On the Yamabe problem and the scalar curvature problems under boundary conditions

Prescribing scalar and boundary mean curvature on the three dimensional half sphere

“…The blow up analysis for the three dimensional case strongly relies on the differentiability of K at blow up points. This implies that blow ups of symmetric functions outside ∂S For the case of any n, some results are proved in [4] when K and H are close to some constants; here we are extending some of those results for n = 3 without the close to constant conditions, see also Remark 7.2. In the paper [13] the case K ≡ 0 and H close to a positive constant is considered, for n ≥ 3.…”

“…This fact, in the case of H ≡ 0, could be roughly explained as follows. Reflecting both K and v evenly to the whole S 3 , one could study symmetric solutions of (4). The blow up analysis for the three dimensional case strongly relies on the differentiability of K at blow up points.…”

Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

“…According to equations (P 1 ), the problem is equivalent to finding a smooth positive solution u of the following equation Such a problem was studied in [1] [14], [15] [16], [17], [18], [19] [21] . Yanyan Li [21], and Djadli-Malchiodi-Ould Ahmedou [15] studied this problem when the manifold is the three dimensional standard half sphere.…”

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

“…According to equation (1), the problem is equivalent to finding a smooth positive solution u of the following equation −c n Δ g u + R g u = Ku (n+2)/(n−2) in M, 2 n−2 ∂u ∂ν + h g u = Hu n/(n− 2) on ∂M.…”