MEAN-VALUE INEQUALITIES IN HILBERT-SPACE

We establish a new mean value theorem applicable to lower semicontinuo us functions on Hilbert space. A novel feature of the result is its '' multidirectionality'': it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequen ces of the theorem, including applications to calculus, flow invarianc e, and generalized solutions to partial differential equations.