On the Uniqueness of Minimizers for a Class of Variational Problems with Polyconvex Integrand
| Autor: | Romeo Awi, Marc Sedjro |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Partial differential equation 35B27 39J40 Applied Mathematics 010102 general mathematics Degenerate energy levels Elasticity (physics) 01 natural sciences Convexity 010101 applied mathematics Displacement mapping Mathematics - Analysis of PDEs Direct methods FOS: Mathematics Uniqueness Minification 0101 mathematics Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Acta Applicandae Mathematicae. 168:137-167 |
| ISSN: | 1572-9036 0167-8019 |
| DOI: | 10.1007/s10440-019-00282-0 |
| Popis: | We prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo to a larger class of integrands. We are interested in Lagrangians of the form \(L(A,u)=f(A)+H(\det A)-F\cdot u \). Here the strict convexity condition on \(f \) and \(H \) have been relaxed to a convexity condition. Meanwhile, we have allowed the map \(F \) to be non-degenerate. First, we study these variational problems over displacements for which the determinant is positive. Second, we consider a limit case in which the functionals are degenerate. In that case, the set of admissible displacements reduces to that of incompressible displacements which are measure preserving maps. Finally, we establish that the minimizer over the set of incompressible maps may be obtained as a limit of minimizers corresponding to a sequence of minimization problems over general displacements provided we have enough regularity on the dual problems. We point out that these results do not rely on the direct methods of the calculus of variations. |
| Databáze: | OpenAIRE |
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