# Bernatskaya J. N.

### On the behavior of a simple-layer potential for a parabolic equation on a Riemannian manifold

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 879–891

On a Riemannian manifold of nonpositive sectional curvature (Cartan-Hadamard-type manifold), we consider a parabolic equation. The second boundary-value problem for this equation is set in a bounded domain whose surface is a smooth submanifold. We prove that the gradient of the simple-layer potential for this problem has a jump when passing across the submanifold, similarly to its behavior in a Euclidean space.

### Perturbation Method for a Parabolic Equation with Drift on a Riemannian Manifold

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 148-159

We construct a fundamental solution of a parabolic equation with drift on a Riemannian manifold of nonpositive curvature by the perturbation method on the basis of a solution of an equation without drift. We establish conditions for the drift field under which this method is applicable.

### Behavior of the Double-Layer Potential for a Parabolic Equation on a Manifold

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 590-603

We prove that, similarly to the double-layer potential in \(\mathbb{R}^n \) , the double-layer potential constructed in a Riemann manifold of nonpositive sectional curvature has a jump in passing through the surface where its density is defined.

### On Stabilization of Energy of a Conservative System Perturbed by a Random Process of “White-Noise” Type in the Itô Form

Bernatskaya J. N., Kulinich G. L.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1429-1435

We investigate the problem of deterministic control over the behavior of the total energy of the simplest conservative nonlinear system with one degree of freedom without friction in the case of random perturbations by a process of the “white-noise” type in the Itô form. These perturbations act under a fixed angle to the vector of phase velocity of the conservative system.