Vol 60, No (2016)

We study various classes of nonlinear equations containing an operator of potential type (Riesz potential). By the monotone operators method in the Lebesgue spaces of real-valued functions Lp(a, b) we prove global theorems on existence, uniqueness, estimates, and methods of obtaining of their solutions. We consider corollaries as applications of our results.

We study local instabilities of a di erentially rotating viscous ow of electrically conducting incompressible uid subject to an external azimuthal magnetic eld. The hydrodynamically stable ow can be destabilized by the magnetic eld both in the ideal and in the viscous and resistive system giving rise to the azimuthal magnetorotational instability. A special solution to the equations of the ideal magnetohydrodynamics characterized by the constant total pressure, the uid velocity parallel to the direction of the magnetic eld, and by the magnetic and kinetic energies that are nite and equal - the Chandrasekhar equipartition solution - is marginally stable in the absence of viscosity and resistivity. Performing a local stability analysis we nd the conditions when the azimuthal magnetorotational instability can be interpreted as a dissipation-induced instability of the Chandrasekhar equipartition solution.

For the Lame´ system from the at anisotropic theory of elasticity, we introduce generalized double-layer potentials in connection with the function-theory approach. These potentials are built both for the translation vector (the solution of the Lame´ system) and for the adjoint vector functions describing the stress tensor. The integral representation of these solutions is obtained using the potentials. As a corolary, the rst and the second boundary-value problems in various spaces (Ho¨lder, Hardy, and the class of functions just continuous in a closed domain) are reduced to the equivalent system of the Fredholm boundary equations in corresponding spaces. Note that such an approach was developed in [13, 14] for common second-order elliptic systems with constant (higher-order only) coe cients. However, due to important applications, it makes sense to consider this approach in detail directly for the Lame´ system. To illustrate these results, in the last two sections we consider the Dirichlet problem with piecewise-constant Lame´ coe cients when contact conditions are given on the boundary between two media. This problem is reduced to the equivalent system of the Fredholm boundary equations. The smoothness of kernels of the obtained integral operators is investigated in detail depending on the smoothness of the boundary contours.