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On total differential inclusions. (English) Zbl 0821.35158
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.

MSC:
35R70 PDEs with multivalued right-hand sides
49J45 Methods involving semicontinuity and convergence; relaxation
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