Skrypnik I. I.

We prove a priori estimates for singular solutions of nonlinear elliptic equations with absorption. Using these estimates, we establish precise conditions for the behavior of the absorption term of the equation under which solutions with point singularities do not exist.

We prove a necessary condition for the regularity of a point on a cylindrical boundary for solutions of second-order quasilinear parabolic equations of divergent form whose coefficients have a superlinear growth relative to derivatives with respect to space variables. This condition coincides with the sufficient condition proved earlier by the author. Thus, we establish a criterion for the regularity of a boundary point similar to the well-known Wiener criterion for the Laplace equation.

We investigate the continuity of solutions of quasilinear parabolic equations near the nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for the regularity of a boundary point, which coincides with the Wiener condition for the Laplace p-operator. The model case of the equations considered is the equation \(\frac{{\partial u}}{{\partial t}} - \Delta _p u = 0\) with the Laplace p-operator Δ _p for 2n / (n + 1) < p < 2.

We investigate the continuity of solutions of quasilinear parabolic equations in the neighborhood of the nonsmooth boundary of a cylindrical domain. As a special case, one can consider the equation \(\frac{{\partial u}}{{\partial t}} - \Delta _p u = 0\) with the p-Laplace operator Δp. We prove a sufficient condition for the regularity of a boundary point in terms of C _p-capacity.

We prove the Hölder regularity of bounded weak solutions of doubly nonlinear degenerate parabolic equations with measurable coefficients.

We establish the inner regularity of solutions and their derivatives with respect to spatial coordinates for a degenerate quasilinear parabolic equation of the second order.

A sequence of solutions of nonlinear elliptic problems is considered in the case where the Dirichlet conditions are given on the one part of the boundary and the Neumann conditions are given on the other part. The boundary-value problem is constructed.

We study the properties of solutions of weakly nonlinear parabolic equations in cylindrical domains. The existence conditions are established for local nontangential limits as $t \rightarrow 0$.

Properties of solutions of parabolic equations in smooth cylindrical domains are studied. Conditions for existence of boundary non-tangents and $L_2$-limits as $t \rightarrow 0$ are found.