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- Title
Improved Trudinger–Moser inequality involving Lp-norm on a closed Riemann surface with isometric group actions.
- Authors
Fang, Yu; Zhang, Mengjie
- Abstract
In this paper, using the method of blow-up analysis, the authors obtain an improved Trudinger–Moser inequality involving L p -norm (p > 1) and prove the existence of its extremal function on a closed Riemann surface (Σ , g) with the action of a finite isometric group G = { σ 1 , σ 2 , ... , σ N }. To be exact, let W 1 , 2 (Σ , g) be the usual Sobolev space, a function space ℋ G = { u ∈ W 1 , 2 (Σ , g) : ∫ Σ u d v g = 0 and u (σ i (x)) = u (x) , ∀ x ∈ Σ , σ i ∈ G } and ℓ = min x ∈ Σ I (x) , where I (x) stands for the number of all distinct points in the set G (x) = { σ 1 (x) , ... , σ N (x) }. Define λ p G = inf u ∈ ℋ G , u ≢ 0 ∥ ∇ g u ∥ 2 2 / ∥ u ∥ p 2 , where ∥ ⋅ ∥ p is the standard L p -norm on (Σ , g). Using blow-up analysis, we prove that if 0 ≤ α < λ p G , the supremum sup u ∈ ℋ G , ∥ ∇ g u ∥ 2 2 − α ∥ u ∥ p 2 ≤ 1 ∫ Σ e 4 π ℓ u 2 d v g < + ∞ , and this supremum can be attained; if α ≥ λ p G , the above supremum is infinite. This kind of inequality will play an important role in the study of prescribing Gaussian curvature problem and mean field equations. In particular, their result generalizes those of Chen [A Trudinger inequality on surfaces with conical singularities, Proc. Amer. Math. Soc. 108 (1990) 821–832], Yang [Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two, J. Differential Equations 258 (2015) 3161–3193] and Fang–Yang [Trudinger–Moser inequalities on a closed Riemannian surface with the action of a finite isometric group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1295–1324].
- Subjects
SOBOLEV spaces; GAUSSIAN curvature; FINITE groups; FUNCTION spaces; RIEMANN surfaces; ISOMETRICS (Mathematics)
- Publication
International Journal of Mathematics, 2022, Vol 33, Issue 5, p1
- ISSN
0129-167X
- Publication type
Article
- DOI
10.1142/S0129167X22500380