Bounded Inhomogeneous Nonlinear Elliptic and Parabolic Equations in the Plane

A study is made of equations of the form F(x, Diju - dδij∂u/∂t, Diu, u) = 0 in a bounded smooth domain in the plane (d = 0) or in a smooth cylinder above the plane (d = 1) with Dirichlet data on the boundary, and also of the problem with a free boundary for these equations. It is proved that if the function tF(x, ξ/t) satisfies an ellipticity condition with respect to ξij, a boundedness condition for the coefficients of ξ and t and a negative condition for the coefficient of u, then all the problems have a solution in the corresponding Sobolev-Slobodeckiĭ space which is unique. Bibliography: 6 items.