Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
E-mail: Keywords: elliptic and parabolic equations; solvability of a boundary value problem; a priori estimate; boundary properties of solutions; embedding theorem; capacity; removable singularities of solutions; maximal function.
Subject:
The class of nondivergent elliptic equations of the second order with Wiener test regularity of a boundary point in terms of introduced function of ellipticity was described. This class ñontain equations with dicontinuous coefficients. The parabolic analog of Cordes condition guaranteeing unique solvability of the first boundary value problem for nondivergent parabolic equations of the second order in the Sobolev space $W^{2,1}_{2,0}$ was found (with I. T. Mamedov). Necessary and sufficient condition on a boundary for unique $L_p$–solvability of the Dirichlet problem together with the corresponding coercive estimate for divergent elliptic equations of the second order was obtained. The smoothness at a point for solutions of parabolic equations of the second order under minimal assumptions on coefficients was investigated. Inner and boundary properties for solutions of quasilinear elliptic equations for integrands $|\xi|^{p(x)}$ were studied. The Holder property for solutions of degenerate elliptic equations of the second order with a weight that is not satisfying neither Muckenhoupt condition nor double condition was proved (with V. V. Zhikov). Interesting feature of these equations is absent of Harnack inequality for positive solutions.
Main publications:
Alkhutov Yu. A., Mamedov I. T. Pervaya kraevaya zadacha dlya nedivergentnykh parabolicheskikh uravnenii vtorogo poryadka s razryvnymi koeffitsientami // Matem. cbornik, 1986, 173(4), 477–500.
