Bounded inhomogeneous nonlinear elliptic and p arabolic equations in the plane

A stilly is made of equations of the form [format text]in a bounded smooth domain in the plane (d=0) or in a smooth cylinder above the plane (d = l) 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, [format text]/t) satisfies an ellipticity condition with respect to [formatted text] boundedness condition for the "coefficients” of [format text] 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.