Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functionals on Banach spaces and applications to harmonic maps

We prove two abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space (stated earlier without proof as Theorem 2.4.5 in Huang (2006)) that generalize previous abstract versions of this inequality, significantly 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 and not necessarily analytic, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). We apply our abstract Lojasiewicz-Simon gradient inequality to prove a Lojasiewicz-Simon gradient inequality for the harmonic map energy functional using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz-Simon gradient inequality for the harmonic map energy functional significantly generalizes those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997).