Reconstruction of the Clarke Subdifferential by the Lasry–Lions Regularizations

Abstract

We prove that the Clarke subdifferential of a locally Lipschitz function with a growth condition, defined on a Hilbert space, can be represented by the derivatives of its Lasry–Lions regularizations. We complete a result of J. Benoist, showing a similar representation for the Clarke subdifferential of a Lipschitz function in an arbitrary Banach space, by the Clarke subdifferentials of its Lasry–Lions regularizations.