We study unilateral problems for a second order parabolic operator with principal part not of divergence form. We show that the supremum of subsolutions is a proper solution when the obstacle is sufficiently regular, as well as an appropriate substitute for a solution when the obstacle is merely continuous. This approach enables us to investigate problems analogous to those called, in the divergence case, "quasi-variational inequalities ~ , concerning which we obtain a regularity result of the Caffarelli-Friedman type.
ABSTRACT.We prove existence of maximal and minimal solutions to bilateral problems for quasilinear elliptic operators with nondivergence principal part independent of the gradient. This result also covers the case of equations, when the obstacles can be taken as lower and upper solutions.
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