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9. 4. 1) f ≤ 0. 1. Area type estimates. We consider the graph of u as a surface in R3 , namely, we set Y = Y (x) := (x, u(x)) ∈ R2 × R, that is, we think Y as a function of x ∈ R2 in such a way Y ∈ R3 belongs to the graph of u. Area estimates are a classical topic in minimal surface theory. 1. There exists a constant C ≥ 1 in such a way that |Y |≤ρ as long as ρ ≥ C. a(|∇u|)|∇u|2 dx ≤ Cρ2 , Proof. This proof is inspired by analogous arguments on page 24 of [Sim07] and page 403 of [GT01]. 3). 2) for any t > 0.

1) div ω(x)B(x)∇ζ(x) = 0 . 2) RN ω(x)ζ 2 (x) B(x)τ (x) · τ (x) dx ≤ C τ L∞ (RN ) R 2 , for any τ ∈ C0∞ (RN , RN ) supported in B2R . Then, ζ is constant. Proof. The proof is a Caccioppoli type argument modified from [BCN97]. We take α ∈ C0∞ (B2 ) so that 0 ≤ α(x) ≤ 1 for any x ∈ RN and α(x) = 1 for any x ∈ B1 . We also set αR (x) := α(x/R), τR (x) := ∇α(x/R) and φR (x) := (αR (x))2 ζ(x). 1), RN α2R ω(B∇ζ) · ∇ζ dx RN ω(B∇ζ) · ∇φR dx − 2 = ≤ 0+2 R≤|x|≤2R RN αR ζω(B∇ζ) · ∇αR dx αR |ζ| ω |(B∇ζ) · ∇αR | dx .

Differential Equations, 206(2):483–515, 2004. [Far02] Alberto Farina. Propri´et´es qualitatives de solutions d’´equations et syst`emes d’´equations non-lin´eaires. 2002. Habilitation ` a diriger des recherches, Paris VI. BERNSTEIN AND DE GIORGI TYPE PROBLEMS [Far03] [Far07] [FCS80] [GG98] [GT01] [LL97] [Mod85] [MP78] [Sav03] [Ser94] [Sim07] [SZ98a] [SZ98b] [Tol84] [Uhl77] [VSS06] 47 Alberto Farina. One-dimensional symmetry for solutions of quasilinear equations in R2 . Boll. Unione Mat. Ital. Sez.