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The mean value theorem and basic properties of the obstacle problem for divergence form elliptic operators. (English) Zbl 1309.35034
Summary: In [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17, 43–77 (1963; Zbl 0116.30302)], W. Littman et al. proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for the same situation, but did not include the details of the proof. We show all of the nontrivial details needed to prove the formula stated by L. A. Caffarelli [The obstacle problem. Rome: Accademia Nazionale dei Lincei; Pisa: Scuola Normale Superiore (1998; Zbl 1084.49001)], and in the course of showing these details we establish some of the basic facts about the obstacle problem for general elliptic divergence form operators, in particular, we show a basic quadratic nondegeneracy property.

MSC:
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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