Related papers: Optimal estimates for transmission problems includ…
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…
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…
When inclusions with extreme conductivity (insulator or perfect conductor) are closely located, the gradient of the solution to the conductivity equation can be arbitrarily large. And computation of the gradient is extremely challenging due…
We consider the field concentration for the transmission problems of the homogeneous and inhomogeneous conductivity equations in the presence of closely located circular inclusions. We revisit these well-studied problems by exploiting the…
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…
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…
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…
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…
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…
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 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…
We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order $\varepsilon$ apart. The solution $u$ represents the electric potential. In dimensions $n \ge 3$ it is an open problem to find…
In this paper, we investigate the gradient estimates for solutions to the perfect conductivity problem with two closely spaced perfect conductors embedded in a homogeneous matrix, modeled by $p$-Laplacian elliptic equations. We first prove…
We consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes…
Connections are found between the two-component percolation problem and the conductor/insulator percolation problem. These produce relations between critical exponents, and suggest formulae connecting the conductivity exponents in different…
We consider the problem of recovering an isotropic conductivity outside some perfectly conducting or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We prove that the conductivity…
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…
In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in $\mathbb{R}^{d}$ with $d\geq3$. We establish the pointwise upper bounds on the gradient of the solution for the generalized…
The inverse problem of electrical impedance tomography is severely ill-posed. In particular, the resolution of images produced by impedance tomography deteriorates as the distance from the measurement boundary increases. Such depth…