On a Generalization of Compact Operators and its Application to the Existence of Critical Points Without Convexity

Abstract

We introduce a certain property for a continuous (non-linear) operator that allows for the existence of critical points for functionals when the derivative complies with such a condition, without the need to check either weak lower semicontinuity or convexity. This condition is formulated in terms of Young measures, and so it requires restricting attention to true spaces of functions. It turns out that this property is a generalization of the standard compactness for a continuous, non-linear operator. We illustrate the relevance of this condition by applying it to the solution of typical Cauchy problems for ODEs, as well as boundary-value problems in one space dimension, and defer the much more complicated situation of PDEs in higher dimensions for a later work.