On one boundary-value problem for a strongly degenerate second-order elliptic equation in an angular domain

We prove the existence and uniqueness of a classical solution of a singular elliptic boundary-value problem in an angular domain. We construct the corresponding Green function and obtain coercive estimates for the solution in the weighted Hölder classes.     

On the solvability of a nonlinear second-order elliptic equation at resonance

We study the existence of solutions of the Neumann problem for semilinear second-order elliptic equations at resonance in which the nonlinear terms may grow superlinearly in one of the directions and , and sublinearly in the other. Solvability results are obtained under assumptions either with or without a Landesman-Lazer condition. The proofs are based on degree-theoretic arguments.

     

On the uniqueness of positive solution of an elliptic equation

This work deals with the uniqueness of positive solution for an elliptic equation whose nonlinearity satisfies a specific monotony property. In order to prove the main result, we employ a change of variable used in previous papers and the maximum principle.     

Unimprovable estimates for solutions of a mixed problem for linear elliptic equations of the second order in a neighborhood of an angular point

Under the minimal conditions for the smoothness of the coefficients of an equation, unimprovable estimates are obtained for solutions of a mixed problem for linear nondivergent elliptic equations of the second order in a neighborhood of an angular point of the boundary of a domain. Lviv University, Lviv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11, pp. 1529–1542, November, 1997.     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

Universal dynamics in a neighborhood of a generic elliptic periodic point

We show that a generic area-preserving two-dimensional map with an elliptic periodic point is C ^ω -universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

On the asymptotics of a solution to an equation with a small parameter at some of the highest derivatives

We study the asymptotic behavior of a solution of the first boundary value problem for a second-order elliptic equation in a nonconvex domain with smooth boundary in the case where a small parameter is a factor at only some of the highest derivatives and the limit equation is an ordinary differential equation. Although the limit equation has the same order as the initial equation, the problem is singulary perturbed. The asymptotic behavior of its solution is studied by the method of matched asymptotic expansions.     

On the asymptotics of the solution of the Dirichlet problem for a fourth-order equation in a layer

The Dirichlet problem for a fourth-order elliptic equation with constant coefficients without first derivatives is considered in the region (layer) $\prod { = \left\{ {(x',x_n ) \in R^n |x' \in R^{n - 1} ,x_n \in (a,b)} \right\}} , - \infty < a < b < + \infty ,n \geqslant 3$ The first term of the asymptotics of the solution at infinity is obtained.