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Let C be a closed bounded convex subset of a Banach space E which has the origin of E as an interior point and let p C denote the Minkowski functional with respect to C. Given a closed set X/E and a point u # E we consider a minimization problem min C (u, X ) which consists in proving the existence of a point x~# X such that p C (x~&u)=* C (u, X ), whereIf such a point is unique and every sequence [x n ]/X satisfying the condition lim n Ä + p C (x n &u)=* C (u, X ) converges to this point, the minimization problem min(u, X ) is called well posed. Under the assumption that the modulus of convexity with respect to p C is strictly positive, we prove that for every closed subset X of E, the set E o (X ) of all u # E for which the minimization problem min C (u, X) is well posed is a residual subset of E. In fact we show more, namely that the set E"E o (X) is _-porous in E. Moreover, we prove that for most closed bounded subsets X of E, the set E "E o (X ) is dense in E.1998 Academic Press

On the Dirichlet problem for a first order partial differential equation 15Furthermore, we wish to point out that the same method, based on Choquet functions and Baire category techniques, seems appropriate to treat other existence problems, for scalar or vectorial partial differential equations in implicit form.The paper is organized as follows. Section 2 contains notation and preliminaries. A fundamental approximation lemma is proved in Section 3. The main results concerning the existence of solutions for the Dirichlet problem (D H ), where H is an admissible Hamiltonian, are established in Section 4.

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