[Paper Review] A sharp Trudinger-Moser inequality on any bounded and convex planar domain

Guozhen Lu, Qiaohua Yang|arXiv (Cornell University)|Dec 22, 2015

Advanced Mathematical Modeling in Engineering12 references3 citations

TL;DR

This paper confirms a conjecture by Wang and Ye regarding a sharp Trudinger-Moser inequality on any bounded, convex planar domain by employing the Riemann mapping theorem to transfer the problem to the unit disk, where a strengthened Trudinger-Moser inequality is established via a rearrangement-free method. The key result is the validity of the inequality with the optimal constant $4 au\pi$ and a finite bound depending only on the domain, even without smooth boundary assumptions.

ABSTRACT

Wang and Ye conjectured in [22]: Let $Ω$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$. There exists a finite constant $C(Ω)>0$ such that \[ \int_Ωe^{\frac{4πu^{2}}{H_{d}(u)}}dxdy\le C(Ω),\;\;\forall u\in C^{\infty}_{0}(Ω), \] where $H_{d}=\int_Ω| abla u|^{2}dxdy-\frac{1}{4}\int_Ω\frac{u^{2}}{d(z,\partialΩ)^{2}}dxdy$ and $d(z,\partialΩ)=\min\limits_{z_{1}\in\partialΩ}|z-z_{1}|$.} The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in $\mathbb{R}^{2}$ via the Riemann mapping theorem (the smoothness of the boundary of the domain is thus irrelevant). We also give a rearrangement-free argument for the following Trudinger-Moser inequality on the hyperbolic space $\mathbb{B}=\{z=x+iy:|z|=\sqrt{x^{2}+y^{2}}<1\}$: \[ \sup_{\|u\|_{\mathcal{H}}\leq 1} \int_{\mathbb{B}}(e^{4πu^{2}}-1-4πu^{2})dV=\sup_{\|u\|_{\mathcal{H}}\leq 1}\int_{\mathbb{B}}\frac{(e^{4πu^{2}}-1-4πu^{2})}{(1-|z|^{2})^{2}}dxdy< \infty, \] by using the method employed earlier by Lam and the first author [9, 10], where $\mathcal{H}$ denotes the closure of $C^{\infty}_{0}(\mathbb{B})$ with respect to the norm $$\|u\|_{\mathcal{H}}=\int_{\mathbb{B}}| abla u|^{2}dxdy-\int_{\mathbb{B}}\frac{u^{2}}{(1-|z|^{2})^{2}}dxdy.$$ Using this strengthened Trudinger-Moser inequality, we also give a simpler proof of the Hardy-Moser-Trudinger inequality obtained by Wang and Ye [22].

Motivation & Objective

• To resolve a conjecture by Wang and Ye on the existence of a finite sharp Trudinger-Moser inequality for all bounded, convex domains in $\mathbb{R}^2$.
• To establish a strengthened Trudinger-Moser inequality on the hyperbolic space $\mathbb{B}$ without relying on symmetrization or rearrangement techniques.
• To provide a new, simpler proof of the Hardy-Moser-Trudinger inequality using the strengthened inequality on $\mathbb{B}$.
• To demonstrate that the sharp constant in the Trudinger-Moser inequality is preserved under conformal mappings, extending the result to general convex domains.

Proposed method

• Utilizes the Riemann mapping theorem to conformally map any bounded, convex planar domain $\Omega$ to the unit disk $\mathbb{B}$, preserving the structure of the functional.
• Applies a rearrangement-free method inspired by Lam and Lu [9,10] to derive a sharp Trudinger-Moser inequality on $\mathbb{B}$ involving the weight $\frac{1}{(1-|z|^2)^2}$.
• Establishes the inequality $\sup_{\|u\|_{\mathcal{H}} \leq 1} \int_{\mathbb{B}} \frac{e^{4\pi u^2} - 1 - 4\pi u^2}{(1-|z|^2)^2} dx dy < \infty$ using integral estimates and conformal invariance.
• Uses the conformal invariance of the Dirichlet energy and the Hardy-type term to relate the weight $\frac{1}{d(z,\partial\Omega)^2}$ on $\Omega$ to the hyperbolic weight on $\mathbb{B}$ via the Jacobian of the conformal map.
• Applies the improved Hardy inequality $\int_{\Omega} |\nabla u|^2 dx dy - \frac{1}{4} \int_{\Omega} \frac{u^2}{d(z,\partial\Omega)^2} dx dy \geq C \int_{\Omega} u^2 dx dy$ to control the $L^2$-norm of $u$.
• Combines the weighted Trudinger-Moser bound on $\mathbb{B}$ with the conformal transformation and pointwise lower bound on the Jacobian to derive the final inequality on $\Omega$.

Experimental results

Research questions

• RQ1 Does the Wang-Ye conjecture on the sharp Trudinger-Moser inequality hold for all bounded, convex domains in $\mathbb{R}^2$, regardless of boundary smoothness?
• RQ2 Can a rearrangement-free method be used to establish sharp Trudinger-Moser inequalities on hyperbolic spaces and related domains?
• RQ3 Is the constant $4\pi$ in the exponent optimal for the Trudinger-Moser inequality when a Hardy-type correction term is included?
• RQ4 Can the conformal invariance of the hyperbolic Trudinger-Moser inequality be leveraged to extend results to arbitrary convex domains?
• RQ5 What is the precise relationship between the distance to the boundary $d(z,\partial\Omega)$ and the hyperbolic metric under conformal mapping?

Key findings

• The Wang-Ye conjecture is confirmed: for any bounded, convex domain $\Omega \subset \mathbb{R}^2$, there exists a finite constant $C(\Omega)$ such that $\int_{\Omega} \exp\left(\frac{4\pi u^2}{H_d(u)}\right) dx dy \leq C(\Omega)$ for all $u \in C_0^\infty(\Omega)$, where $H_d(u) = \int_{\Omega} |\nabla u|^2 dx dy - \frac{1}{4} \int_{\Omega} \frac{u^2}{d(z,\partial\Omega)^2} dx dy$.
• A strengthened Trudinger-Moser inequality is proven on the hyperbolic disk: $\sup_{\|u\|_{\mathcal{H}} \leq 1} \int_{\mathbb{B}} \frac{e^{4\pi u^2} - 1 - 4\pi u^2}{(1-|z|^2)^2} dx dy < \infty$, with the constant $C_2$ independent of $u$.
• The proof avoids symmetrization and relies on conformal invariance and pointwise estimates, providing a new, simpler approach to the Hardy-Moser-Trudinger inequality.
• The optimal constant $4\pi$ in the exponent is preserved under conformal mapping, and the inequality holds uniformly across all bounded, convex domains.
• The improved Hardy inequality $\int_{\Omega} |\nabla u|^2 dx dy - \frac{1}{4} \int_{\Omega} \frac{u^2}{d(z,\partial\Omega)^2} dx dy \geq C_6 \int_{\Omega} u^2 dx dy$ is used to control the $L^2$-norm and close the estimate.
• The final bound on $\int_{\Omega} e^{4\pi u^2} dx dy$ is shown to be finite and uniformly bounded by a constant depending only on $\Omega$, via the chain of inequalities involving the transformed domain and the hyperbolic weight.