Morse-Smale index theorems for elliptic boundary deformation problems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:253

Article

Dalbono, Francesca¹  Portaluri, Alessandro² 

[1] Univ Lisbon, Fac Ciencias, CMAF Ctr Matemat & Aplicacoes Fundamentais, P-1649003 Lisbon, Portugal

[2] Univ Salento, Ex Coll Fiorini, Dipartimento Matemat Ennio De Giorgi, I-73100 Lecce, Italy

关键词: Elliptic boundary deformation problems;    Maslov index;    Spectral flow;   

DOI  :  10.1016/j.jde.2012.04.008

来源: Elsevier

【 摘 要 】

Morse-type index theorems for self-adjoint elliptic second order boundary value problems arise as the second variation of an energy functional corresponding to some variational problem. The celebrated Morse index theorem establishes a precise relation between the Morse index of a geodesic (as critical point of the geodesic action functional) and the number of conjugate points along the curve. Generalization of this theorem to linear elliptic boundary value problems appeared since seventies. (See, for instance. Smale (1965) [12], Uhlenbeck (1973) [15] and Simons (1968) [11] among others.) The aim of this paper is to prove a Morse-Smale index theorem for a second order self-adjoint elliptic boundary value problem in divergence form on a star-shaped domain of the N-dimensional Euclidean space with Dirichlet and Neumann boundary conditions. This result will be achieved by generalizing a recent new idea introduced by authors in Deng and Jones (2011) [5], based on the idea of shrinking the boundary. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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