Taylor’s Theorem for Functionals on BMO With Application to BMO Local Minimizers

Daniel E. Spector*, Scott J. Spector

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this note two results are established for energy functionals that are given by the integral of W(x, ∇u(x)) over Ω ⊂ (Formula presented) with ∇u ∈ BMO(Ω; (Formula presented)), 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 has polynomial growth. This result is then used to demonstrate that every Lipschitz-continuous solution of the corre-sponding Euler-Lagrange equations at which the second variation of the energy is uni-formly positive is a strict local minimizer of the energy in W1,BMO(Ω; (Formula presented) ), the subspace of the Sobolev space W1,1(Ω; (Formula presented) ) for which the weak derivative ∇u ∈ BMO(Ω; (Formula presented)).

Original languageEnglish
Pages (from-to)409-417
Number of pages9
JournalQuarterly of Applied Mathematics
Volume79
Issue number3
DOIs
Publication statusPublished - 2021 Sep
Externally publishedYes

Keywords

  • BMO local minimizers
  • Bounded mean oscillation
  • Taylor’s theorem

ASJC Scopus subject areas

  • Applied Mathematics

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