Related papers: The insulated conductivity problem, effective grad…
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 $\epsilon$, the distance between the inclusions, where $\epsilon$ approaches to…
We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$,…
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb R^n$, for $n \ge 3$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. We…
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. It was known that the…
In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in $\mathbb{R}^n, n\ge 2$. For these two extreme cases of the conductivity problems, the gradients of their…
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. An upper bound for the…
In high-contrast composites, the electric (or stress) field may exhibit significant amplification in the narrow region between inclusions. The behavior of the solution depends on the distance $\epsilon$ between the inclusions, which tends…
We consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions. When inclusions with an extreme conductivity (insulators or perfect conductors) are closely located, the…
We study the insulated conductivity problem which involves two adjacent convex insulators embedded in a bounded domain. It is known that the gradient of solutions may blow up as the distance between the two inclusions tends to zero.…
We study the gradient and higher order derivative estimates for the transmission problem in the presence of closely located inclusions. We show that in two dimensions, when relative conductivities of circular inclusions have different…
This paper concerns optimal gradient estimates of solutions for the perfect conductivity problem with closely spaced interfacial boundaries. The problem arises from composite material. Our estimates exhibit different blow up rates of the…
The purpose of this paper is to set out optimal gradient estimates for solutions to the isotropic conductivity problem in the presence of adjacent conductivity inclusions as the distance between the inclusions goes to zero and their…
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…
In this paper we study the boundary gradient estimate of the solution to the insulated conductivity problem with the Neumann boundary data when a convex insulating inclusion approaches the boundary of the matrix domain. The gradient of…
This paper studies field concentration between two nearly touching conductors separated by imperfect low-conductivity interfaces, modeled by Robin boundary conditions. It is known that for any sufficiently small interfacial bonding…
The insulated and perfect conductivity problems arising from high-contrast composite materials are considered in all dimensions. The solution and its gradient, respectively, represent the electric potential and field. The novelty of this…
We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension…
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…
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…
We study the perfect conductivity problem with closely spaced perfect conductors embedded in a homogeneous matrix where the current-electric field relation is the power law $J=\sigma|E|^{p-2}E$. The gradient of solutions may be arbitrarily…