# Borsuk M. V.

### Estimates of generalized solutions of the Dirichlet problem for quasilinear elliptic equations of the second order in a domain with conical boundary point

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1299–1309

We obtain *a priori* estimates for generalized second derivatives (in the Sobolev weighted norm) of solutions of the Dirichlet problem for the elliptic equation $$\frac{d}{{dx_i }}a_i (x,u,u_x ) + a(x,u,u_x ) = 0,x \in G,$$ in the neighborhood of a conical boundary point of the domain *G*. We give an example that demonstrates that the estimates obtained are almost exact.

### On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 13-24

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.

### Behavior of solutions of the Dirichlet problem for a second-order quasilinear elliptic equation of general form close to a corner point

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 167–173

The Dirichlet problem for the uniformly elliptic equation $$a_{ij} (x,u,u_x )u_{x_i x_j } + a(x,u,u_x ) = 0$$ is considered in a bounded plane region. It is assumed that there is a corner point on the boundary of the region (the origin), and that the coefficients of the equation satisfy minimal smoothness conditions and appropriate conditions of growth on the gradient (not greater than quadratic). For a smooth solution, it is shown that, in a neighborhood of the corner point,$$u(x) = O(|x|^{\pi /\omega } ),\nabla u(x) = O(|x|^{\pi /\omega - 1} ),$$ where $ω$ is the angle in which the two arcs of the boundary of the region intersect at the origin.