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

Abstract

We consider the poisson problem $\wt{\mc{L}} u = E$ for the operator $\wt{\mc{L}} = Δ_{\mb{R}^2} + 2\cosh^{-2}(s)$ on domains of the form $\Lambda :=\{(s,z):|s|\le \ell (z)\}$, 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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