On total differential inclusions (Q1891513)

Summary: We show the existence of a solution of the total differential inclusion: \[ \nabla u(x)\in \text{ext } F(x, u(x)), \quad x\in \Omega, \qquad u= u_ 0 \quad \text{on } \partial \Omega, \] assuming that the convexified problem \[ \nabla u(x)\in \text{int } \overline {\text{co }} F(x, u(x)), \quad x\in \Omega, \qquad u= u_ 0 \quad \text{on } \partial\Omega, \] admits a smooth solution. The proof relies on a Baire category argument. Some examples are given, showing that in general our hypotheses cannot be relaxed.