Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field blows up in the L-infinity norm as the distance between the conductors tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of the distance between the conductors.

Blow-up of solutions to a p-Laplace equation

. In high-contrast composite materials, the electric (or stress) ﬁeld may blow up in the narrow region between inclusions. The gradient of solutions depend on ε , the distance between the inclusions, where ε approaches to 0. By using the maximum principle techniques, we give another proof of the Dong- Li-Yang estimates [17] for any convex inclusions of arbitrary shape with n ≥ 3. This result solves the problem raised by [31], where the spherical inclusions with n ≥ 4 is considered. Moreover, we also generalize the above results with ﬂatter boundaries near touching points.

Optimal gradient estimates for the insulated conductivity problem with dimensions more than two

. To study the ﬁeld enhancements in the narrow region between two inclusions is vital important in material sciences and ﬂuid mechanics. Complex ﬂuids including particle suspensions usually result in complicated ﬂow behavior. In this paper we establish the pointwise upper bounds of the gradient estimates for the Stokes ﬂow when two rigid particles are closely spaced suspending in an open bounded domain and away from the boundary in dimensions two and three. Moreover, the lower bounds at the narrowest place of the neck region show the optimality of these blow-up rates. These results are valid for inclusions with arbitrary shape.

Stress Concentration between Two Adjacent Rigid Inclusions for the Stokes Flow in 2D and 3D

This paper is devoted to establishing the pointwise upper and lower bounds estimates of the gradient of the solutions to a class of general elliptic systems with H¨older continuous coe ﬃ cients in a narrow region where the upper and lower boundaries is C 1 ,γ , 0 < γ < 1, weaker than the previous C 2 ,γ assumption. These estimates play a key role in the damage analysis of composite materials. From our results, the damage may initiate from the narrowest place.

Gradient estimates for elliptic systems from composite materials with closely spaced stiff $C^{1,\gamma}$ inclusions

In this paper, we establish the asymptotic expressions for the gradient of a solution to the Lame systems with partially infinity coefficients as two rigid $C^{1,\gamma}$-inclusions are very close but not touching. The novelty of these asymptotics, which improve and make complete the previous results of Chen-Li (JFA 2021), lies in that they show the optimality of the gradient blow-up rate in dimensions greater than two.

Yu Chen

Papers

Estimates and asymptotics for the stress concentration between closely spaced stiff C1,γ inclusions in linear elasticity