Related papers: Exact solutions for the insulated and perfect cond…
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…
In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we…
We describe a new type of three material microstructures which we call wheel assemblages, that correspond to extremal conductivity and extremal bulk modulus for a composite made of two materials and an ideal material. The exact lower bounds…
We show that the exact solution of the Schr\"odinger equation for two electrons confined to two distinct concentric rings or spheres can be found in closed form for particular sets of the ring or sphere radii. In the case of two concentric…
We study the asymptotic behaviour of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of…
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 $\varepsilon$,…
Inverse problems of electric conductivity are studied that arise in the design of spherical shielding or cloaking shells and other functional devices used to control DC electric fields. The shells are considered consisting of a finite…
Closed formulas are derived for the field in the focal region of a diffraction limited lens, such that the electric field component in a given direction at the focal point is larger than that of all other focused fields with the same power…
The experimental realization of cloaks for electromagnetic fields has been possible only in reduced schemes because of extreme material conditions required in exact cloaks. Here we analytically demonstrate that a simple arrangement of two…
The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural…
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…
The main objective of this paper is to study the regularity and stability for solutions to the conductivity problems with degenerate coefficients in the presence of two rigid conductors, as one conductor keeps motionless and another…
Solutions to the complementarity problem constructed in [1], generally, possess non-zero total charge. In natural sciences, bodies possessing non-zero total charge (ions and similar object) are considered as specific objects. Bodies…
The new generation of manufacturing technologies such as e.g. additive manufacturing and automated fiber placement has enabled the development of material systems with desired functional and mechanical properties via particular designs of…
The fusion created by magnetically confined plasma is a promising clean and essentially unlimited future energy source. However, net energy generation has not been yet demonstrated in fusion experiments. Some of the main problems hindering…
We consider the neutral inclusion problem in three dimensions which is to prove if a coated structure consisting of a core and a shell is neutral to all uniform fields, then the core and the shell must be concentric balls if the matrix is…
The aim of the paper is twofold. Firstly, we would like to derive quantitative uniqueness estimates for solutions of the general complex conductivity equation. It is still unknown whether the \emph{strong} unique continuation property holds…
The idea of replacing an edgy perfectly conducting boundary by the corresponding interface filled with a dielectric material of extreme complex permittivities, is examined in the present work. A semi-analytical solution to the corresponding…
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…
A new exact static interior solution of the Einstein equations is obtained for a gravitating ball filled with a Pascal perfect fluid . The solution is an extension of the well-known interior solution with a parabolic distribution of mass…