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Łojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces. (English) Zbl 1447.58018
Summary: We prove several abstract versions of the Łojasiewicz-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 S. Łojasiewicz [in: Actes Congr. internat. Math. 1970, 2, 237–241 (1971; Zbl 0241.32005)] (LaTeX version by M. Coste, August 29, 2006 based on mimeographed course notes by S. Łojasiewicz) and proved by L. Simon [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071)]. We prove that the optimal exponent of the Łojasiewicz-Simon gradient inequality is obtained when the function is Morse-Bott, improving on similar results due to [R. Chill, J. Funct. Anal. 201, No. 2, 572–601 (2003; Zbl 1036.26015); “The Łojasiewicz-Simon gradient inequality in Hilbert spaces”, in: Proceedings of the 5th European-Maghrebian workshop on semigroup theory, evolution equations, and applications, 2006. Carthage: Université de Carthage. 25–36 (2007); A. Haraux and M. A. Jendoubi, J. Evol. Equ. 7, No. 3, 449–470 (2007; Zbl 1145.35033); L. Simon, Theorems on regularity and singularity of energy minimizing maps. Based on lecture notes by Norbert Hungerbühler. Basel: Birkhäuser (1996; Zbl 0864.58015)]. In [“Łojasiewicz-Simon gradient inequalities for harmonic maps”, Preprint, arXiv:1903.01953], we apply our abstract gradient inequalities to prove Łojasiewicz-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 H. Kwon [Asymptotic convergence of harmonic map heat flow. Ann Arbor: ProQuest LLC, Stanford University (PhD thesis) (2002)], Q. Liu and Y. Yang [Ark. Mat. 48, No. 1, 121–130 (2010; Zbl 1191.53028)], L. Simon [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071); “Isolated singularities of extrema of geometric variational problems”, Lect. Notes Math. 1161, 206–277 (2006; doi:10.1007/BFb0075139)], and P. M. Topping [J. Differ. Geom. 45, No. 3, 593–610 (1997; Zbl 0955.58013)]. In [“Łojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions”, Preprint, arXiv:1510.03815], we prove Łojasiewicz-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 [“Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow”, Preprint, arXiv:1409.1525] for base manifolds of arbitrary dimension and due to J. Råde [J. Reine Angew. Math. 431, 123–163 (1992; Zbl 0760.58041)] for dimensions two and three.

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
46B25 Classical Banach spaces in the general theory
58E20 Harmonic maps, etc.
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