2017/09/27 by Van Hoang Nguyen, Nguyen, Van Hoang
Computer Science · Mathematics · #26D10 #46E35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1709.09608
openalex publication_date 2017/09/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We establish an improved version of the Moser-Trudinger inequality in the\nhyperbolic space mathbb Hn, n\≥ 2. Namely, we prove the following\nresult: for any 0 \≤ \λ < \( fracn-1n\)n, then we have \n
sup_
substacku
in C0^
infty(
mathbb Hn)
int
mathbb Hn |
nablagν|gn d
textVolg -
lambda
int
mathbb Hn |u|n d
text Volg
leq 1\n
int
mathbb Hn
Phin(
alphan |u|^
fracnn-1) d
text Volg lt;\n
infty, where \αn = n \ωn-1^ frac1n-1, \ωn-1\ndenotes the surface area of the unit sphere in mathbb Rn and \Φn(t) =≠t -\∑j=0n-2\(tj)/(j!). This improves the Moser-Trudinger\ninequality in hyperbolic spaces obtained recently by Mancini and Sandeep, by\nMancini, Sandeep and Tintarev and by Adimurthi and Tintarev. In the limiting\ncase \λ =( fracn-1n)n, we prove a Moser-Trudinger inequality with\nexact growth in mathbb Hn,
sup_
substacku
in C0^
infty(
mathbb Hn)\n
int
mathbb Hn |
nablag u|gn d
text Volg -(
fracn-1n)n\n
int
mathbb Hn |u|n d
text Volg
leq 1
frac1
int
mathbb Hn\n|u|n d
text Volg
int
mathbb Hn
frac
Phin(
alphan\n|u|^
fracnn-1)(1+ |u|)^
frac nn-1 d
text Volg lt;
infty. This\nimproves the Moser-Trudinger inequality with exact growth in mathbb Hn\nestablished by Lu and Tang.\n