2018/03/31 by Paul M. N. Feehan
Mathematics · #math.DG #math.AG #math.AP #math.GT #msc:32B20 #msc:32C05 #msc:32C18 #msc:32C25 #msc:58E05 #msc:14E15 #msc:32S45 #msc:57R45 #msc:58A07 #msc:58A35
paper · pdf · doi:10.1007/s00526-020-01734-4
published as Calculus of Variations and Partial Differential Equations 59 (2020), article no. 87, 50 pages · 47 pages, incorporating final galley proof corrections. Background material drawn from arXiv:1708.09775, arXiv:1706.09349, and arXiv:1510.03817v5. To appear in Calculus of Variations and Partial Differential Equations
arxiv created 2020/04/12 · arxiv updated 2020/08/17
It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space and the Morse Lemma for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Lojasiewicz-Simon gradient inequalities for analytic functions on Banach spaces.