A sharp Trudinger–Moser inequality on any bounded and convex planar domain

• Published in: Calculus of variations and partial differential equations 2016-12, Vol.55 (6), p.1-16, Article 153
• Main Authors: Lu, Guozhen, Yang, Qiaohua
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Inequality; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Smoothness; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-016-1077-5

Wang and Ye conjectured in (Adv Math 230:294–320, 2012 ): Let Ω be a regular, bounded and convex domain in R 2 . There exists a finite constant C ( Ω ) > 0 such that ∫ Ω e 4 π u 2 H d ( u ) d x d y ≤ C ( Ω ) , ∀ u ∈ C 0 ∞ ( Ω ) , where H d = ∫ Ω | ∇ u | 2 d x d y - 1 4 ∫ Ω u 2 d ( z , ∂ Ω ) 2 d x d y and d ( z , ∂ Ω ) = min z 1 ∈ ∂ Ω | z - z 1 | . The main purpose of this paper is to confirm that this conjecture indeed holds for any bounded and convex domain in 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 B = { z = x + i y : | z | = x 2 + y 2 < 1 } : sup ‖ u ‖ H ≤ 1 ∫ B ( e 4 π u 2 - 1 - 4 π u 2 ) d V = 4 sup ‖ u ‖ H ≤ 1 ∫ B ( e 4 π u 2 - 1 - 4 π u 2 ) ( 1 - | z | 2 ) 2 d x d y < ∞ , by using the method employed earlier by Lam and the first author (Adv Math 231(6):3259–3287, 2012 , J Differ Equ 255:298–325, 2013 ), where H denotes the closure of C 0 ∞ ( B ) with respect to the norm ‖ u ‖ H = ∫ B | ∇ u | 2 d x d y - ∫ B u 2 ( 1 - | z | 2 ) 2 d x d y . Using this strengthened Trudinger–Moser inequality, we also give a simpler proof of the Hardy–Moser–Trudinger inequality obtained by Wang and Ye (Adv Math 230:294–320, 2012 ).