It is a consequence of the Morse–Bott Lemma (see Theorems 2.10 and 2.14) that a C2 Morse–Bott function on an open neighborhood of a critical point in a Banach space obeys a Łojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses (Theorems 1, 2, and Corollary 3) for analytic functions on Banach spaces: If the Łojasiewicz 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 Łojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space (Łojasiewicz in Ensembles Semi-analytiques, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 1965) and the Morse Lemma (Theorems 4 and 5) 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 Łojasiewicz–Simon gradient inequalities for analytic functions on Banach spaces (Theorems 9 and 10).
|Original language||English (US)|
|Journal||Calculus of Variations and Partial Differential Equations|
|State||Published - Apr 1 2020|
All Science Journal Classification (ASJC) codes
- Applied Mathematics