TY - JOUR
T1 - Moser-Trudinger Inequality on Compact Riemannian Manifolds of Dimension Two
AU - Yuxiang Li 
JO - Journal of Partial Differential Equations
VL - 2
SP - 163
EP - 192
PY - 2001
DA - 2001/05
SN - 14
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5478.html
KW - Moser-Trüdinger inequality
KW - extremal function
AB -  ln this paper, we prove Moser-Trüdinger inequality in any two dimensional manifolds. Let (M,g_M,) be a two dimensional manifold without boundary and (g, g_N) with boundary, we shall prove the following three inequalities: u∈H¹(M), \sup\limits_{and ||u||_{H¹(M)}}=1∫_M^{e^{4\pi u²}<+∞} u∈H¹(M), \sup\limits_{∫_M u=0, and} ∫_M|∇u|²=1∫_M^{e^{4\pi u²}<+∞} u∈H¹_0(N), \sup\limits_{and ∫_M|∇u²|=1∫_M^{e^{4\pi u²}<+∞} Moreover, we shall show that there exist of extremal functions which at tain the above three inequalities.