| 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 |
|
|
