| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! |
| # Variational isotropic elasticity |
|
|
| H¹ is the closure of smooth value/gradient jets in L². Boundary data are |
| the quotient by the closure of compactly supported smooth jets, and the |
| Dirichlet-to-Neumann map is the physical variational displacement-to-traction |
| pairing. Arbitrary-data Dirichlet well-posedness and independence of |
| representatives are proved, not assumed. Identification of the quotient |
| norm with boundary-chart fractional Sobolev norms is not established here. |
| -/ |
|
|
| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators |
|
|
| namespace Elasticity |
|
|
| abbrev X := EuclideanSpace ℝ (Fin 3) |
| abbrev V := X |
|
|
| def coordVector (i : Fin 3) : X := EuclideanSpace.single i 1 |
|
|
| def SmoothBoundary (Ω : Set X) : Prop := |
| ∀ x ∈ frontier Ω, ∃ e : OpenPartialHomeomorph X X, |
| x ∈ e.source ∧ |
| ContDiffOn ℝ (⊤ : ℕ∞) e e.source ∧ |
| ContDiffOn ℝ (⊤ : ℕ∞) e.symm e.target ∧ |
| e x 0 = 0 ∧ |
| ∀ y ∈ e.source, y ∈ Ω ↔ 0 < e y 0 |
|
|
| def Domain (Ω : Set X) : Prop := |
| IsOpen Ω ∧ IsConnected Ω ∧ Bornology.IsBounded Ω ∧ SmoothBoundary Ω |
|
|
| def SmoothUpTo (Ω : Set X) (f : X → ℝ) : Prop := |
| ∃ U : Set X, IsOpen U ∧ closure Ω ⊆ U ∧ ContDiffOn ℝ (⊤ : ℕ∞) f U |
|
|
| def Admissible (Ω : Set X) (lam mu : X → ℝ) : Prop := |
| SmoothUpTo Ω lam ∧ SmoothUpTo Ω mu ∧ |
| ∀ x ∈ closure Ω, 0 < mu x ∧ 0 < 3 * lam x + 2 * mu x |
|
|
| abbrev Ambient (Ω : Set X) := |
| Lp V 2 (volume.restrict Ω) × (Fin 3 → Lp V 2 (volume.restrict Ω)) |
|
|
| def coordDeriv (f : X → V) (i : Fin 3) (x : X) : V := |
| fderiv ℝ f x (coordVector i) |
|
|
| def SmoothJets (Ω : Set X) : Set (Ambient Ω) := |
| {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω)) |
| (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)), |
| ContDiff ℝ (⊤ : ℕ∞) f ∧ |
| j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)} |
|
|
| def TestJets (Ω : Set X) : Set (Ambient Ω) := |
| {j | ∃ (f : X → V) (hf : MemLp f 2 (volume.restrict Ω)) |
| (hd : ∀ i, MemLp (coordDeriv f i) 2 (volume.restrict Ω)), |
| ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ tsupport f ⊆ Ω ∧ |
| j.1 = hf.toLp f ∧ j.2 = fun i => (hd i).toLp (coordDeriv f i)} |
|
|
| abbrev H1 (Ω : Set X) := {j : Ambient Ω // j ∈ closure (SmoothJets Ω)} |
|
|
| def HasZeroTrace (Ω : Set X) (u : H1 Ω) : Prop := |
| u.val ∈ closure (TestJets Ω) |
|
|
| def SameTrace (Ω : Set X) (u v : H1 Ω) : Prop := |
| u.val - v.val ∈ closure (TestJets Ω) |
|
|
| / |
| abbrev BoundaryData (Ω : Set X) := Quot (SameTrace Ω) |
|
|
| def trace (Ω : Set X) (u : H1 Ω) : BoundaryData Ω := Quot.mk _ u |
|
|
| def div (Ω : Set X) (u : H1 Ω) (x : X) : ℝ := |
| ∑ i, (u.val.2 i x) i |
|
|
| def strain (Ω : Set X) (u : H1 Ω) (x : X) (i j : Fin 3) : ℝ := |
| ((u.val.2 j x) i + (u.val.2 i x) j) / 2 |
|
|
| def energy (Ω : Set X) (lam mu : X → ℝ) (u v : H1 Ω) : ℝ := |
| ∫ x, (lam x * div Ω u x * div Ω v x + |
| 2 * mu x * ∑ i, ∑ j, strain Ω u x i j * strain Ω v x i j) |
| ∂(volume.restrict Ω) |
|
|
| def WeakSolution (Ω : Set X) (lam mu : X → ℝ) (u : H1 Ω) : Prop := |
| ∀ v : H1 Ω, HasZeroTrace Ω v → energy Ω lam mu u v = 0 |
|
|
| / |
| irrelevant once the Dirichlet problem has been proved solvable. -/ |
| def dirichletSolution (Ω : Set X) (lam mu : X → ℝ) |
| (f : BoundaryData Ω) : H1 Ω := by |
| classical |
| exact if h : ∃ u : H1 Ω, trace Ω u = f ∧ WeakSolution Ω lam mu u |
| then Classical.choose h |
| else f.out |
|
|
| / |
| def DN (Ω : Set X) (lam mu : X → ℝ) : BoundaryData Ω → BoundaryData Ω → ℝ := |
| fun f g => energy Ω lam mu (dirichletSolution Ω lam mu f) g.out |
|
|
| def MainClaim : Prop := |
| ∀ (Ω : Set X), Domain Ω → |
| ∀ (lam₁ mu₁ lam₂ mu₂ : X → ℝ), |
| Admissible Ω lam₁ mu₁ → Admissible Ω lam₂ mu₂ → |
| DN Ω lam₁ mu₁ = DN Ω lam₂ mu₂ → |
| (∀ x ∈ Ω, lam₁ x = lam₂ x) ∧ (∀ x ∈ Ω, mu₁ x = mu₂ x) |
|
|
| end Elasticity |
|
|
| end |
|
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| end OAI |
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|
