Weighted estimates for the stability operator of the helicoid on slowly varying domains

Stephen J. Kleene

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arXiv · 2607.03379

Weighted estimates for the stability operator of the helicoid on slowly varying domains

Stephen J. Kleene

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Abstract

We consider the poisson problem $\wt{\mc{L}} u = E$ for the operator $\wt{\mc{L}} = \Delta_{\mb{R}^2} + 2\cosh^{-2}(s)$ on domains of the form $\Lambda : = \{ (s, z): |s| \leq \ell(z) \}$, where the width $\ell(z)$ of the domain $\Lambda$ varies with $z$. We prove the existence of solutions satisfying weighted estimates when the source term satisfies certain natural orthogonality conditions, and when $e^{-\ell}$ is small and slowly varying.

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Stephen J. Kleene. 2026-07-03. Weighted estimates for the stability operator of the helicoid on slowly varying domains. https://arxiv.org/abs/2607.03379

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