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PreprintarXiv licencearXiv

The Solution to Agmon's Conjecture for the polylaplacian

  • Artur Andrade
  • Dorina Mitrea
  • Irina Mitrea
  • Marius Mitrea
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Abstract

In his 1957 paper on the Dirichlet problem for higher-order elliptic equations in the plane, Agmon conjectured that there should exist higher-order equations in dimensions greater than two for which a potential-theoretic approach based on multi-layer integral operators can be successfully implemented. We show that this is indeed the case for the polyharmonic operator Δm of arbitrary order m∈ℕ in ℝn, n≥ 2, even in geometrically rough domains permitting singularities beyond the scope of the existing theory of higher-order elliptic boundary value problems. Central to our approach is a new genre of multi-layer singular integral operators, which play for Δm the role assumed by the classical harmonic double layer potential in the treatment of the Dirichlet problem for harmonic functions.

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Cite

BibTeXRIS
BibTeX
@misc{andrade2026solution,
  title = {{The Solution to Agmon's Conjecture for the polylaplacian}},
  author = {Artur Andrade and Dorina Mitrea and Irina Mitrea and Marius Mitrea},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25401},
  eprint = {2609.25401},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25401},
}
RIS
TY  - UNPB
TI  - The Solution to Agmon's Conjecture for the polylaplacian
AU  - Artur Andrade
AU  - Dorina Mitrea
AU  - Irina Mitrea
AU  - Marius Mitrea
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25401
UR  - https://arxiv.org/abs/2609.25401
AB  - In his 1957 paper on the Dirichlet problem for higher-order elliptic equations in the plane, Agmon conjectured that there should exist higher-order equations in dimensions greater than two for which a potential-theoretic approach based on multi-layer integral operators can be successfully implemented. We show that this is indeed the case for the polyharmonic operator Δ^m of arbitrary order m∈ℕ in ℝ^n, n≥ 2, even in geometrically rough domains permitting singularities beyond the scope of the existing theory of higher-order elliptic boundary value problems. Central to our approach is a new genre of multi-layer singular integral operators, which play for Δ^m the role assumed by the classical harmonic double layer potential in the treatment of the Dirichlet problem for harmonic functions.
ER  -