Łojasiewicz–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 function 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 [62] proved by Simon as [75, Theorem 3]. We prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the function is Morse-Bott, improving on similar results due to Chill [17, Corollary 3.12], [18, Corollary 4], Haraux and Jendoubi [44, Theorem 2.1], and Simon [77, Lemma 3.13.1]. 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, Theorem 4.2], Liu and Yang [60, Lemma 3.3], Simon [75, Theorem 3], [76, Equation (4.27)], and Topping [83, Lemma 1]. 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, Theorems 23.1 and 23.17] for base manifolds of arbitrary dimension and due to Råde [69, Proposition 7.2] for dimensions two and three. Contents

Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds

“…A well-known result due to Hélein [17, Theorem 4.1.1] tells us that if M has dimension d = 2, then f ∈ C ∞ (M ; N ); for d ≥ 3, regularity results are far more limited -see, for example, [17, Theorem 4.3.1] due to Bethuel. From [12], we recall the…”

“…Theorem 2.2 is proved by the author and Maridakis in [12] as a consequence of a more general abstract Lojasiewicz-Simon gradient inequality for an analytic function on a Banach space, namely [12, Theorem 2], while Theorem 2.4 may be deduced as a consequence of [12,Theorem 2].…”

“…Outline of the article. In Section 2, we review the Lojasiewicz-Simon gradient inequality for the harmonic map energy function based on results of the author and Maridakis [12] and Simon [38,39]. In Section 3, we prove certain a priori estimates for the difference of two harmonic maps and, with the aid of the Lojasiewicz-Simon gradient inequality, complete the proof of Theorem 1.…”

“…Lojasiewicz-Simon gradient inequality for analytic functions on Banach spaces). (See[12, Corollary 3].) Let X andX be Banach spaces with continuous embeddings, X ⊂ X ⊂ X * , and such that the embedding, X ⊂ X * , is definite.…”

Convergence of solutions for the fractional Cahn–Hilliard system

“…The Lojasiewicz-Simon ( LS) inequality is an infinite-dimensional extension of the Lojasiewicz inequality, which is a gradient inequality for analytic functions defined on an open set U ⊂ R d (see [26,27] and Proposition 2.3 below). The LS inequality was first introduced by L. Simon [35] and has been subsequently applied to various PDEs with gradient(-like) structures by Haraux, Jendoubi, Chill and many other authors (see, e.g., without any claim of completeness, [25,22,19,14,21,23,8,24,9,20,13]). A general form of the Lojasiewicz-Simon inequality reads as follows: Let E : X → R be a "smooth" functional defined on a Banach space X and let φ be a critical point of E, i.e., E ′ (φ) = 0 in the dual space X * , where E ′ : X → X * denotes the Fréchet derivative of E. Then there exist constants θ ∈ (0, 1/2] and ω, δ > 0 such that…”