[Paper Review] Some properties of a class of elliptic partial differential operators

Shaolin Chen, Матти Вуоринен|arXiv (Cornell University)|Dec 26, 2014

Analytic and geometric function theory18 references3 citations

TL;DR

This paper establishes Schwarz-Pick type estimates and coefficient bounds for solutions to a class of second-order elliptic partial differential operators $ T_{eta} $, generalizing classical results for harmonic functions. It derives a Landau-type theorem for univalent mappings, proving the existence of a univalent disk in the image of such solutions under $ L^p $-norm constraints.

ABSTRACT

We prove Schwarz-Pick type estimates and coefficient estimates for a class of elliptic partial differential operators introduced by Olofsson. Then we apply these results to obtain a Landau type theorem.

Motivation & Objective

• To extend classical Schwarz-Pick and coefficient estimates from harmonic functions to solutions of a broader class of elliptic PDEs defined by the operator $ T_{\alpha} $.
• To investigate the geometric and analytic properties of solutions to the Dirichlet problem $ T_{\alpha}(f) = 0 $ in the unit disk $ \mathbb{D} $, with distributional boundary data.
• To derive a Landau-type theorem ensuring the existence of a univalent disk in the image of solutions under $ L^p $-norm constraints.
• To generalize Heinz-type inequalities for the operator norm $ \|D_f(z)\| $ in terms of $ \alpha $-dependent bounds.
• To characterize extremal functions achieving sharp estimates via integral representations involving hypergeometric functions.

Proposed method

• Utilizes the Poisson-type integral representation (1.3) for solutions $ f $ to $ T_{\alpha}(f) = 0 $, with kernel $ K_{\alpha}(z) = c_{\alpha} \frac{(1-|z|^2)^{\alpha+1}}{|1-z|^{\alpha+2}} $.
• Applies Jensen’s inequality to $ L^p $-norms of boundary data to derive pointwise bounds on $ |f(z)| $ and $ \|D_f(z)\| $, leveraging convexity of $ x \mapsto x^p $.
• Employs the hypergeometric function representation $ \mathcal{M}_{\alpha}(r) = \frac{[\Gamma(1+\alpha/2)]^2}{\Gamma(1+\alpha)} F(-\alpha/2, -\alpha/2; 1; r^2) $ to express average kernel values.
• Derives sharp bounds on $ \|D_f(z)\| $ via the expression $ \|D_f(z)\| \leq \frac{M \mathcal{M}_{\alpha}(|z|) [2+\alpha + (4+3\alpha)|z|]}{1-|z|^2} $, with $ M = \sup |f| $.
• Applies conformal invariance and normalization via $ Q(\zeta) = f(\gamma \zeta)/\gamma $ to reduce univalence to extremal coefficient estimates.
• Uses the condition $ \frac{\lambda}{M^*(2+\alpha)} - \frac{4M^*\rho_0}{\pi} \left[ \frac{2-\rho_0}{(1-\rho_0)^2} + \none \right] = 0 $ to determine the maximal univalent disk radius $ \rho_0 $.

Experimental results

Research questions

• RQ1 What are the sharp Schwarz-Pick type estimates for solutions to the elliptic PDE $ T_{\alpha}(f) = 0 $ in the unit disk?
• RQ2 How do coefficient estimates for $ T_{\alpha} $-harmonic functions depend on the parameter $ \alpha $ and the $ L^p $-norm of the boundary data?
• RQ3 Can a Landau-type theorem be established for $ T_{\alpha} $-harmonic mappings, guaranteeing a univalent image disk of positive radius?
• RQ4 What is the optimal radius $ \rho_0 $ of the largest disk $ \mathbb{D}_{\rho_0} $ on which the normalized mapping $ Q(\zeta) = f(\gamma\zeta)/\gamma $ is univalent?
• RQ5 What are the extremal functions achieving the sharp bounds in the estimates for $ \|D_f(z)\| $ and $ |f(z)| $?

Key findings

• The paper establishes the sharp estimate $ \left|f(z) - \frac{(1-|z|)^{\alpha+1}}{1+|z|} f(0) \right| \leq M \left[ \frac{1}{2\pi} \int_0^{2\pi} K_{\alpha}(ze^{-it}) dt - \frac{(1-|z|)^{\alpha+1}}{1+|z|} K_{\alpha}(0) \right] $ for $ \alpha > -1 $.
• It proves $ \|D_f(z)\| \leq \frac{M \mathcal{M}_{\alpha}(|z|) [2+\alpha + (4+3\alpha)|z|]}{1-|z|^2} $, with $ \mathcal{M}_{\alpha}(r) = \frac{[\Gamma(1+\alpha/2)]^2}{\Gamma(1+\alpha)} F(-\alpha/2, -\alpha/2; 1; r^2) $, and this bound is sharp.
• For $ f \in \mathcal{C}^2(\mathbb{D}) $ with $ \|f\|_p < \infty $, the estimate $ |f(z)| \leq c_{\alpha}^{1/p} \|f\|_p \frac{(1+|z|)^{\frac{\alpha+1}{p}}}{(1-|z|)^{1/p}} $ holds for all $ z \in \mathbb{D} $, with $ c_{\alpha} = \frac{[\Gamma(1+\alpha/2)]^2}{\Gamma(1+\alpha)} $.
• The normalized mapping $ Q(\zeta) = f(\gamma\zeta)/\gamma $ is univalent in $ \mathbb{D}_{\rho_0} $, where $ \rho_0 $ satisfies $ \frac{\lambda}{M^*(2+\alpha)} - \frac{4M^*\rho_0}{\pi} \left[ \frac{2-\rho_0}{(1-\rho_0)^2} + \frac{2\rho_0}{(1-\rho_0)(1-\rho_0^2)^2} \right] = 0 $, and $ M^* = c_{\alpha}^{1/p} \|f\|_p \mu(\gamma_0) $.
• The image $ f(\mathbb{D}_{\gamma_0 \rho_0}) $ contains a univalent disk $ \mathbb{D}_{R_0} $ with $ R_0 \geq \frac{2\rho_0}{3} \left[ \frac{\lambda}{M^*(2+\alpha)} - \frac{M^*\rho_0(2-\rho_0)}{\pi(1-\rho_0)^2} \right] $, providing a quantitative lower bound on the univalent image radius.
• Extremal functions achieving the sharpness of the estimates are of the form $ f(z) = \frac{2\gamma}{\pi} \arg\left( \frac{1+\psi(z)}{1-\psi(z)} \right) $, where $ \gamma \in \partial\mathbb{D} $, $ \psi $ is a conformal automorphism of $ \mathbb{D} $, and the bounds are sharp for $ \alpha = 0 $, recovering classical results of Heinz and Colonna.