A note on the Morse homology for a class of functionals in Banach spaces involving the $p$-Laplacian

Author:

L. Asselle, M. Starostka

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

date:

2023-08-21 16:00:00

Abstract

In this paper we show how to construct Morse homology for an explicit class of functionals involving the $p$-Laplacian. The natural domain of definition of such functionals is the Banach space $W^{1,2p}_0(\Omega)$, where $p>n/2$ and $\Omega \subset \mathbb R^n$ is a bounded domain with sufficiently smooth boundary. As $W^{1,2p}_0(\Omega)$ is not isomorphic to its dual space, critical points of such functionals cannot be non-degenerate in the usual sense, and hence in the construction of Morse homology we only require that the second differential at each critical point be injective. Our result upgrades a result of Cingolani and Vannella, where critical groups for an analogous class of functionals are computed, and provides in this special case a positive answer to Smale's suggestion that injectivity of the second differential should be enough for Morse theory.

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