DeGiorgi.lean`

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14 declarations, 9 contain sorry, 0 declared as axioms

De Giorgi's conjecture

This file states a conjecture of De Giorgi about entire solutions to $Δ u + u - u^3 = 0$. The conjecture is a rigidity theorem: in spatial dimension $n ≤ 8$, the level sets of bounded solutions which satisfy $∂₁u > 0$ everywhere are hyperplanes. It has been shown that the condition $n ≤ 8$ is sharp.

The main theorems are:

DeGiorgi_le_eight: the conjecture holds in dimension $n ≤ 8$.
DeGiorgi_ge_nine: the conclusion of the conjecture does not hold if $n ≥ 9$.

The cases $1 ≤ n ≤ 8$ are also listed individually to enable partial solutions. The cases $1 ≤ n ≤ 3$ are solved, while $4 ≤ n ≤ 8$ remains open.

Existing results

The case $n = 1$ trivially holds ($u$ is injective since $∂_1 u > 0$).
The case $n = 2$ was proven by Ghoussoub and Gui.
The case $n = 3$ was proven by Ambrosio and Cabré.
The case $4 ≤ n ≤ 8$ was proven under an extra assumption by Savin.
The counterexample for $n ≥ 9$ was proven by Del Pino, Kowalczyk, and Wei.

References

Ghoussoub, Gui, Mathematische Annalen 311 (1998) proves the conjecture for $n = 2$.
Ambrosio, Cabré, Journal of the American Mathematical Society 13 (2000) proves the conjecture for $n = 3$.
Savin, Annals of Mathematics 169 (2009) proves the case $4 ≤ n ≤ 8$ under an additional assumption.
Del Pino, Kowalczyk, Wei, Annals of Mathematics 174 (2011) shows that the condition $n ≤ 8$ is sharp.

Imports (1)

FormalConjecturesUtil

structure DeGiorgi.IsBoundedSolution The function $u : ℝ^n → ℝ$ is a bounded classical solution to $Δ u + u - u^3 = 0$ line 64 source
Source documentation
The function $u : ℝ^n → ℝ$ is a bounded classical solution to $Δ u + u - u^3 = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, EuclideanSpace, Real, Fin
```
structure IsBoundedSolution (u : ℝ^n → ℝ) : Prop where
…
```
def DeGiorgi.HasPositiveDeriv The first partial derivative of $u : ℝ^n → ℝ$ is strictly positive line 72 source
Source documentation
The first partial derivative of $u : ℝ^n → ℝ$ is strictly positive.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, NeZero, EuclideanSpace, Real, Fin
Proof uses: LT.lt, OfNat.ofNat, lineDeriv, ENNReal, EuclideanSpace.single
37 instances, coercions and auxiliary constants left out.
```
def HasPositiveDeriv (u : ℝ^n → ℝ) : Prop :=
…
```
def DeGiorgi.HasHyperplaneLevelSets The level sets of $u : ℝ^n → ℝ$ are hyperplanes line 79 source
Source documentation
The level sets of $u : ℝ^n → ℝ$ are hyperplanes. This is expressed by stating that there exists some affine subspace with rank $n - 1$ which coincides with the level set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, EuclideanSpace, Real, Fin
Proof uses: Membership.mem, Set, Set.range, Exists, AffineSubspace, OfNat.ofNat, ENNReal, And, Eq, Set.preimage, Singleton.singleton, Module.finrank, Subtype, Submodule, AffineSubspace.direction, HSub.hSub
38 instances, coercions and auxiliary constants left out.
```
def HasHyperplaneLevelSets (u : ℝ^n → ℝ) : Prop :=
…
```
def DeGiorgi.DeGiorgi_conclusion Documented declaration: DeGiorgi_conclusion line 87 source
Source documentation
The conclusion to De Giorgi's conjecture: if $u : ℝ^n → ℝ$ is a bounded classical solution to $Δ u + u - u^3 = 0$ satisfying $∂₁u > 0$ everywhere, then the level sets of $u$ are hyperplanes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, NeZero
Proof uses: EuclideanSpace, Real, Fin, And, DeGiorgi.IsBoundedSolution, DeGiorgi.HasPositiveDeriv, DeGiorgi.HasHyperplaneLevelSets
4 instances, coercions and auxiliary constants left out.
```
def DeGiorgi_conclusion (n : ℕ) [NeZero n] : Prop :=
…
```
theorem DeGiorgi.DeGiorgi_le_eight De Giorgi's conjecture holds in dimension $n ≤ 8$ sorry line 94 source
Source documentation
De Giorgi's conjecture holds in dimension $n ≤ 8$.

depends on sorry what this means
Statement uses: Nat, NeZero, LE.le, OfNat.ofNat, DeGiorgi.DeGiorgi_conclusion
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_le_eight (hn : n ≤ 8) : (DeGiorgi_conclusion n) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_ge_nine In dimension $n ≥ 9$, the conclusion of De Giorgi's conjecture does not hold sorry line 101 source
Source documentation
In dimension $n ≥ 9$, the conclusion of De Giorgi's conjecture does not hold.

depends on sorry what this means
Statement uses: Nat, NeZero, GE.ge, OfNat.ofNat, Not, DeGiorgi.DeGiorgi_conclusion
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_ge_nine (hn : n ≥ 9) : ¬(DeGiorgi_conclusion n) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_one De Giorgi's conjecture trivially holds in dimension $n = 1$ line 108 source
Source documentation
De Giorgi's conjecture trivially holds in dimension $n = 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: EuclideanSpace, Real, Fin, And, DeGiorgi.IsBoundedSolution, DeGiorgi.HasPositiveDeriv, DeGiorgi.HasHyperplaneLevelSets, Membership.mem, Set, Set.range, Eq, Exists, AffineSubspace, ENNReal, Set.preimage, Singleton.singleton, Module.finrank, Subtype, Submodule, AffineSubspace.direction and 60 more
127 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_one : (DeGiorgi_conclusion 1) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_two De Giorgi's conjecture holds in dimension $n = 2$ sorry line 128 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 2$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_two : (DeGiorgi_conclusion 2) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_three De Giorgi's conjecture holds in dimension $n = 3$ sorry line 135 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 3$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_three : (DeGiorgi_conclusion 3) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_four De Giorgi's conjecture holds in dimension $n = 4$ sorry line 142 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 4$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_four : (DeGiorgi_conclusion 4) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_five De Giorgi's conjecture holds in dimension $n = 5$ sorry line 149 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 5$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_five : (DeGiorgi_conclusion 5) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_six De Giorgi's conjecture holds in dimension $n = 6$ sorry line 156 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 6$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_six : (DeGiorgi_conclusion 6) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_seven De Giorgi's conjecture holds in dimension $n = 7$ sorry line 163 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 7$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_seven : (DeGiorgi_conclusion 7) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion
theorem DeGiorgi.DeGiorgi_eight De Giorgi's conjecture holds in dimension $n = 8$ sorry line 170 source
Source documentation
De Giorgi's conjecture holds in dimension $n = 8$.

depends on sorry what this means
Statement uses: DeGiorgi.DeGiorgi_conclusion, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem DeGiorgi_eight : (DeGiorgi_conclusion 8) := by
…
```
Mentions: DeGiorgi.DeGiorgi_conclusion