(untitled) (Journal of Partial Differential Equations, 2001)
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∈H^1(M), and ||u||_H^1(M)=_1_M e^4π u^2<+∞ u∈H^1(M), ∫_M u=0, and ∫_M|∇u|^2=_1∫_M e^4π u^2 <+∞ u∈H^1_0(N), and ∫_M|∇u^2|=_1∫_M e^4π u^2<+∞ Moreover, we shall show that there exist of extremal functions which at tain the above three inequalities.