Related papers: Gradient estimates for the insulated conductivity …
In the perfect conductivity problem of composite material, the electric field concentrates in a narrow region in between two inclusions and always becomes arbitrarily large when the distance between inclusions tends to zero. To characterize…
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…
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…
If two conducting or insulating inclusions are closely located, the gradient of the solution may become arbitrarily large as the distance between inclusions tends to zero, resulting in high concentration of stress in between two inclusions.…
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…
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…
A high-contrast two-phase nonlinear composite material with adjacent inclusions of $m$-convex shapes is considered for $m>2$. The mathematical formulation consists of the insulated conductivity problem with $p$-Laplace operator in…
We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and…
We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We…
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 $\varepsilon$,…
When perfectly conducting or insulating inclusions are closely located, stress which is the gradient of the solution to the conductivity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important…
We consider a boundary value problem for the conductivity equation in a bounded domain containing an inclusion which is nearly touching to the domain's boundary. We assume that the domain and the inclusion are disks with conductivity jump…
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…
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 establish upper bounds on the blow up rate of the gradients of solutions of the Lam\'e system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system…
In this paper, we are concerned with the gradient estimate of the electric field due to two nearly touching dielectric inclusions, which is a central topic in the theory of composite materials. We derive accurate quantitative…
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…
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result…
In this paper we analyze the gradient blow-up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core-shell geometry. Assuming that the core and shell have circular boundaries that…
We are concerned with the field concentration between two nearly-touching inclusions with high-contrast material parameters, which is a central topic in the theory of composite materials. The degree of concentration is characterised by the…