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Haigang Li

Publications and source records attributed to Haigang Li.

At least 37 records · Page 2Linked to original sources
arXiv2020

Asymptotics of the stress concentration in high-contrast elastic composites

A long-standing area of materials science research has been the study of electrostatic, magnetic, and elastic fields in composite with densely packed inclusions whose material properties differ from that of the background. For a general elliptic system, when the coefficients are piecewise Hölder continuous and uniformly bounded, an $\varepsilon$-independent bound of the gradient was obtained by Li and Nirenberg \cite{ln}, where $\varepsilon$ represents the distance between the interfacial surfaces. However, in high-contrast composites, when $\varepsilon$ tends to zero, the stress always concentrates in the narrow regions. As a contrast to the uniform boundedness result of Li and Nirenberg, in order to investigate the role of $\varepsilon$ played in such kind of concentration phenomenon, in this paper we establish the blow-up asymptotic expressions of the gradients of solutions to the Lamé system with partially infinite coefficients in dimensions two and three. We discover the relationship between the blow-up rate of the stress and the relative convexity of adjacent surfaces, and find a family of blow-up factor matrices with respect to the boundary data. Therefore, this work completely solves the Babuuska problem on blow-up analysis of stress concentration in high-contrast composite media. Moreover, as a byproduct of these local analysis, we establish an extended Flaherty-Keller formula on the global effective elastic property of a periodic composite with densely packed fibers, which is related to the "Vigdergauz microstructure" in the shape optimization of fibers.

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arXiv2020

Estimates and Asymptotics for the stress concentration between closely spaced stiff $C^{1, γ}$ inclusions in linear elasticity

This paper is concerned with the stress concentration phenomenon in elastic composite materials when the inclusions are very closely spaced. We investigate the gradient blow-up estimates for the Lamé system of linear elasticity with partially infinite coefficients to show the dependence of the degree of stress enhancement on the distance between inclusions in a composite containing densely placed stiff inclusions. In this paper we assume that the inclusions to be of $C^{1, γ}$, weaker than the previous $C^{2, γ}$ assumption. To overcome this new difficulty, we make use of $W^{1, p}$ estimates for elliptic system with right hand side in divergence form, instead of a direct $W^{2, p}$ argument for $C^{2, γ}$ inclusion case, and combine with the Campanato's approach to establish the optimal gradient estimates, including upper and lower bounds. Moreover, an asymptotic formula of the gradient near the blow-up point is obtained for some symmetric $C^{1, γ}$ inclusions.

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arXiv2020

Hessian quotient equations on exterior domains

It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability of the exterior Dirichlet problem of Hessian quotient equations $σ_k(λ(D^2u))/σ_l(λ(D^2u))=1$ for any $1\leq l<k\leq n$ in all dimensions $n\geq 2$. By introducing the concept of generalized symmetric subsolutions and then using the Perron's method, we establish the existence theorem for viscosity solutions, with prescribed asymptotic behavior which is close to some quadratic polynomial at infinity.

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arXiv2020

On the exterior Dirichlet problem for a class of fully nonlinear elliptic equations

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

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arXiv2020

An extended Flaherty-Keller formula for an elastic composite with densely packed convex inclusions

In this paper, we are concerned with the effective elastic property of a two-phase high-contrast periodic composite with densely packed inclusions. The equations of linear elasticity are assumed. We first give a novel proof of the Flaherty-Keller formula for elliptic inclusions, which improves a recent result of Kang and Yu (Calc.Var.Partial Differential Equations, 2020). We construct an auxiliary function consisting of the Keller function and an additional corrected function depending on the coefficients of Lamé system and the geometry of inclusions, to capture the full singular term of the gradient. On the other hand, this method allows us to deal with the inclusions of arbitrary shape, even with zero curvature. An extended Flaherty-Keller formula is proved for m-convex inclusions, m > 2, curvilinear squares with round off angles, which minimize the elastic modulus under the same volume fraction of hard inclusions.

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arXiv2020

Asymptotics for the electric field concentration in the perfect conductivity problem

In the perfect conductivity problem of composite material, the electric field concentrates in a narrow region in between two inclusions and always becomes arbitrarily large when the distance between inclusions tends to zero. To characterize such singular behavior, we capture the leading term of the gradient and reveal that the blow-up rates are determined by their relative convexity of the two adjacent inclusions. On the other hand, a blow-up factor, which is a linear functional of boundary data, is found to determine the blow-up will occur or not.

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arXiv2019

Boundary blow-up analysis of gradient estimates for Lamé systems in the presence of $M$-convex hard inclusions

In high-contrast elastic composites, it is vitally important to investigate the stress concentration from an engineering point of view. The purpose of this paper is to show that the blowup rate of the stress depends not only on the shape of the inclusions, but also on the given boundary data, when hard inclusions are close to matrix boundary. First, when the boundary of inclusion is partially relatively parallel to that of matrix, we establish the gradient estimates for Lamé systems with partially infinite coefficients and find that they are bounded for some boundary data $φ$ while some $φ$ will increase the blow-up rate. In order to identify such novel blowup phenomenon, we further consider the general $m$-convex inclusion cases and uncover the dependence of blow-up rate on the inclusion's convexity $m$ and the boundary data's order of growth $k$ in all dimensions. In particular, the sharpness of these blow-up rates is also presented for some prescribed boundary data.

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arXiv2019

The Optimal Gradient Estimates for Perfect Conductivity Problem with C^{1,α} inclusions

In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive the upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from C^{2,α} to C^{1,α}. To obtain an pointwise upper bound of the gradient, we follow an iteration technique developed by Bao-Li-Li (ARMA 2015), who mainly deal with the system of linear elasticity. However, when the inclusions are of C^{1, α}, we can not use W^{2,p} estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. Moreover, the establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.

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arXiv2018

Lower bounds of gradient's blow-up for the Lamé system with partially infinite coefficients

In composite material, the stress may be arbitrarily large in the narrow region between two close-to-touching hard inclusions. The stress is represented by the gradient of a solution to the Lamé system of linear elasticity. The aim of this paper is to establish lower bounds of the gradients of solutions of the Lamé system with partially infinite coefficients as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. Combining it with the pointwise upper bounds obtained in our previous work, the optimality of the blow-up rate of gradients is proved for inclusions with arbitrary shape in dimensions two and three. The key to show this is that we find a blow-up factor, a linear functional of the boundary data, to determine whether the blow-up will occur or not.

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arXiv2018

Estimates for Elliptic Systems in a Narrow Region arising from Composite Materials

In this paper, we establish the pointwise upper and lower bounds of the gradients of solutions to a class of elliptic systems, including linear systems of elasticity, in a general narrow region and in all dimensions. This problem arises from the study of damage analysis of high-contrast composite materials. Our results show that the damage may initiate from the narrowest place.

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arXiv2018

Characterization of Electric Fields for Perfect Conductivity Problems in 3D

In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions. An explicit blowup factor relying on radii is obtained, which also involves the digamma function and Euler-Mascheroni constant, and so the role of inclusions' radii played in such blowup analysis is identified.

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arXiv2018

Blowup Analysis for the Perfect Conductivity Problem with convex but not strictly convex inclusions

In the perfect conductivity problem, it is interesting to study whether the electric field can become arbitrarily large or not, in a narrow region between two adjacent perfectly conducting inclusions. In this paper, we show that the relative convexity of two adjacent inclusions plays a key role in the blowup analysis of the electric field and find some new phenomena. By energy method, we prove the boundedness of the gradient of the solution if two adjacent inclusions fail to be locally relatively strictly convex, namely, if the top and bottom boundaries of the narrow region are partially "flat". The boundary estimates when an inclusion with partially "flat" boundary is close to the "flat" matrix boundary and estimates for the general elliptic equation of divergence form are also established in all dimensions.

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arXiv2018

The convexity of inclusions and gradient's concentration for Lamé systems with partially infinite coefficients

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially "flat". This is contrary to our expectation. In order to further explore the blow-up mechanism of the gradient, we next investigate two adjacent inclusions with relative convexity of order m and finally reveal an underlying relationship between the blow-up rate of the stress and the order of the relative convexity of the subdomains in all dimensions.

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arXiv2017

Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients

In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lamé systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blow-up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.

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arXiv2017

Optimal estimates for the perfect conductivity problem with inclusions close to the boundary

When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and the lower bound of the gradient estimate for this perfect conductivity problem by using the energy method. These results give the optimal blow-up rates of electric field for conductors with arbitrary shape and in all dimensions. A particular case when a circular inclusion is close to the boundary of a circular matrix domain in dimension two is studied earlier by Ammari,Kang,Lee,Lee and Lim(2007). From the view of methodology, the technique we develop in this paper is significantly different from the previous one restricted to the circular case, which allows us further investigate the general elliptic equations with divergence form.

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arXiv2016

Optimal estimates for the conductivity problem by Green's function method

We study a class of second-order elliptic equations of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. We establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions. The novelty of these estimates is that they unify the known results in the literature and answer open problem (b) proposed by Li-Vogelius (2000) for the isotropic conductivity problem. We also obtain more interesting higher-order derivative estimates, which answers open problem (c) of Li-Vogelius (2000). It is worth pointing out that the equations under consideration in this paper are nonhomogeneous.

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arXiv2015

Global solutions and exterior Dirichlet problem for Monge-Ampere equation in $\mathbb R^2$

Monge-Ampère equation $\det(D^2u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Ampère equation defined either globally in $\mathbb R^2$ or outside a convex set. First we prove the existence of a global solution that satisfies a prescribed asymptotic behavior at infinity, if $f$ is asymptotically close to a positive constant. Then we solve the exterior Dirichlet problem if data are given on the boundary of a convex set and at infinity.

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