Bounded inhomogeneous nonlinear elliptic and parabolic equations in the plane

A study is made of equations of the form $F\bigl(x,D_{ij}u-d\delta_{ij}\frac{\partial u}{\partial t},D_iu,u\bigr)=0$ 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 $tF\bigl(x,\frac\xi t\bigr)$ satisfies an ellipticity condition with respect to $\xi_{ij}$, a boundedness condition for the “coefficients” of $\xi$ and $t$ and a negative condition for the “coefficient” of $u$, then all the problems have a solution in the corresponding Sobolev–Slobodetskii space which is unique.
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