Skip to content
DraconiticResearch
Search Sources API
203,948 studies31 sourcesUpdated 20 min ago
PreprintCC BY 4.0arXiv

An Alt-Caffarelli problem for elastic graphs

  • Marius Müller
Open at sourcePDFDOICite

Abstract

We study an Alt-Caffarelli-type free boundary problem for symmetric elastic graphs with fixed boundary height. Our results concern existence of minimizers and their description in terms of regularity and shape. It turns out that the weight coefficient in front of the measure term is decisive for existence. We identify a sharp threshold for this coefficient below which existence is guaranteed. Moreover, we find that all sign-changing minimizers are described by a system of three Eulerian elasticae, none of which are lines. Another result shows non-flatness of free boundary points, a classical question in the study of Alt-Caffarelli problems. These properties lead to a better understanding of the nonexistence phenomenon above the aforementioned weight threshold.

Related

  1. A minimizing cone from Cartan polynomials in the Alt-Caffarelli problem
    Matteo Carducci, Bozhidar Velichkov
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. The inverse elastic scattering problem in the cavity with impedance boundary
    Shiyu Xu, Huiling Zheng, Fang Zeng
    arXivPreprintarXiv licenceFull textAnalysis & PDEsNumerical analysis & scientific computingarXiv
  3. On Equilibrium Problem for Elastic Body with Neumann Boundary Condition and Delaminated Timoshenko Inclusion
    Alexander Khludnev, Evgeny Rudoy
    AxiomsPeer-reviewedCC BY 4.0General mathematicsCrossref
  4. Equal Lagrange Numbers with Non-Isomorphic Band Graphs: A Counterexample to Schiffler's Problem 6.3
    Qiyue Tang, Yizhi Zhang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. A deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs
    Zhiqiang Xu
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{muller2026altcaffarelli,
  title = {{An Alt-Caffarelli problem for elastic graphs}},
  author = {Marius Müller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34447},
  eprint = {2610.34447},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34447},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Alt-Caffarelli problem for elastic graphs
AU  - Marius Müller
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34447
UR  - https://arxiv.org/abs/2610.34447
AB  - We study an Alt-Caffarelli-type free boundary problem for symmetric elastic graphs with fixed boundary height. Our results concern existence of minimizers and their description in terms of regularity and shape. It turns out that the weight coefficient in front of the measure term is decisive for existence. We identify a sharp threshold for this coefficient below which existence is guaranteed. Moreover, we find that all sign-changing minimizers are described by a system of three Eulerian elasticae, none of which are lines. Another result shows non-flatness of free boundary points, a classical question in the study of Alt-Caffarelli problems. These properties lead to a better understanding of the nonexistence phenomenon above the aforementioned weight threshold.
ER  -