Weak and strong maximum principles for semicontinuous functions

Abstract

A local differential criterion is shown to control maximum principles for many known classes of functions satisfying such a principle. In spite of its differential nature, this criterion requires only the upper semicontinuity of the function of interest. It is satisfied by the solutions of nonlinear second order elliptic problems incorporating various types of degeneracy, for which it yields generalizations of many known results. The approach developed here also reveals that the difference between a strong and a weak principle for a given function is in fact a matter of invariance or noninvariance, for that function, of the criterion under changes of variable. The invariance property can often be translated in more familiar terms, which however completely conceal its actual nature.