Geodesic
A Mean Value Theorem for non Differentiable Mappings in Banach Spaces
Serdica Mathematical Journal, Tome 21 (1995) no. 1, pp. 59-66.

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We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.
Keywords: Mean Value Theorem, Smooth Variational Principle, Non Smooth Analysis, Viscosity Solutions 
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Deville, Robert. A Mean Value Theorem for non Differentiable Mappings in Banach Spaces. Serdica Mathematical Journal, Tome 21 (1995) no. 1, pp. 59-66. http://geodesic.mathdoc.fr/item/SMJ2_1995_21_1_a3/