A three-dimensional model of the varistor device is proposed. The thermal and electric conductivity of the material are taken to depend, in addition to the electric potential, on the temperature. Two theorems of existence and uniqueness of solutions for the boundary-value problem which determine the potential and the temperature inside the device are proposed. Levy–Caccioppoli global inversion theorem is used for the proof.

A method is proposed for reducing autonomous nonlinear boundary-value problems of partial differential equations with a particular structure and constant boundary conditions to the simple problem

This requires the solution of a non-standard two-point system of ordinary differential equations. Some applications are also presented for classical problems of mathematical physics.

This paper deals with a new method to determine the dependence of the electrical conductivity of metals or semiconductors on temperature. It is based on the fact that the current–voltage relationship is easily measurable. This inverse problem is solved by the classical Abel integral equation.

The theorem of bifurcation from a simple eigenvalue is applied to prove non-uniqueness for the problem of a layer of a dielectric liquid subjected to an electric field and to injection of charges on the electrodes.

The electrical heating of a conductor connected with a current limiting device of total resistance R is studied under mixed boundary conditions. A theorem of existence is given using a transformation which permits the reduction of the nonlinear system governing the problem to the classical mixed boundary value problem for the Laplace equation.