Taylor’s theorem for functionals on BMO with application to BMO local minimizers
| Autor: | Scott J. Spector, Daniel Spector |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Physics
Polynomial (hyperelastic model) Applied Mathematics Mathematics::Classical Analysis and ODEs Mathematics::Analysis of PDEs 01 natural sciences Weak derivative Bounded mean oscillation Dirichlet distribution 010101 applied mathematics Combinatorics Sobolev space symbols.namesake symbols Nabla symbol 0101 mathematics Energy (signal processing) Taylor's theorem |
| Zdroj: | Quarterly of Applied Mathematics. 79:409-417 |
| ISSN: | 1552-4485 0033-569X |
| DOI: | 10.1090/qam/1586 |
| Popis: | In this note two results are established for energy functionals that are given by the integral of W ( x , ∇ u ( x ) ) W({\mathbf x},\nabla {\mathbf u}({\mathbf x})) over Ω ⊂ R n \Omega \subset {\mathbb R}^n with ∇ u ∈ B M O ( Ω ; R N × n ) \nabla {\mathbf u}\in \mathrm {BMO}(\Omega ;{\mathbb R}^{N\times n}) , the space of functions of Bounded Mean Oscillation of John and Nirenberg. A version of Taylor’s theorem is first shown to be valid provided the integrand W W has polynomial growth. This result is then used to demonstrate that every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the second variation of the energy is uniformly positive is a strict local minimizer of the energy in W 1 , B M O ( Ω ; R N ) W^{1,\mathrm {BMO}}(\Omega ;{\mathbb R}^N) , the subspace of the Sobolev space W 1 , 1 ( Ω ; R N ) W^{1,1}(\Omega ;{\mathbb R}^N) for which the weak derivative ∇ u ∈ B M O ( Ω ; R N × n ) \nabla {\mathbf u}\in \mathrm {BMO}(\Omega ;{\mathbb R}^{N\times n}) . |
| Databáze: | OpenAIRE |
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