[Paper Review] A Hardy-Moser-Trudinger inequality

Guofang Wang, Dong Ye|arXiv (Cornell University)|Dec 27, 2010

Nonlinear Partial Differential Equations17 references4 citations

TL;DR

This paper establishes a sharp Hardy-Moser-Trudinger inequality on the unit disc in ℝ² by combining the classical Moser-Trudinger and Hardy inequalities, proving that the supremum of the exponential integral is finite and achieved in a weighted Hilbert space. The key result is the existence of a finite sharp constant $ C_0 $ such that $ \int_B e^{\frac{4\pi u^2}{H(u)}} dx \leq C_0 $ for all $ u \in \mathcal{H} $, where $ H(u) $ is the energy functional incorporating the Hardy potential.

ABSTRACT

In this paper we obtain an inequality on the unit disc $B$ in the plane, which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant $C_0>0$ such that \[ \int_B e^{\frac {4πu^2}{H(u)}} dx \le C_0 < \infty, \quad \forall\; u\in C^\infty_0(B),\] where $$H(u) := \int_B | u|^2 dx - \int_B \frac {u^2}{(1-|x|^2)^2} dx.$$ This inequality is a two dimensional analog of the Hardy-Sobolev-Maz'ya inequality in higher dimensions, which was recently intensively studied. We also prove that the supremum is achieved in a suitable function space, which is an analog of the celebrated result of Carleson-Chang for the Moser-Trudinger inequality.

Motivation & Objective

• To unify the classical Moser-Trudinger and Hardy inequalities in two dimensions by introducing a combined functional framework.
• To establish a sharp inequality of the form $ \int_B e^{\frac{4\pi u^2}{H(u)}} dx \leq C_0 < \infty $ for all $ u \in \mathcal{H} $, where $ \mathcal{H} $ is the completion of $ C_0^\infty(B) $ under the norm $ \|u\| = \sqrt{H(u)} $.
• To prove that the best constant in this inequality is achieved by a maximizer in $ \mathcal{H} $, extending the Carleson-Chang result to the Hardy-modified setting.
• To conjecture a generalization of this inequality to more general domains $ \Omega \subset \mathbb{R}^2 $, with a Hardy-type potential based on distance to the boundary.

Proposed method

• Define the functional $ H(u) = \int_B |\nabla u|^2 dx - \int_B \frac{u^2}{(1-|x|^2)^2} dx $, which is non-negative and defines a Hilbert space norm on $ \mathcal{H} $.
• Prove the inequality $ \int_B e^{\frac{4\pi u^2}{H(u)}} dx \leq C_0 < \infty $ for all $ u \in \mathcal{H} $, using blow-up analysis and test functions concentrating near the origin.
• Construct a family of test functions $ f_\varepsilon $ with controlled $ H(f_\varepsilon) = 1 $, using a Green's function and cutoff functions to localize the concentration.
• Estimate the $ L^1 $-norm of $ e^{4\pi f_\varepsilon^2} $ from below using asymptotic expansions of logarithmic and Green's function terms.
• Use the asymptotic behavior of $ \beta_\varepsilon $ and $ \gamma_\varepsilon $ as $ \varepsilon \to 0 $ to show that the integral exceeds a value strictly greater than $ \pi + \pi e^{4\pi C_G + 1} $, proving the supremum is strictly larger than in the classical case.
• Apply variational methods and compactness arguments to show that the supremum is attained in $ \mathcal{H} $, with the maximizer not lying in $ H_0^1(B) $.

Experimental results

Research questions

• RQ1 Can the classical Moser-Trudinger inequality be improved by incorporating the Hardy potential in two dimensions?
• RQ2 Is the sharp constant in the improved inequality finite and attainable in a suitable function space?
• RQ3 Does the maximizer of the exponential functional lie in $ H_0^1(B) $, or is it strictly in the larger space $ \mathcal{H} $?
• RQ4 Can the inequality be extended to general bounded, convex domains in $ \mathbb{R}^2 $ with a distance-based Hardy potential?
• RQ5 What is the asymptotic behavior of the energy functional $ H(u) $ under blow-up sequences, and how does it affect the exponential integral?

Key findings

• The inequality $ \int_B e^{\frac{4\pi u^2}{H(u)}} dx \leq C_0 < \infty $ holds for all $ u \in \mathcal{H} $, with $ C_0 > 0 $ a finite sharp constant.
• The supremum of $ \int_B e^{4\pi u^2} dx $ over the unit ball in $ \mathcal{H} $ is achieved by a function $ u_0 \in \mathcal{H} $ with $ \|u_0\| = 1 $, extending the Carleson-Chang result to the Hardy-modified setting.
• The maximizer $ u_0 $ does not belong to $ H_0^1(B) $, indicating that the space $ \mathcal{H} $ is strictly larger than $ H_0^1(B) $, and the inequality is strictly stronger.
• The proof shows that the integral $ \int_B e^{4\pi f_\varepsilon^2} dx $ exceeds $ \pi + \pi e^{4\pi C_G + 1} $ for small $ \varepsilon $, proving the sharpness of the constant.
• The asymptotic analysis confirms $ \beta_\varepsilon = O(|\ln \varepsilon|^{1/2}) $, and $ \gamma_\varepsilon \to 1 $ as $ \varepsilon \to 0 $, which is crucial for the lower bound on the exponential integral.
• The result is a two-dimensional analog of the Hardy-Sobolev-Maz’ya inequality in higher dimensions, with the same sharp constant $ 4\pi $ as in the classical Moser-Trudinger inequality.