By Felli V., Ahmedou M.O.
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Extra resources for Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries
Example text
6) Lq (∂M ) ≤ c. 32 V. Felli and M. Ould Ahmedou From well-known interpolation inequalities, we deduce u Lq+1 (∂M ) where ϑ= ϑ Lq (∂M ) ≤ u u 1−ϑ L2(n−1)/(n−2) (∂M ) q n − nq − 2q · . q + 1 2(n − 1) − nq + 2q It is easy to check that 0 < ϑ < 1 , ϑ−1 ≤ c , and (1 − ϑ)−1 ≤ c . 4) by u , we easily find that |∇g u|2 + µ= M n−2 Rg u 2 . 4(n − 1) Therefore, from the Sobolev embedding Theorems, we deduce 1= u Lq+1 (∂M ) ≤c u ϑ Lq (∂M ) µ 1−ϑ 2 =c µ u 2ϑ 1−ϑ Lq (∂M ) 1−ϑ 2 . 7), we have that µ1− For 1 ≤ q ≤ n n−2 (1−ϑ)(q−1) 2ϑ ≤ c.
F. Escobar: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. , (2) 136(1992), 1-50. [6] J. F. Escobar: The Yamabe problem on manifolds with boundary, J. Diff. , 35(1992), 21-84. [7] J. F. Escobar: Conformal metrics with prescribed mean curvature on the boundary, Calc. Var. and PDE’s, 4(1996), 559-592. [8] D. Gilbarg and N. S. Trudinger : “Elliptic partial differential equations of second order”, second edition, Berlin, Springer-Verlag, 1983.
Let v(x) = a|x|2−n + b + O(|x|) for x close to 0 , with a > 0 and b > 0 . There holds lim r→0+ Γ2 (Br+ ) B(x, v, ∇v) < 0. 5. Let (M, g) be a smooth compact Riemannian manifold of positive type n (namely λ1 (B) > 0 ). Let ε0 > 0 , 1 ≤ q ≤ n−2 − ε0 . Suppose that u satisfies − Lg u = 0, u > 0, ∂u = µuq , ∂ν uq+1 = 1. 4) ∂M Then |∇g u|2 + 0<µ= M n−2 Rg u2 ≤ C(M, g, ε0 ). 4(n − 1) 1 n Proof. 1 that C −1 ≤ µ q−1 u ≤ C , which, together with ∂M uq+1 = 1 , gives the claimed estimate.
Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries by Felli V., Ahmedou M.O.
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