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

作者：Lu, Guozhen; Yang, Qiaohua^*

来源：Calculus of Variations and Partial Differential Equations, 2016, 55(6): 153.

DOI：10.1007/s00526-016-1077-5

• HARDY-SOBOLEV INEQUALITIES
• SPACE H-N
• RIEMANNIAN-MANIFOLDS
• NEGATIVE CURVATURE
• HYPERBOLIC SPACES
• OPERATORS

Wang and Ye conjectured in (Adv Math 230:294-320, 2012): Let Omega be a regular, bounded and convex domain in R-2. There exists a finite constant C(Omega) > 0 such that
integral(Omega) 4 pi u(2)/e pi(d)(u) <= C (Omega), for all u is an element of C0 infinity(Omega)
Where H-d = integral(Omega) vertical bar del u vertical bar(2) dxdy - 1/4 integral(Omega) u(2/)d(z, partial derivative Omega = min z1 is an element of partial derivative Omega vertical bar Z-Z(1)vertical bar. 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 + iy : vertical bar Z vertical bar = root x(2) + y(2) < 1}:
sup parallel to u parallel to H <= 1 integral(B) (e(4 pi u2) - 1 - 4 pi u(2))dv = 4
sup parallel to u parallel to H <= 1 integral(B) (e(4 pi u2) - 1 - 4 pi u(2))/(1-vertical bar Z vertical bar(2))(2)dxdy < infinity,
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(infinity)(B) with respect to the norm

• 出版日期2016-12
• 单位武汉大学