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4, A∇uN · ∇ψ − f ′ (u)uN ψ dx 0 = φ2 uN 2φ A∇uN · ∇φ − dx A∇uN · ∇uN − f ′ (u)φ2 uN + ǫ (uN + ǫ)2 uN + ǫ uN dx . 1). 2. Degenerate examples. Our scope is now to show by examples that interesting degenerate cases are covered by our setting. This part is not used in the proofs of the main results, and it may thus be skipped by the uninterested reader. 2. Let p > 2. Then, there exist w ∈ C 2 (RN ) and f ∈ C 1 (R) in such a way that • w is a stable solution of ∆p w + f (w) = 0 having one-dimensional symmetry, • 0 ≤ w(x) ≤ 1 and ∂xN w(x) ≥ 0 for any x ∈ RN , • w(x) = 0 if xN ≤ 0 and w(x) = 1 if xN ≥ 1.
7). 10) |∇v(x)|4 λ1 (|∇v(x)|) ≤ const Ξ + |∇v(x)|2 a(|∇v(x)|) for any x ∈ RN . 11) |∇v(x)|2 |A(∇v(x))| ≤ const Ξ + |∇v(x)|2 a(|∇v(x)|) for any x ∈ RN . 12) BR \B√R Ξ + |∇v(x)|2 a(|∇v|) dx ≤ C ln R , |Y |2 as long as R is large enough. Then, given R > 0 (to be taken appropriately large in what follows) and x ∈ RN , we now define √ 1 if |Y | ≤ R, √ |) ϕR (x) := 2 ln(R/|Y if R < |Y | < R, ln R 0 if |Y | ≥ R. By construction, ϕR is a Lipschitz function and const x + v(x)∇v(x) ∇ϕR (x) = − |Y |2 ln R √ for any x ∈ RN such that R < |Y | < R.
Then, by (A1), R2 λ2 (|∇u(x)|) |∇u(x)|2 κ2 (x)ϕ2 (x) dx ≤ K R2 |∇ϕ(x)|2 dx . 6, since ∇u never vanishes, we conclude that the level sets are regular curves with vanishing curvatures, thence straight lines. ♦ References [AAC01] Giovanni Alberti, Luigi Ambrosio, and Xavier Cabr´e. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. , 65(1-3):9–33, 2001. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday.
Self organize branching by Milovanov