Gradient estimates for the conductivity problem with imperfect bonding interfaces
-
Hongjie Dong
,
Zhuolun Yang
and
Hanye Zhu
Abstract
We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. We discover a new dichotomy for the field concentration depending on the bonding parameter 𝛾. Specifically, we show that the gradient of solution is uniformly bounded independent of 𝜀 (the distance between two inclusions) when 𝛾 is sufficiently small. However, the gradient may blow up when 𝛾 is large. Moreover, we identify the threshold of 𝛾 and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.
1 Introduction and main results
Our study is motivated by the phenomenon of field concentration in high-contrast composites, a central topic in composite material research. When two inclusions within a composite material are positioned very close to each other and their material properties significantly differ from the surrounding matrix, it may lead to a significant build-up of stress or field intensity in the narrow gap between the inclusions. Understanding the field concentration phenomenon quantitatively is important since it may cause material failure (see, for example, [5, 26]). In some other cases, the inclusions are designed to create the field concentration to achieve desired enhancement of the field (see, for example, [33, 24]). Theoretical analysis related to the field concentration phenomenon originated from the effective medium theory. In the pioneering work [25], the electrical or thermal resistance of the narrow gaps between the inclusions was analyzed in order to estimate the effective conductivity of a medium containing a dense (nearly touching) array of perfectly conducting or insulating cylinders (2D) or perfectly conducting spheres (3D), all with perfect interfaces. See also [18] in the setting of elasticity. There has been significant progress over the past three decades in quantitatively analyzing the field concentration phenomenon in scenarios where the inclusions and the background matrix are perfectly bonded: optimal estimates for the local fields in the narrow gaps between the inclusions and asymptotic characterizations capturing the field concentration phenomenon have been derived.
Motivated by the significant physics work [34] and recent pioneering paper [19], we investigate the case of non-ideal bonding. This model holds significant practical relevance, as it takes into account the contact resistance due to roughness and possible debonding at the interface, and it also approximates the membrane structure in biological systems. It is known in physics literature that interfacial effects are very important in a variety of systems. For example, they may dramatically alter the effective properties [34, 32, 12, 8, 17, 21]. In [34, 32], lower and upper bounds for the effective conductivity of a medium containing randomly packed equisized spheres with imperfect bonding interfaces were deduced by constructing trial fields that account for the complex interactions between the spheres and using minimum energy principles. It is noteworthy that the work [34, 32] could not provide estimates for the local fields or capture the field concentration phenomenon when the spheres are closely located, because, as stated in [34], “the complexity of the microstructure prohibits one from obtaining the local fields exactly”. However, in this paper, we will establish optimal estimates for the local fields under certain symmetry assumptions.
Let us describe the mathematical setup.
Since the region of interest is the local narrow area in between two inclusions which are closely located, the mathematical problem is formulated with two inclusions.
Let
where 𝑎 represents the conductivity distribution, given by
Here and throughout the paper, the subscripts
where
see [2, 3, 35, 36, 6, 7].
These bounds were shown to be optimal.
In the insulated case where
While the optimal blow-up rate in two dimensions was captured about two decades ago in [3, 2], the optimal rate in dimensions
When the bonding between the inclusions and the background matrix is non-ideal, at least one of the transmission conditions (1.2) fails.
We consider the inclusion with imperfect bonding interfaces of low conductivity type (LC-type), which can be understood as an approximation of an inclusion with a low conductivity shell as the shell thickness approaches zero.
Specifically, consider an inclusion with a core-shell structure, where the shell has a thickness denoted by 𝑡.
Let 𝑘 and
exists and is positive, then the transmission conditions become
and we say that the inclusion has an imperfect bonding interface of LC-type, where 𝛾 is called the bonding parameter. In this scenario, the continuity of potential no longer holds. It is important to note that there is another type of imperfect bonding interface known as the high conductivity type (HC-type), where the continuity of flux fails. In two dimensions, The HC-type problem is indeed the dual problem of the LC-type problem. For a detailed discussion and derivation of the transmission conditions, we refer the reader to [19, 1, 9].
Throughout this article, we consider
where 𝜈 is the inward normal vector on
In [19], it was proved that, in two dimensions, the gradient of the solution to
is bounded independent of 𝜀 when
In this article, we prove three main results.
Upper bound for gradient. We establish in Theorem 1.1 an upper bound of order
Absence of field concentration for small 𝛾. We show that, when the bonding parameter 𝛾 is sufficiently small, the gradient of solution is bounded independent of 𝜀. This result confirms the conjecture raised in [19] under the assumption that 𝛾 is sufficiently small.
Dichotomy for field concentration. We show that, when 𝛾 is large, the conjecture does not hold and the boundedness result in Corollary 1.2 is highly unstable, in the sense that if the boundary data 𝜑 is slightly tilted away from being odd in
We use the notation
where
and
Even though the constants
Therefore, by the maximum principle and classical gradient estimates (cf. [31, Chapters 1 and 4], the solution
As such, we focus on the following problem near the origin:
where 𝜈 is the unit normal vector pointing upward on
Our first main result is the following pointwise gradient estimate of order
Let
A quick corollary of the theorem is the boundedness of the gradient under some additional symmetry assumptions. This generalizes one of the results in [19].
For
It is important to note that the factor on the right-hand side of estimate (1.13) is 𝑟 instead of
Our second result concerns the local problem (1.12) with sufficiently small 𝛾.
We show that
Let 𝑓, 𝑔 be
where
As
For the next result, we further assume that
We also assume that, for some
Again, we focus on the local problem of (1.17), that is,
where 𝜈 is the unit normal vector pointing upward on
Note that 𝛼 is always positive, and
Our third result is as follows.
For
when
(1.20)when
(1.21)
Estimate (1.21) indicates that, when
For
where 𝛼 is given in (1.19).
The fact that
It is noteworthy that Theorems 1.4 and 1.5 are the first blow-up results in the context of field concentration for imperfect bonding interfaces. Note that
Therefore, the blow-up exponent in (1.21), (1.22) converges to the optimal blow-up exponent
When
The rest of the paper is organized as follows.
In the next section, we prove the uniform boundedness of
2 Preliminary
In this section, we prove the uniform boundedness of
Let
Proof
By taking the average of both sides of equality (1.5)2 on
Thus it suffices to prove (2.1), since (2.2) follows directly from (2.1) and (2.3).
Without loss of generality, we only show that
Let
Here we used the fact
Let
Proof
First, we prove that (1.5) has at most one solution
First we take
This implies
Next, we take
Since
Finally, we choose
Similarly, by choosing
By (2.6) and (2.8)–(2.10), we can conclude that the minimizer 𝑢 of (1.6) is a solution to (1.5) with
3 A flattening and extension argument
In this section, we use a flattening and extension argument to prove Theorem 1.1. Even though we are working on the Laplace equation with Robin boundary conditions, the argument we present here is very robust, and can be applied to more general settings.
Without loss of generality, we assume
By the triangle inequality, for any
Combining (3.1) and (3.2), we obtain
for any
Let
Step 1: The flattening map
To ensure that the solution after the change of variables still satisfies Robin boundary conditions, we adapt the flattening map introduced in [16, Section 3] to our setting, which preserves the normal directions on the upper and lower boundaries.
More precisely, we define
We denote
and for
where
where 𝜑 is a positive smooth function with unit integral supported in
As in [16, Section 3], the mollification (3.5) was introduced to overcome the lack of regularities of 𝑓 and 𝑔.
Note that, for any
and thus
Thus, by (1.8), for
Similar to [16, Lemma 3.1], one can obtain several properties for the change of variables Φ as in the lemma below. The proof will be given in the appendix for completeness.
There exists an
There exists a positive constant 𝐶, independent of 𝜀 and 𝑟, such that
and hence Φ is invertible.
Let
(3.7)and
(3.8)where
satisfy
(3.9)and
(3.10)for any
satisfying
(3.11)
Step 2: Reducing to the homogeneous Robin boundary condition
Next, we construct an auxiliary function
Let
Then
Now we choose
We use the ansatz
where
Let
Let
Moreover, we have
Proof
By direct computations, using (1.8)–(1.10), we obtain
Thus, by noting that
Note that, for any
Therefore, by (1.8)–(1.10) and (3.11), we have
Moreover, by using (1.8)–(1.10) and (3.6), we also have
By direct computations and using (3.18), (3.19), (3.21), and (3.22), we have the following estimates for
By (3.20) and (3.27), we have
Step 3: Reducing to the homogeneous Neumann boundary condition
We make another transformation
Thus if we choose
namely,
then by (3.16),
We have the following lemma.
Let 𝜓,
where
Proof
By (3.11) and direct computations, we have
For
where
Step 4: Extension and estimates
Next we extend
Note that, because of (3.10), after the extension, we have that
Thus we have the following gradient estimate for
Let
Proof
Observe that, after the extensions defined as above,
Then
where
By (3.30) and (3.33), we know that
Then, by the classical
Here
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1
As is explained at the beginning of this section, we prove the theorem at
Since
Next, since
Here we used (3.17) in the last line.
Finally, since
Next, we give the proof of Corollary 1.2.
Proof of Corollary 1.2
By classical gradient estimates (cf. [31, Chapter 4]), it suffices to prove (1.14) for the case when
Then, by the uniqueness of solutions (Lemma 2.2), we know that
and
By Theorem 1.1, (3.39), and (2.2), for any
Using (1.11), (3.41) directly implies
By the mean value theorem and (3.40), for any
The last equality together with (3.42), (1.8), and (1.10) yields
for any
4 Proof of Theorem 1.3
In this section, we prove Theorem 1.3.
Proof of Theorem 1.3
As in the proof of Theorem 1.1, without loss of generality, we may assume that
and
Similarly, on
We denote
Let 𝜂 be a smooth cut-off function in the
which implies
By the fundamental theorem of calculus, we have
which implies
Combining (4.1), (4.2), and (4.3), we have
where 𝐶 is some positive constant independent of 𝛾.
Let 𝑁 be a large constant to be determined later.
We can choose
where 𝐶 is some positive constant that may depend on 𝛾.
For
Then (4.4) implies that
By (4.2) with
Now we choose
which implies, by choosing 𝑘 such that 𝑠 and
Then, by estimate (1.13) with
To obtain the estimate for
Then (4.3) with
which gives, by estimate (1.13) with
This concludes the proof. ∎
5 A dimensional reduction argument
In this section, we adapt a dimensional reduction argument in [14] to reduce equation (1.18) to
The weak formulation of (1.18) is given by
for all smooth 𝜑 that vanishes on
and let
Similarly,
Therefore, 𝑣 satisfies
where the coefficient matrix
Note that there is an extra
By (5.2),
where
Summarizing this section, we have shown the following lemma.
Let
where
and 𝐶 is some positive constant depending only on 𝑛, 𝛾,
Proof
The derivation of equation (5.6) follows from the argument above.
It remains to prove estimates (5.7).
Since
For the 𝐺 term, we have
This completes the proof. ∎
6 Proof of Theorem 1.4
In this section, we prove Theorem 1.4.
Without loss of generality, we may assume
In this case,
For general
For
Throughout this section,
the average of 𝑢 over
First, we establish estimates for
For
where
Then, for any
where
For the proof, we use an iteration argument based on the following lemmas.
For
Then, for any
where
Proof
By the elliptic theory,
Take the spherical harmonic decomposition
where
and
and satisfies
for each
This implies
Next, we consider two separate cases
Case 1. When
Since
By the maximum principle,
Case 2. When
Note that
where in the second line, we used the equality
following from the definition of
since
Therefore,
and clearly,
By the maximum principle,
Combining these two cases, it follows from (6.6), (6.7), and (6.9),
This completes the proof for
When
The lemma is proved. ∎
For
Then we have
where
Proof
Without loss of generality, we assume
By assumptions,
For
By the definition in (6.2),
To estimate the first term on the right-hand side of (6.12), we use Young’s inequality,
Then, by Hölder’s inequality and
where
To estimate the last term of (6.12), we have
Therefore,
Hence, from (6.12), we have
We use the following version of the Caffarelli–Kohn–Nirenberg inequality (see [11]):
Taking
For
By Young’s inequality,
For
Iterating the relations above, we have, by (6.16),
where 𝐶 is a positive constant depending on 𝑛 and 𝜎, and is in particular independent of 𝑘.
The lemma is concluded by taking
Now we are in a position to prove Proposition 6.1.
Proof of Proposition 6.1
Without loss of generality, we assume that
We denote
and
and
where
We apply Lemma 6.3 to
By Lemma 6.2,
Combining (6.19) and (6.18) yields, using
For a positive integer 𝑘, we take
It follows that
For any
Proof of Theorem 1.4
Without loss of generality, we assume that
where
By the harmonicity of
where
By decreasing 𝜎 if necessary, we may assume that
By Proposition 6.1 and (6.23), we have
where
Similarly, for any
which implies, by (1.13) with
Therefore, we have improved the upper bound
where
7 Proof of Theorem 1.5
In this section, we prove Theorem 1.5.
Without loss of generality, we assume
will play an important role in our proof. First, we prove the following ODE lemma.
For
satisfying
where 𝛼 is given by (6.1) and ℎ is strictly increasing in
Proof
Because of the singularity of 𝐿 at
Hence
where
Now, we show the uniqueness of ℎ.
Let
where
By (7.3), we have
When
as
as
Finally, we show that ℎ is strictly increasing in
Next, we prove a lower bound of ℎ by constructing a subsolution.
For
where 𝛼 is given by (6.1).
Proof
Denote
Therefore, for any
Now we are ready to prove Theorem 1.5.
Proof of Theorem 1.5
Without loss of generality, we assume that
Step 1. By the elliptic theory, the fact that
By (1.13) with
where
Similar to the proof of Lemma 6.2, we denote by
where
where 𝐿 is the differential operator defined in (7.1),
Step 2. We will prove, for some 𝜀-dependent constant
where
Let
By a direct computation,
where
Therefore, using (7.3),
Since
Step 3. We will show that
for some positive 𝜀-independent constant 𝐶.
By a direct computation,
Recall that
Combining with (7.3) and (7.9), we have
where
Step 4. Completion of the proof of Theorem 1.5. In view of (7.3), (7.9), and (7.10), it follows that
for some small 𝜀-independent constant
where 𝑐 is the constant in Lemma 7.2. By (7.7), (7.11), and (7.12), we have
for some small positive
Estimate (1.22) follows from (7.13) and
When
where 𝐹 and 𝐺 satisfy the same bounds as in (7.6).
By using
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2006372
Award Identifier / Grant number: DMS-2055244
Award Identifier / Grant number: DMS-2306726
Award Identifier / Grant number: DMS-2350129
Award Identifier / Grant number: DMS-2510251
Funding statement: Hongjie Dong was partially supported by the NSF under agreements DMS-2055244 and DMS-2350129. Zhuolun Yang was partially supported by the AMS-Simons Travel Grant and the NSF under agreement DMS-2510251. Hanye Zhu was partially supported by the NSF under agreements DMS-2006372 and DMS-2306726.
A Appendix
Here we prove Lemma 3.1. The proof essentially follows that of [16, Lemma 3.1].
Proof of Lemma 3.1
Following the proof of [16, Lemma 3.1], with (3.6) in place of [16, (3.9)], for
Since
by (3.3), for
Then (a) follows by shrinking
Since
To verify (c), note that 𝑢 is smooth by the classical elliptic theory. By the chain rule,
For
Then we have
and
We then deduce the boundary condition for
where
we know that
when
Similarly,
By (A.2), (A.5), (A.6), and (A.3), we know that
Next, we prove (3.9). From part (a), we have
which directly implies that
The last three inequalities directly imply (3.9).
Now, we prove (3.10).
Let
When
Therefore, by (A.4), when
Similarly, we also have
Since
Finally, we prove (3.11). Using (1.8)–(1.10), and (3.6), one can directly obtain
Similar to the proof of [16, Lemma 3.1], we have
The inequalities above also hold with
Thus (3.11) holds. The lemma is proved. ∎
References
[1] H. Ammari, T. Boulier and J. Garnier, Modeling active electrolocation in weakly electric fish, SIAM J. Imaging Sci. 6 (2013), no. 1, 285–321. 10.1137/12086858XSearch in Google Scholar
[2] H. Ammari, H. Kang, H. Lee, J. Lee and M. Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl. (9) 88 (2007), no. 4, 307–324. 10.1016/j.matpur.2007.07.005Search in Google Scholar
[3] H. Ammari, H. Kang and M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), no. 2, 277–286. 10.1007/s00208-004-0626-ySearch in Google Scholar
[4] F. F. T. Araujo and H. M. Rosenberg, The thermal conductivity of epoxy-resin/metal-powder composite materials from 1.7 to 300k, J. Phys. D 9 (1976), no. 4, Paper No. 665. 10.1088/0022-3727/9/4/017Search in Google Scholar
[5] I. Babuška, B. Andersson, P. J. Smith and K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg. 172 (1999), no. 1–4, 27–77. 10.1016/S0045-7825(98)00225-4Search in Google Scholar
[6] E. S. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 195–226. 10.1007/s00205-008-0159-8Search in Google Scholar
[7] E. S. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Comm. Partial Differential Equations 35 (2010), no. 11, 1982–2006. 10.1080/03605300903564000Search in Google Scholar
[8] Y. Benveniste, Effective thermal conductivity of composites with a thermal contact resistance between the constituents: Nondilute case, J. Appl. Phys. 61 (1987), no. 8, 2840–2843. 10.1063/1.337877Search in Google Scholar
[9] Y. Benveniste and T. Miloh, Neutral inhomogeneities in conduction phenomena, J. Mech. Phys. Solids 47 (1999), no. 9, 1873–1892. 10.1016/S0022-5096(98)00127-6Search in Google Scholar
[10] E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. 10.1137/S0036141098333980Search in Google Scholar
[11] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), no. 3, 259–275. Search in Google Scholar
[12] Y. C. Chiew and E. D. Glandt, The effect of structure on the conductivity of a dispersion, J. Coll. Interface Sci. 94 (1983), no. 1, 90–104. 10.1016/0021-9797(83)90238-2Search in Google Scholar
[13] H. Dong, Y. Li and Z. Yang, Gradient estimates for the insulated conductivity problem: The non-umbilical case, J. Math. Pures Appl. (9) 189 (2024), Article ID 103587. 10.1016/j.matpur.2024.06.002Search in Google Scholar
[14] H. Dong, Y. Li and Z. Yang, Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 8, 3275–3296. 10.4171/jems/1432Search in Google Scholar
[15] H. Dong and Z. Yang, Optimal estimates for transmission problems including relative conductivities with different signs, Adv. Math. 428 (2023), Article ID 109160. 10.1016/j.aim.2023.109160Search in Google Scholar
[16] H. Dong, Z. Yang and H. Zhu, The insulated conductivity problem with 𝑝-Laplacian, Arch. Ration. Mech. Anal. 247 (2023), no. 5, Paper No. 95. 10.1007/s00205-023-01926-0Search in Google Scholar
[17] A. G. Every, Y. Tzou, D. P. H. Hasselman and R. Raj, The effect of particle size on the thermal conductivity of zns/diamond composites, Acta Metall. Mater. 40 (1992), no. 1, 123–129. 10.1016/0956-7151(92)90205-SSearch in Google Scholar
[18] J. E. Flaherty and J. B. Keller, Elastic behavior of composite media, Comm. Pure Appl. Math. 26 (1973), 565–580. 10.1002/cpa.3160260409Search in Google Scholar
[19] S. Fukushima, Y.-G. Ji, H. Kang and X. Li, Finiteness of the stress in presence of closely located inclusions with imperfect bonding, Math. Ann. 391 (2025), no. 2, 1753–1778. 10.1007/s00208-024-02968-9Search in Google Scholar
[20] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Class. Math., Springer, Berlin 2001. 10.1007/978-3-642-61798-0Search in Google Scholar
[21] Z. Hashin, Extremum principles for elastic heterogenous media with imperfect interfaces and their application to bounding of effective moduli, J. Mech. Phys. Solids 40 (1992), no. 4, 767–781. 10.1016/0022-5096(92)90003-KSearch in Google Scholar
[22] Y.-G. Ji and H. Kang, Spectrum of the Neumann–Poincaré operator and optimal estimates for transmission problems in the presence of two circular inclusions, Int. Math. Res. Not. IMRN 2023 (2023), no. 9, 7638–7685. 10.1093/imrn/rnac057Search in Google Scholar
[23] H. Kang, Quantitative analysis of field concentration in presence of closely located inclusions of high contrast, ICM—International Congress of Mathematicians. Vol. 7. Sections 15–20, European Mathematical Society, Zürich (2023), 5680–5699. 10.4171/icm2022/83Search in Google Scholar
[24] H. Kang and K. Yun, Precise estimates of the field excited by an emitter in presence of closely located inclusions of a bow-tie shape, J. Math. Anal. Appl. 479 (2019), no. 2, 1670–1707. 10.1016/j.jmaa.2019.07.018Search in Google Scholar
[25] J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. 34 (1963), no. 4, 991–993. 10.1063/1.1729580Search in Google Scholar
[26] J. B. Keller, Stresses in narrow regions, J. Appl. Mech. 60 (1993), no. 4, 1054–1056. 10.1115/1.2900977Search in Google Scholar
[27] H. Li and Y. Zhao, Optimal gradient estimates for the insulated conductivity problem with general convex inclusions case, preprint (2024), https://arxiv.org/abs/2404.17201. Search in Google Scholar
[28] Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925. 10.1002/cpa.10079Search in Google Scholar
[29] Y. Li and Z. Yang, Gradient estimates of solutions to the insulated conductivity problem in dimension greater than two, Math. Ann. 385 (2023), no. 3–4, 1775–1796. 10.1007/s00208-022-02368-xSearch in Google Scholar
[30] Y. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. 10.1007/s002050000082Search in Google Scholar
[31] G. M. Lieberman, Oblique derivative problems for elliptic equations, World Scientific, Hackensack 2013. 10.1142/8679Search in Google Scholar
[32] R. Lipton and B. Vernescu, Composites with imperfect interface, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1945, 329–358. 10.1098/rspa.1996.0018Search in Google Scholar
[33] V. Pacheco-Peña, M. Beruete, A. I. Fernández-Domínguez, Y. Luo and M. Navarro-Cía, Description of bow-tie nanoantennas excited by localized emitters using conformal transformation, ACS Photonics 3 (2016), no. 7, 1223–1232. 10.1021/acsphotonics.6b00232Search in Google Scholar
[34] S. Torquato and M. D. Rintoul, Effect of the interface on the properties of composite media, Phys. Rev. Lett. 75 (1995), 4067–4070. 10.1103/PhysRevLett.75.4067Search in Google Scholar PubMed
[35] K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math. 67 (2007), no. 3, 714–730. 10.1137/060648817Search in Google Scholar
[36] K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl. 350 (2009), no. 1, 306–312. 10.1016/j.jmaa.2008.09.057Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- A new energy method for shortening and straightening complete curves
- Exceptional Tannaka groups only arise from cubic threefolds
- Gradient estimates for the conductivity problem with imperfect bonding interfaces
- On the structure of singularities of weak mean curvature flows with mean curvature bounds
- Second adjointness and cuspidal supports at the categorical level
- Transverse surfaces and pseudo-Anosov flows
- On the minimal number of closed geodesics on positively curved Finsler spheres
- On the geometry of spaces of filtrations on local rings
Articles in the same Issue
- Frontmatter
- A new energy method for shortening and straightening complete curves
- Exceptional Tannaka groups only arise from cubic threefolds
- Gradient estimates for the conductivity problem with imperfect bonding interfaces
- On the structure of singularities of weak mean curvature flows with mean curvature bounds
- Second adjointness and cuspidal supports at the categorical level
- Transverse surfaces and pseudo-Anosov flows
- On the minimal number of closed geodesics on positively curved Finsler spheres
- On the geometry of spaces of filtrations on local rings
