## 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

### Keywords:

existence; convexified problem; Baire category
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### References:

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