File size: 2,408 Bytes
59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | import OAI.Geometry.IsometricImmersion.Energy.DirectedIntegralEstimate
namespace OAI
noncomputable section
open scoped ContDiff
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
theorem directedQuadratic_edge_zero
(A B C I u : Coord → ℝ) (edge lambda epsilon : ℝ) (p : Coord)
(hA : DifferentiableAt ℝ A p) (hI : DifferentiableAt ℝ I p) (hedge : p 1 = edge) :
multiplierQuadratic A B C (directedM edge lambda I) (directedN edge lambda epsilon I) u p = 0 := by
have hW := directedWeight_edge_zero edge lambda I p hedge
have hWd (i : Fin 2) := directedWeight_partial_edge_zero edge lambda hI hedge i
have hm := directedM_differentiableAt edge lambda hI
have hn := directedN_differentiableAt edge lambda epsilon hI
obtain ⟨hm0, hn0⟩ := directed_mn_edge_zero edge lambda epsilon I p hedge
have hmd (i : Fin 2) : coordPartial i (directedM edge lambda I) p = 0 := by
rw [directedM_partial, hWd i, neg_zero]
have hnd (i : Fin 2) : coordPartial i (directedN edge lambda epsilon I) p = 0 := by
rw [directedN_partial edge lambda epsilon hI i, hW, hWd i]
ring
have hAm (i : Fin 2) : coordPartial i (fun q => A q * directedM edge lambda I q) p = 0 := by
rw [HessianCalculus.coordPartial_mul_at hA hm i, hm0, hmd i]
ring
have hAn (i : Fin 2) : coordPartial i (fun q => A q * directedN edge lambda epsilon I q) p = 0 := by
rw [HessianCalculus.coordPartial_mul_at hA hn i, hn0, hnd i]
ring
simp only [multiplierQuadratic, multiplierT, multiplierS, multiplierJ,
hmd, hnd, hAm, hAn, hm0, hn0]
ring
theorem directedCutoffError_edge_zero
(A chi I u : Coord → ℝ) (edge lambda epsilon : ℝ) (p : Coord) (hedge : p 1 = edge) :
directedCutoffError A chi I u edge lambda epsilon p = 0 := by
obtain ⟨ht, hs⟩ := directed_flux_edge_zero A u I edge lambda epsilon p hedge
simp [directedCutoffError, multiplierCutoffError, ht, hs]
theorem directedCoercivityDefect_edge_zero
(A B C chi I u : Coord → ℝ) (edge lambda epsilon c : ℝ) (p : Coord)
(hA : DifferentiableAt ℝ A p) (hI : DifferentiableAt ℝ I p) (hedge : p 1 = edge) :
directedCoercivityDefect A B C chi I u edge lambda epsilon c p = 0 := by
rw [directedCoercivityDefect, directedWeight_edge_zero edge lambda I p hedge,
directedQuadratic_edge_zero A B C I u edge lambda epsilon p hA hI hedge]
simp
end SmoothLocal.Weighted
end
end OAI
|