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Transactions of the American Mathematical Society

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Paths to uniqueness of critical points and applications to partial differential equations


Authors: Denis Bonheure, Juraj Földes, Ederson Moreira dos Santos, Alberto Saldaña and Hugo Tavares
Journal: Trans. Amer. Math. Soc. 370 (2018), 7081-7127
MSC (2010): Primary 46N10, 49K20, 35J10, 35J15, 35J47, 35J62
DOI: https://doi.org/10.1090/tran/7231
Published electronically: June 7, 2018
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Abstract: We prove a general criterion for the uniqueness of critical points of a functional in the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove the uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, providing the uniqueness of solutions to the corresponding Euler-Lagrange equations. For functionals emerging from elliptic problems, the assumptions of our abstract theorems follow from maximum principles, decay properties, and novel general inequalities. To illustrate our method we present a unified proof of known results, as well as new theorems for mean-curvature type operators, fractional Laplacians, Hamiltonian systems, Schrödinger equations, and Gross-Pitaevskii systems.


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Additional Information

Denis Bonheure
Affiliation: Département de Mathématique, Université libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
Email: denis.bonheure@ulb.ac.be

Juraj Földes
Affiliation: Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, Charlottesville, Virginia 22904
Email: foldes@virginia.edu

Ederson Moreira dos Santos
Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, CEP 13560-970, São Carlos - SP, Brazil
Email: ederson@icmc.usp.br

Alberto Saldaña
Affiliation: CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: alberto.saldana@tecnico.ulisboa.pt

Hugo Tavares
Affiliation: CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal – and – Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Edifício C6, Piso 1, Campo Grande 1749-016 Lisboa, Portugal
Email: hrtavares@ciencias.ulisboa.pt

DOI: https://doi.org/10.1090/tran/7231
Keywords: Uniqueness of critical points, non-convex functionals, uniqueness of positive solutions to elliptic equations and systems
Received by editor(s): August 16, 2016
Received by editor(s) in revised form: February 24, 2017
Published electronically: June 7, 2018
Additional Notes: The first author was supported by INRIA - Team MEPHYSTO and the projects PDR T.1110.14F (FNRS) and ARC AUWB-2012-12/17-ULB1- IAPAS
The first, second, and fourth authors were supported by the project MIS F.4508.14 (FNRS)
The first and third authors were partially supported by a bilateral agreement FNRS/CNPq
The first and fifth authors were supported by the project ERC Advanced Grant 2013 no. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”
The third author was partially supported by CNPq projects 309291/2012-7, 490250/2013-0, and 307358/2015-1 and FAPESP projects 2014/03805-2 and 2015/17096-6
The fifth author was supported by FCT/Portugal through the program Investigador FCT and through UID/MAT/04459/2013
Article copyright: © Copyright 2018 American Mathematical Society

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