A Decay Estimate For The Stability Operator Of The Helicoid

Technical Summary: A Decay Estimate for the Stability Operator of the Helicoid

Problem Statement
This article addresses the Poisson problem for the stability operator of the helicoid on a vertical strip domain $R = [-\ell, \ell] \times \mathbb{R} \subset \mathbb{R}^2$. The operator is defined as $\tilde{L} = \Delta_{\mathbb{R}^2} + 2\cosh^{-2}(s)$, where $(s, z)$ are coordinates on the plane. The problem seeks to solve the equation $\tilde{L}u = E$, where the source term $E$ is a $C^{0,\alpha}$ function supported on a fixed-height strip $[-\ell, \ell] \times [-2\pi, 2\pi]$.

The motivation for this study arises from geometric problems involving both the helicoid and the catenoid. While previous works (e.g., [BK], [BK2]) have established invertibility properties for this operator under strong symmetry assumptions (reducing the domain to long cylinders $[-\ell, \ell] \times S^1$), this paper aims to relax those symmetry assumptions. The goal is to replace symmetry with decay estimates, specifically investigating how solutions behave away from the support of the source term when the source satisfies specific orthogonality conditions.

Methodology
The proof strategy involves a limiting process on finite domains. The author constructs solutions $u_N$ on bounded rectangles $[-\ell, \ell] \times [-N, N]$ with specific boundary conditions and then proves uniform estimates in $N$ to extract a limiting solution $u$ on the infinite strip.

1. Boundary Conditions and Orthogonality: To handle the kernel of the operator, the problem is split into odd ($u_-$) and even ($u_+$) parts with respect to $s$.

   • The odd part satisfies a Robin boundary condition at $s=\ell$: $\partial_s u_-(\ell, z)\tanh(\ell) - u_-(\ell, z)/\cosh^2(\ell) = 0$.
   • The even part satisfies a Robin boundary condition at $s=\ell$: $\partial_s u_+(\ell, z)\cosh^{-1}(\ell) - u_+(\ell, z)(\cosh^{-1})'(\ell) = 0$.
   • Neumann conditions are imposed at $z = \pm N$.
   • Crucially, the source term $E$ is assumed to satisfy "strong orthogonality conditions" (Equation 3): $\int_{-\ell}^{\ell} E(s, z)\cosh^{-1}(s)ds = 0$ and $\int_{-\ell}^{\ell} E(s, z)\tanh(s)ds = 0$ for all $z$. These conditions are shown to be inherited by the solutions $u_N$.

2. A Priori Estimates and Spectral Analysis:

   • The core analytical tool is a one-dimensional weighted Poincaré inequality (Propositions 2 and 3). These inequalities bound the energy of functions orthogonal to the kernel elements ($\tanh(s)$ for the odd case, $\cosh^{-1}(s)$ for the even case) in terms of their weighted $L^2$ norms.
   • Lemma 1 characterizes the lower spectrum of the weighted eigenvalue problem $\tilde{L}\phi = \lambda \cosh^{-2}(s)\phi$. It establishes that the only non-positive eigenvalues are $\lambda = -2$ (constant eigenfunctions) and $\lambda = 0$ (eigenfunctions proportional to $\tanh(s)$). This spectral gap is essential for proving the Poincaré inequalities.
   • Proposition 1 establishes $L^2$ and $C^0$ bounds for solutions on the finite rectangles, showing that $\|u_N\| \leq C \ell^2 \|E\|_{L^2}$.

3. Existence and Convergence:

   • Proposition 5 demonstrates the existence of solutions $u_N$ on the finite rectangles satisfying the orthogonality conditions and uniform bounds.
   • Using Schauder estimates and a bootstrapping argument, the sequence $u_N$ is shown to converge on compact subsets to a unique locally $C^{2,\alpha}$ function $u$ solving the Poisson problem on the infinite strip.

4. Decay Estimation:

   • The decay is derived by analyzing the convexity of the integral $\int_{-\ell}^{\ell} u^2 ds$ for $|z| > 2\pi$ (away from the source support). The Poincaré inequality implies this integral is convex and, combined with the uniform bounds, leads to a decay rate of $O((1+|z|)^{-1})$.
   • A bootstrapping argument using the $C^{2,\alpha}$ regularity and the integral decay is then employed to improve the pointwise decay rate. By iterating the estimate, the decay exponent is shown to converge to any $\xi \in (0, 1/2)$.

Key Contributions and Results
The main result is Theorem 1, which states that for any $\xi \in (0, 1/2)$, there exists a constant $C$ such that the unique solution $u$ to the Poisson problem $\tilde{L}u = E$ (subject to the specified boundary and orthogonality conditions) satisfies the weighted pointwise estimate:
$$\|u\|_{2,\alpha} \leq C \ell^{5/2} \beta \left( \frac{1}{1+|z|} \right)^\xi$$
where $\beta$ bounds the source term $E$.

The paper provides:

• A rigorous characterization of the kernel of the stability operator on finite rectangles with mixed boundary conditions (Corollaries 1 and 2).
• The derivation of one-dimensional weighted Poincaré inequalities tailored to the specific geometry of the helicoid stability operator.
• A proof that strong orthogonality conditions on the source term are sufficient to guarantee the existence of solutions with definite decay rates, even in the absence of rotational symmetry.

Significance and Claims
The paper claims to provide a necessary analytical tool for geometric problems where symmetry assumptions cannot be made. By establishing that solutions decay at a definite rate away from the source support under specific orthogonality conditions, the work suggests that symmetry can be effectively replaced by decay estimates in the study of the helicoid and related minimal surfaces.

The author notes modestly that while the arguments presented are specific to the stability operator of the helicoid, they "can likely be generalized to operators other than stability of the Helicoid," though this generalization is not pursued in the current text. The work serves as a record of these specific decay estimates, filling a gap in the understanding of the operator's invertibility on non-compact, non-symmetric domains.