Variational composition of a monotone operator and a linear mapping with applications to elliptic PDEs with singular coefficients

This paper proposes a regularized notion of a composition of a monotone operator with a linear mapping. This new concept, called variational composition, can be shown to be maximal monotone in many cases where the usual composition is not. The two notions coincide, however, whenever the latter is maximal monotone. The utility of the variational composition is demonstrated by applications to subdifferential calculus, theory of measurable multifunctions, and elliptic PDEs with singular coefficients.