358 research outputs found
Lower bounds of gradient's blow-up for the Lam\'{e} 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 Lam\'{e} system of linear elasticity. The
aim of this paper is to establish lower bounds of the gradients of solutions of
the Lam\'{e} 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.Comment: 33 pages; submitte
Gradient Estimates for Parabolic Systems from Composite Material
In this paper we derive and piecewise estimates
for solutions, and their derivatives, of divergence form parabolic systems
with coefficients piecewise H\"older continuous in space variables and
smooth in . This is an extension to parabolic systems of results of Li and
Nirenberg on elliptic systems. These estimates depend on the shape and the size
of the surfaces of discontinuity of the coefficients, but are independent of
the distance between these surfaces.Comment: A new result is added which extends an estimate of Campanato
for strongly parabolic systems to rather weak parabolic systems, see Appendi
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 Lam\'e 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.Comment: 26 pages, 5 figure
Harmonic maps on domains with piecewise Lipschitz continuous metrics
For a bounded domain equipped with a piecewise Lipschitz continuous
Riemannian metric g, we consider harmonic map from to a compact
Riemannian manifold without boundary. We generalize
the notion of stationary harmonic map and prove the partial regularity. We also
discuss the global Lipschitz and piecewise -regularity of
harmonic maps from manifolds that support convex distance
functions.Comment: 24 page
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.Comment: 23 pages, submitte
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.Comment: to appear in SIAM J. Math. Ana
Asymptotics of the gradient of solutions to the perfect conductivity problem
In the perfect conductivity problem of composite material, the gradient of
solutions can be arbitrarily large when two inclusions are located very close.
To characterize the singular behavior of the gradient in the narrow region
between two inclusions, we capture the leading term of the gradient and give a
fairly sharp description of such asymptotics.Comment: The exposition is improved. to appear in Multiscale Modeling and
Simulatio
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.Comment: 23 page
Gradient estimates for solutions of the Lam\'e system with partially infinite coefficients in dimensions greater than two
We establish upper bounds on the blow-up rate of the gradients of solutions
of the Lam\'{e} system with partially infinite coefficients in dimensions
greater than two as the distance between the surfaces of discontinuity of the
coefficients of the system tends to zero.Comment: 35 pages. arXiv admin note: text overlap with arXiv:1311.127
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.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1305.0921 by
other author
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