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

Abstract

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 Łojasiewicz [S. Łojasiewicz, Ensembles semi-analytiques, (1965), Publ. Inst. Hautes Etudes Sci., Bures-sur-Yvette. LaTeX version by M. Coste, August 29, 2006 based on mimeographed course notes by S. Łojasiewicz, https://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf] and proved by Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525-571]. 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 Chill [R. Chill, On the Łojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 2003, 2, 572-601], [R. Chill, The Łojasiewicz-Simon gradient inequality in Hilbert spaces, Proceedings of the 5th European-Maghrebian workshop on semigroup theory, evolution equations, and applications 2006, 25-36], Haraux and Jendoubi [A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 2007, 3, 449-470], and Simon [L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lect. Math. ETH Zürich, Birkhäuser, Basel 1996]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz-Simon gradient inequalities for harmonic maps, preprint 2019, https://arxiv.org/abs/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 Kwon [H. Kwon, Asymptotic convergence of harmonic map heat flow, ProQuest LLC, Ann Arbor 2002; Ph.D. thesis, Stanford University, 2002], Liu and Yang [Q. Liu and Y. Yang, Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds, Ark. Mat. 48 2010, 1, 121-130], Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525-571], [L. Simon, Isolated singularities of extrema of geometric variational problems, Harmonic mappings and minimal immersions (Montecatini 1984), Lecture Notes in Math. 1161, Springer, Berlin 1985, 206-277], and Topping [P. M. Topping, Rigidity in the harmonic map heat flow, J. Differential Geom. 45 1997, 3, 593-610]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions, preprint 2019, https://arxiv.org/abs/1510.03815v6; to appear in Mem. Amer. Math. Soc.], 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 [P. M. N. Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow, preprint 2016, https://arxiv.org/abs/1409.1525v4] for base manifolds of arbitrary dimension and due to Råde [J. Råde, On the Yang-Mills heat equation in two and three dimensions, J. reine angew. Math. 431 1992, 123-163] for dimensions two and three.