Paths to uniqueness of critical points and applications to partial differential equations
| Autor: | Juraj Földes, Ederson Moreira dos Santos, Hugo Tavares, Alberto Saldaña, Denis Bonheure |
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| Rok vydání: | 2018 |
| Předmět: |
Flexibility (engineering)
Partial differential equation Applied Mathematics General Mathematics 46N10 49K20 35J10 35J15 35J47 35J62 010102 general mathematics EQUAÇÕES DIFERENCIAIS PARCIAIS ELÍTICAS DE 2ª ORDEM Type (model theory) 01 natural sciences Domain (mathematical analysis) Convexity Schrödinger equation Hamiltonian system 010101 applied mathematics symbols.namesake Mathematics - Analysis of PDEs FOS: Mathematics symbols Applied mathematics Uniqueness 0101 mathematics Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Repositório Institucional da USP (Biblioteca Digital da Produção Intelectual) Universidade de São Paulo (USP) instacron:USP |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/tran/7231 |
| Popis: | We prove a unified and 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 uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide 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, Schr\"odinger equations, and Gross-Pitaevski systems. Comment: 41 pages |
| Databáze: | OpenAIRE |
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