| import Mathlib |
| import OAI.Geometry.PrescribedPotential.RealFrameScalar |
| import OAI.Geometry.PrescribedPotential.WirtingerCalculus |
|
|
| namespace OAI |
|
|
| /-! Bernstein Product Maximum. -/ |
|
|
| section |
|
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| |
| noncomputable section |
| open Set Filter Topology Finset |
| open scoped ContDiff |
| namespace BernsteinEstimate |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
|
|
| lemma product_gradient_cross_identity {η a T : ℝ} (hη : η ≠ 0) (ha : a ≠ 0) |
| (X Y Z : V) (h : (a*T) • X+(η*T) • Y+(η*a) • Z=0) : |
| 2*η*inner ℝ Y Z+2*a*inner ℝ X Z+2*T*inner ℝ X Y = |
| -2*η*T/a*‖Y‖^2-2*a*T/η*‖X‖^2-2*T*inner ℝ X Y := by |
| have hx := congrArg (fun W => inner ℝ X W) h |
| have hy := congrArg (fun W => inner ℝ Y W) h |
| simp only [inner_add_right,inner_smul_right,inner_zero_right,real_inner_self_eq_norm_sq] at hx hy |
| rw [real_inner_comm X Y] at hy |
| have hx' := congrArg (fun r : ℝ => 2*a*r) hx |
| have hy' := congrArg (fun r : ℝ => 2*η*r) hy |
| field_simp |
| nlinarith only [hx',hy'] |
| end BernsteinEstimate |
|
|
| namespace HigherJet |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {ι : Type*} [Fintype ι] |
|
|
| lemma quartic_test_at_max {U : Set E} (hU : IsOpen U) {η S T : E → ℝ} |
| (hηs : ContDiffOn ℝ ∞ η U) (hSs : ContDiffOn ℝ ∞ S U) (hTs : ContDiffOn ℝ ∞ T U) |
| {x : E} (hx : x ∈ U) (v : ι → E) (A : ℝ) |
| (hηx : η x ≠ 0) (hax : A+S x ≠ 0) |
| (hmax : IsLocalMax (fun y => η y*(A+S y)*T y) x) : |
| η x*(A+S x)*frameLaplace v T x+η x*T x*frameLaplace v S x+ |
| (A+S x)*T x*frameLaplace v η x - |
| 2*η x*T x/(A+S x)*‖frameGradient v S x‖^2 - |
| 2*(A+S x)*T x/η x*‖frameGradient v η x‖^2 - |
| 2*T x*inner ℝ (frameGradient v η x) (frameGradient v S x) ≤ 0 := by |
| let a : E → ℝ := fun y => A+S y |
| have has : ContDiffOn ℝ ∞ a U := contDiffOn_const.add hSs |
| have hηa := (hηs.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp) |
| have hSa := (hSs.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp) |
| have hTa := (hTs.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp) |
| have haa := (has.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp) |
| have hg : frameGradient v a x = frameGradient v S x := by |
| rw [show a = fun y => A+S y from rfl,frameGradient_add (differentiableAt_const A) hSa,frameGradient_const,zero_add] |
| have hl : frameLaplace v a x = frameLaplace v S x := by |
| rw [show a = fun y => A+S y from rfl,frameLaplace_add hU contDiffOn_const hSs hx,frameLaplace_const,zero_add] |
| have hgrad := frameGradient_eq_zero_of_localMax hmax v |
| have hm := frameLaplace_nonpos_of_localMax |
| (((hηs.mul has).mul hTs).contDiffAt (hU.mem_nhds hx)) hmax v |
| change frameGradient v (fun y => (η y*a y)*T y) x = 0 at hgrad |
| rw [frameGradient_mul (f := fun y => η y*a y) (hηa.mul haa) hTa, |
| frameGradient_mul hηa haa,hg,smul_add,smul_smul,smul_smul] at hgrad |
| have hcross := BernsteinEstimate.product_gradient_cross_identity (T := T x) hηx hax |
| (frameGradient v η x) (frameGradient v S x) (frameGradient v T x) |
| (by simpa only [mul_comm,mul_left_comm,mul_assoc,a] using hgrad) |
| change frameLaplace v (fun y => (η y*a y)*T y) x ≤ 0 at hm |
| rw [frameLaplace_mul hU (f := fun y => η y*a y) (hηs.mul has) hTs hx, |
| frameLaplace_mul hU hηs has hx,frameGradient_mul hηa haa,hg,hl, |
| inner_add_left,inner_smul_left,inner_smul_left] at hm |
| dsimp only [a] at hm |
| simp only [starRingEnd_apply,star_trivial] at hm |
| simp only [neg_mul,neg_div] at hcross |
| nlinarith only [hm,hcross] |
| end HigherJet |
|
|
| end |
| end |
|
|
| end OAI |
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|
