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 $Λ: = \{ (s, z): |s| \leq \ell(z) \}$, where the width $\ell(z)$ of the domain $Λ$ 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.