By John Bonnycastle
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Additional info for An Introduction To Mensuration And Practical Geometry; With Notes, Containing The Reason Of Every Rule
Connectivity of phase boundaries in strictly convex domains. Arch. Rational Mech. , 141(4):375–400, 1998. Peter Sternberg and Kevin Zumbrun. A Poincar´e inequality with applications to volume-constrained area-minimizing surfaces. J. Reine Angew. , 503:63– 85, 1998. Peter Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51(1):126–150, 1984. K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. , 138(3-4):219–240, 1977. Enrico Valdinoci, Berardino Sciunzi, and Vasile Ovidiu Savin.
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. Then, we take ζ ∈ C0∞ (B2ρ , [0, 1]) such that ζ(x) = 1 if x ∈ Bρ , 0 ≤ ζ ≤ 1 and |∇ζ| ≤ const /ρ. For any t ∈ R, we also define if t ≥ ρ, 1 t+ρ if |t| < ρ, γ(t) := 2ρ 0 if t ≤ −ρ, and ϕ(x) := γ(u(x))ζ(x). 2), R2 a(|∇u(x)|) γ(u(x)) ∇u(x) · ∇ζ(x) dx ≤ const ρ . 1), const a(|∇u(x)|)|∇u|2 dx |x|≤ρ ρ |u|≤ρ ≤ R2 a(|∇u(x)|) γ ′ (u(x)) ζ(x) |∇u(x)|2 dx R2 f (u(x)) ϕ(x) − a(|∇u(x)|) γ(u(x)) ∇u(x) · ∇ζ(x) dx = ≤ const ρ , which yields the desired result.
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 . 3) R≤|x|≤2R α2R ω(B∇ζ) · ∇ζ dx + δ−1 R≤|x|≤2R ωζ 2 (B∇αR ) · ∇αR dx . 2), R≤|x|≤2R ωζ 2 (B∇αR ) · ∇αR dx = R−2 RN ωζ 2 (B∇τR ) · ∇τR dx ≤ C ′ , for a suitable C ′ > 0.