Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions

Abstract

We continue the systematic study of first-order Hamilton-Jacobi equations in infinite dimensions begun in Parts I-IV of this series [ 121. Our approach relies on, as before, an appropriate interpretation of the notion of viscosity solutions as introduced in [ 11, lo]. In the interim, the strong development of “viscosity solution” theory in many directions has continued unabated. In addition to the many references given in earlier papers in this series (which we will not repeat here; see [12]) we would like to point out that a complete theory of second-order fully nonlinear and possibly degenerate elliptic equations in finite dimensions has been made possible by the use of viscosity solutions [S, 9, 18, 20-22, 19, 32, 331 for example) and there is now a substantial body of work concerning second order equations in infinite dimensions [25-271. However, in this work we come back to the particular situation studied in [ 12, Part IV]; namely, we come back to Hamilton-Jacobi equations in