\unbounded" Second Order Partial Differential Equations in Infinite Dimensional Hilbert Spaces

0. Introduction. In this paper we study fully nonlinear stationary partial diierential equations (PDE's) having the form (S) as well as Cauchy problems (CP) (u t + hA x; Dui + F(t; x; Du; D 2 u) = 0 for (t; x) 2 (0; T) H (E) u(0; x) = (x) for x 2 H; where H is a real, separable Hilbert space, A is a linear, densely deened, maximal monotone operator on H and u is a real valued function. Above Du(x) and D 2 u(x) correspond respectively to the rst and second order Fr echet derivatives of u. Identifying H with its dual, Du(x) corresponds to an element of H and D 2 u(x) to an element of S(H), the space of bounded, self-adjoint operators on H. Therefore,

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