-integral. The asymptotic analysis was performed not only for hyperbolic but also for elliptic conditions, and the energy release rate was calculated as a function of the crack-tip velocity in both regimes. These results are very different from those predicted by classical singular elastodynamics, where the dynamic J-integral is zero when super-shear is attained and there can be only an elliptic solution. Moreover, the results are very useful for couple-stress elastodynamics where equivalent length scales are present due to the analogy with flexoelectricity.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

AbstractIn this paper, we show that within the class of isotropic Mindlin‐Toupin gradient theories there exist a particular variant of the theory, which allows a completely simplified form of traction boundary value problems. This theory can be obtained assuming that the high‐grade part of the strain energy density depends only on the vector‐type quantities, that are the gradient of dilatation and the curl of small rotation. Such incomplete gradient theory becomes positive semi‐definite and at the same time it obeys the strong ellipticity conditions of the general Mindlin‐Toupin first strain gradient elasticity. Based on the variational approach it is shown that the equilibrium equations of the developed theory and its definition for the surface traction can be given only in terms of the total stresses (like in classical elasticity). Such formulation can be useful for derivation of the closed form solutions for the problems with traction‐type boundary conditions. Examples of the solutions for the problem of cylindrical bending and for the inplane crack tip fields are presented. It is shown that considered theory allows to obtain a regularized solution for the crack problems and at the same time it does not predict a non‐physical infinite increase of the material's rigidity under bending.

Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor

Strong ellipticity condition (SE-condition) of the equilibrium equations for general nonlinear elastic materials can be equivalently transformed into the SE-condition of a partially symmetric tensor. Qi et al. in 2009 proved that the SE-condition of a partially symmetric tensor holds if and only if its smallest M-eigenvalue is positive. In this paper, a criterion for the SE-condition of a partially symmetric tensor is first given. And then, an alternative method to compute all M-eigentriples of a partially symmetric tensor is presented. Finally, two numerical examples show the effectiveness of the proposed criterion and the calculation of M-eigenvalues in judging the SE-condition of the equilibrium equations.

Constructing a model of the influence of structural and technological inhomogeneities on electromagnetic energy losses in cables based on a twisted pair

The objects of research reported here were cables based on twisted pairs of various designs. The issue of the emergence of additional losses of electromagnetic energy is related to structural and technological inhomogeneities at the technological stage of cable production. The influence of the working capacitance of the twisted pair on the energy losses in the cables has been substantiated. A methodology was proposed for the numerical calculation of the electric field under the condition of ellipticity of the structural elements of the twisted pair. That has made it possible to determine the distortion of the electric field and the effect of inhomogeneities on the working capacitance of a twisted pair of different designs. Specifically, it was shown that for shielded structures with continuous polymer insulation, the distortion of the electric field and the growth of the working capacitance are observed to a greater extent. The need to find technological solutions to reduce the effect of screen ellipticity on cable capacitance has been emphasized. The effectiveness of capacitance regulation in the presence of inhomogeneities has been confirmed, by using foamed insulation to reduce the loss of electromagnetic energy in the cable. The effect of reducing the working capacitance and increasing additional losses under the simultaneous influence of the ellipticity of the electrical insulation and the protective polymer shell has been established, which necessitates the technological process adjustment in the manufacture of twisted pair. An applied aspect of using the results is the possibility of improving the typical technological process of manufacturing twisted pairs to reduce additional losses of electromagnetic energy at the final technological stage of cable production. The adequacy of the numerical studies was confirmed by experimental dependences of the attenuation coefficient and additional energy losses of the unshielded cable in the frequency range up to 100 MHz

FORWARD MODELING THE GROUP AND PHASE VELOCITIES OF RAYLEIGH AND LOVE SURFACE WAVES BENEATH THE CENTRAL ANATOLIA: FIFTH PARAMETER FOR TRANSVERSE ISOTROPY

The complex dynamic processes such as the magma movements in the crust and uppermost mantle result in the anisotropic wave propagation. In case of the Rayleigh and Love surface waves, this anisotropy is known as the Rayleigh-Love wave discrepancy. The surface wave propagation beneath the Central Anatolia shows this discrepancy for which we utilize the vertical transverse isotropy in the form of forward solutions. We use single-station and two-station methods to attain the observed surface wave dispersion curves in the period range 7-40 s and then apply the two-dimensional (2-D) tomography to convert these curves into the velocity maps defined through a 0.05o x 0.05o – sized grid. The damped least-squares technique is used to invert the individual group and phase velocity curves for the one-dimensional (1-D) structure. The latter 1-D inversion provides the depth profiles for the Voigt isotropic average shear-wave velocity (𝑉𝑆) and the radial anisotropy (𝜉). The 𝑉𝑆 and 𝜉 depth profiles are employed to construct the vertical transverse isotropic (VTI) velocity structure beneath a grid point. Through the forward modeling, the VTI velocity structure is revised to jointly fit the observed Rayleigh and Love group and phase velocities. In the forward modeling, the fifth VTI parameter (𝜂𝐾) measuring the departure from the elliptic condition (𝜂𝐾=1.0) and the anisotropic velocity perturbations (𝑎𝑃2⁄ and 𝑎𝑆2⁄) are primarily adjusted to achieve the fit between the observed and theoretical dispersion curves. In the depth range ~20-30 km, the 𝜂𝐾 is generally found to be 𝜂𝐾<1.0 changing in the range 0.94≤𝜂𝐾≤0.95, which is consistent with the PREM and the expression 𝑎𝑃2⁄=0.5𝑎𝑆2⁄ appears to be valid for the mid-to-lower crust beneath the studied region. In particular, the Rayleigh group velocities require the setting 𝜂𝐾<1.0 in the mid-to-lower crustal depth range.