By Felli V., Ahmedou M.O.
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Additional resources for Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries
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Suppose ψ ∈ C 2 (Ω) ∩ C 1 (Ω) ∆ψ + V ψ ≤ 0, ∂ψ ≥ hψ, ∂ν in Ω, on Σ, ¯ satisfies and v ∈ C 2 (Ω) ∩ C 1 (Ω) ∆v + V v ≤ 0, ∂v ≥ hv, ∂ν v ≥ 0, in Ω, on Σ, on Γ. ¯. Then v ≥ 0 in Ω We now derive a Pohozaev-type identity for our problem; its proof is quite standard (see ). 3. 3) on Γ1 (Br+ ) = ∂Br+ ∩ ∂Rn+ , and c(n) is constant depending on n . Then n−1 n−2 − q−1 2 − in Br+ , c(n)r q+1 hv q+1 dσ + Γ1 (Br+ ) c(n) q+1 v q+1 h dσ = ∂Γ1 (Br+ ) Γ2 (Br+ ) n−1 v q+1 Γ1 (Br+ ) i=1 B(x, r, v, ∇v) dσ ∂h xi dσ ∂xi 31 Compactness results in conformal deformations on manifolds with boundaries where Γ2 (Br+ ) = ∂Br+ ∩ Rn+ and n − 2 ∂v 1 B(x, r, v, ∇v) = v+ r 2 ∂ν 2 ∂v ∂ν 2 1 − r|∇tan v|2 2 where ∇tan v denotes the component of the gradient ∇v which is tangent to Γ2 (Br+ ) .