The Kurdyka–Łojasiewicz–Simon inequality and stabilisation in nonsmooth infinite-dimensional gradient systems

Abstract

We state and reprove a stabilisation result for solutions of abstract gradient systems associated with nonsmooth energy functions on infinite- dimensional Hilbert spaces. The main feature is the introduction of a convenient topology on the effective domain of the energy function, and that in this general setting the usual assumption on the relative compactness of the range of the solution in the energy space can be considerably relaxed to relative compactness of the range in the ambient Hilbert space. This simplifies the applicability of the stabilisation result even in the case of smooth energies.