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

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic function on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, that of the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz [60] and proved by Simon [75]. We prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the function is Morse-Bolt, improving on similar results due to Chill [17,18], Haraux and Jendoubi [44], and Simon [77]. In [33], we apply our abstract gradient inequalities to prove Lojasiewicz-Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities generalize those of Kwon [58], Liu and Yang [63], Simon [75, 76], and Topping [83]. In [32], 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 [26] for base manifolds of arbitrary dimension and due to Rade [69] for dimensions two and three.