Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces

Abstract

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz proved by Simon (1983). We also prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the functional is Morse-Bott, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). In our article arXiv:1903.01953, we apply our abstract Lojasiewicz-Simon gradient inequalities to prove a Lojasiewicz-Simon gradient inequalities for the harmonic map energy functional using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities for the harmonic map energy functional generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997). In our monograph arXiv:1510.03815, we prove Lojasiewicz--Simon gradient inequalities for coupled Yang--Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang--Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of arbitrary dimension and due to Rade (1992) for dimensions two and three.