By the Ambrosetti-Prodi theorem, the map F(u) = \Delta u - f (u) between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function f is required. We show in two different ways that convexity is indeed necessary. If f is not convex, there is a point with at least four preimages under F. Even more, F generically admits cusps among its critical points. We present a larger class of nonlinearities f for which the critical set of F has cusps. The results are true for Dirichlet, Neumann and periodic boundary conditions, among others.
Cusps and a converse to the Ambrosetti-Prodi Theorem / M. Calanchi, C. Tomei, A. Zaccur. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - 18:2(2018), pp. 483-507.
Titolo: | Cusps and a converse to the Ambrosetti-Prodi Theorem | |
Autori: | ||
Parole Chiave: | Ambrosetti-Prodi Theorem; Cusps | |
Settore Scientifico Disciplinare: | Settore MAT/05 - Analisi Matematica | |
Data di pubblicazione: | 2018 | |
Rivista: | ||
Tipologia: | Article (author) | |
Digital Object Identifier (DOI): | http://dx.doi.org/10.2422/2036-2145.201511_005 | |
Appare nelle tipologie: | 01 - Articolo su periodico |
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