Uniform Stability of Linearized Quasicontinuum Operators

Definitions and mathematical statement

Write \[ A:=\phi''(1)+4\phi''(2), \] where the source presupposes that these second derivatives exist and that the inequality $A>0$ is meaningful. The function $\phi$, its domain, and its relation to the underlying interaction model are not specified. Let $L_N$ denote the source's linearized force-based quasicontinuum (QCF) operator $L$, with its dependence on the parameter $N$ made explicit; the operator formula and the meaning of $N$ are not supplied.

The source asks whether \[ A>0 \quad\Longrightarrow\quad \text{$L_N$ inherits uniform stability or positive definiteness as $N\to\infty$.} \] This is an incompletely specified question: the reference operator from which stability is to be inherited, the admissible models, the spaces, the norms, the boundary conditions, and the treatment of possible null modes are absent.

For clarity, possible meanings can be distinguished without selecting one as the intended statement. If $L_N:V_N\to W_N$ acts between specified normed spaces, a uniform stability estimate could mean that there exist $c>0$ and $N_0$, independent of $N$, such that \[ \|L_Nu\|_{W_N}\ge c\|u\|_{V_N} \qquad(N\ge N_0,\ u\in V_N). \] The spaces, their norm scaling, and any additional requirements such as invertibility would have to be specified. If instead $L_N$ acts on a real inner product space $V_N$, quadratic-form positive definiteness means \[ \langle L_Nu,u\rangle_N>0 \qquad(0\ne u\in V_N), \] whereas uniform coercivity would require \[ \langle L_Nu,u\rangle_N\ge c\|u\|_{V_N}^{2} \qquad(N\ge N_0,\ u\in V_N) \] for an $N$-independent $c>0$. These are conditional definitions only; the source does not identify the intended criterion. It also does not specify whether positive definiteness refers to a quadratic form or whether a spectral property is intended. Consequently, the supplied scalar inequality alone does not determine a precise operator proposition to assess.

The supplied dated curation describes proved one-dimensional QCF stability and spectral results and an unresolved general inheritance question, especially for multidimensional models. This qualification is retained as a supplied assessment, not as an independently verified mathematical or literature conclusion; the intended dimensional scope remains unspecified.

Short English statement

Assuming $\phi''(1)+4\phi''(2)>0$, the source asks whether linearized force-based quasicontinuum operators retain stability or positive definiteness as $N\to\infty$. The operator, model, boundary conditions, norms, and intended positivity criterion are missing, so a unique precise formulation cannot yet be given. The supplied classification is partially solved, with a dated distinction between one-dimensional results and the broader unresolved question.

Sources

• Supplied local problem registry record: permanent UnsolvedMath ID $30001339$, titled Uniform Stability of Linearized Quasicontinuum Operators; exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2009 and names no proposer.
• Supplied report citation: PDE and Materials (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/42, pp. 2335--2404. https://doi.org/10.4171/owr/2009/42. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. The citation's 2010 title date and its 2009 DOI component are retained without reconciliation; no exact problem page is supplied.
• Supplied literature reference, labeled "force-based quasicontinuum stability": https://arxiv.org/abs/0903.0610.
• Supplied literature reference, labeled "QCF spectrum": https://arxiv.org/abs/1004.3435.
• Supplied literature reference, labeled "B-QCF stability": https://arxiv.org/abs/1112.2528. The record does not specify its precise relationship to the unqualified QCF question.
• Literature assessment checked at 2026-08-21: the supplied status is partially_solved. The supplied assessment states that one-dimensional QCF stability and spectral results are proved, while the general inheritance question, especially for multidimensional models, remains unresolved. This is supplied dated curation derived from the OWR record, not independently verified here; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its accompanying verification describes only enclosing TeX notation in mathematical delimiters without changing mathematical content. It does not establish mathematical correctness or resolve the missing hypotheses.