Behaviour of solutions of the Dirichlet problem for quasilinear elliptical equations of the second order of a general kind near angular point - Borsuk , M. V.; Борсук , М. В. (Institute of Mathematics, NAS of Ukraine, 1992)
The Dirichlet problem for the uniformly elliptic equation aij (x,u,ux)uxixj + a (x,u,ux)=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, и (х) =О (| х | π/ω), ∇u (х) = О (| х | π/ω-1), where ω is the angle in which the two arcs of the boundary of the region intersect at the origin.