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We study fully nonlinear uniformly elliptic equations with measurable ingredients. Recently signiicant progress has been made in this area due to fundamental work of Caaarelli on W 2;p estimates for viscosity solutions. Here we present a uniied treatment of this theory based on an appropriate notion of viscosity solution. For instance, it is shown that… (More)
We consider a two-player, zero-sum diierential game governed by an abstract nonlinear diierential equation of accretive type in an innnite dimensional space. We prove that the value function of the game is the unique viscosity solution of the corresponding Hamilton-Jacobi-Isaacs equation in the sense of Crandall-Lions 12]. We also discuss some properties of… (More)
We consider nonlinear optimal control problems with state constraints and nonneg-ative cost in infinite dimensions, where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower… (More)
We prove existence and uniqueness of viscosity solutions of Cauchy problems for fully nonlinear unbounded second order Hamilton-Jacobi-Bellman-Isaacs equations deened on the product of two innnite dimensional Hilbert spaces H 0 H 00 , where H 00 is separable. The equations have a special \separated" form in a sense that the terms involving second… (More)
This paper provides a numb e r o f w orking tools for the discussion of fully nonlinear parabolic equations. These include: a proof that the maximum principle which provides L 1 estimates of strong" solutions of extremal equations by L n+1 norms of the forcing term over the contact set" remains valid for viscosity solutions in an L n+1 sense, a gradient… (More)
We introduce the notion of a good" solution of a fully nonlinear uniformly elliptic equation. It is proven that good" solutions are equivalent to L p-viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic… (More)
This paper studies the H 1 control problem for a nonlinear, unbounded, innnite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton-Jacobi-Isaacs equation, proving that the H 1 problem can be solved if and only if the HJI equation has a nonnegative viscosity supersolution, vanishing at the origin.