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10), although we could not find a reference. 4. Let u ∈ C0+ (RN ), and let G be a function satisfying + N −1 G ∈ L∞ RN −1 × R+ , 0 × R0 × R G = G(x , v, z1 , z ), G is continuous in (v, z1 , z ), nondecreasing in z1 and convex in (z1 , z ), G(x , 0, 0, 0) = 0 ∀x ∈ RN −1 . Then RN G x , u, ∂u , ∇ u dx ∂x1 RN G x , u∗ , ∂u∗ , ∇ u∗ dx. 10) (Here ∇ u = (ux2 , . . ) Furthermore, if u ∈ W 1,p (RN ) ∩ S+ for some p ∈ [1, +∞], then u∗ ∈ W 1,p (RN ) ∩ S+ , ∇u ∂u ∂xi p ∇u∗ ∂xi p and p, ∂u∗ , i = 1, . .

The proof for the Schwarz symmetrization is analogous and will be omitted. 1. 5 have many applications. g. [109,21,113]). Here is an example (see [109]): Let p ∈ (1, N ). Then there exists a constant C > 0 such that ∇v p C v Np/(N −p) ∀v ∈ W 1,p RN . 13) becomes an equality. 18). 19) lim u(x) = 0, |x|→∞ for some d > 0. 18) holds true. 21) 32 F. g. [44,66,113,18,37]). 9) for p = 1 implies the Isoperimetric inequality in RN : Among all smooth sets Ω ⊂ RN with given (fixed) LN -measure, the ball yields the smallest perimeter (= (N − 1)-surface measure).

Hence we have that u = uλ for λ λ0 and σλ u = uλ for λ λ0 , which means that u is nonincreasing in x1 for x1 λ0 . Since u decays at infinity this in particular implies that u 0, and the first part of (1) follows. (2) Proceeding as in part (1) of the proof for any of the xi -directions, we find that u ∈ S+ , and that there is point ξ = (ξ1 , . . , ξN ) ∈ RN such that the function v(·) := u(· − ξ ) is even in the variables xi and nonincreasing in xi for xi 0 (i = 1, . . , N ). This implies that if H ∈ H0 then v = vH , and if H ∈ H \ H0 then σH v = vH .

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4-dimensional anti-Kahler manifolds and Weyl curvature by Kim J.

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