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arXiv · 2609.25401

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 $Δ^m$ of arbitrary order $m\in\mathbb{N}$ in $\mathbb{R}^n$, $n\geq 2$, 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.

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Artur Andrade, Dorina Mitrea, Irina Mitrea, Marius Mitrea. 2026-09-21. The Solution to Agmon's Conjecture for the polylaplacian. https://arxiv.org/abs/2609.25401

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