regularized distances and harmonic measure in co-dimension
TRANSCRIPT
![Page 1: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/1.jpg)
Regularized Distances and HarmonicMeasure in Co-Dimension Greater than One
Max Engelstein(joint work with G. David (Allez les Bleus!) and S. Mayboroda)
Massachusetts Institute of Technology
July 16th, 2018
This research was partially supported by an NSF Postdoctoral Research Fellowship, DMS 1703306 and NSF DMS 1500771.
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Definition of Harmonic Measure
X ∈ Ω ⊂ Rn. E ⊂ ∂Ω. ωX (E ) = Probability a B.M. exits Ω first in E .
Figure: Brownian Motion exiting adomain (figure credit Matthew Badger)
(D) =
∆uf =0 x ∈ Ω
uf (Q) =f (Q) Q ∈ ∂Ω
uf (X ) =
ˆ∂Ω
fdωX .
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Definition of Harmonic Measure
X ∈ Ω ⊂ Rn. E ⊂ ∂Ω. ωX (E ) = Probability a B.M. exits Ω first in E .
Figure: Brownian Motion exiting adomain (figure credit Matthew Badger)
(D) =
∆uf =0 x ∈ Ω
uf (Q) =f (Q) Q ∈ ∂Ω
uf (X ) =
ˆ∂Ω
fdωX .
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Definition of Harmonic Measure
X ∈ Ω ⊂ Rn. E ⊂ ∂Ω. ωX (E ) = Probability a B.M. exits Ω first in E .
Figure: Brownian Motion exiting adomain (figure credit Matthew Badger)
(D) =
∆uf =0 x ∈ Ω
uf (Q) =f (Q) Q ∈ ∂Ω
uf (X ) =
ˆ∂Ω
fdωX .
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Three Examples: ω vs Hn−1|∂Ω
A disk, Lipschitz domain and Snowflake (figure from Matthew Badger)
For the disk, ω0 = σ2πr .
For a Lipschitz domain, ω0 << σ << ω0 (in a scale invariant way!)
For the snowflake ω0 ⊥ σ.
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Three Examples: ω vs Hn−1|∂Ω
A disk, Lipschitz domain and Snowflake (figure from Matthew Badger)
For the disk, ω0 = σ2πr .
For a Lipschitz domain, ω0 << σ << ω0 (in a scale invariant way!)
For the snowflake ω0 ⊥ σ.
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Three Examples: ω vs Hn−1|∂Ω
A disk, Lipschitz domain and Snowflake (figure from Matthew Badger)
For the disk, ω0 = σ2πr .
For a Lipschitz domain, ω0 << σ << ω0 (in a scale invariant way!)
For the snowflake ω0 ⊥ σ.
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Three Examples: ω vs Hn−1|∂Ω
A disk, Lipschitz domain and Snowflake (figure from Matthew Badger)
For the disk, ω0 = σ2πr .
For a Lipschitz domain, ω0 << σ << ω0 (in a scale invariant way!)
For the snowflake ω0 ⊥ σ.
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Relationship between Geometry and H.M.
Two big questions (ω is H.M., σ = Hn−1|∂Ω):
Q1 (direct): Does smoothness of Ω imply smoothness of dωdσ ?
Q2 (free boundary): Does smoothness of dωdσ imply smoothness of Ω?
(some a priori topology)
Lots of work: Badger, Bortz, David, E., Feneuil, Hofmann, Kenig, Martell,Mayboroda, Tolsa, Toro, Uriarte-Tuero, Zhao...and that’s just the peoplehere at PCMI! Many more also!
Connections to the Dirichlet problem, probability, potential theory...
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Relationship between Geometry and H.M.
Two big questions (ω is H.M., σ = Hn−1|∂Ω):
Q1 (direct): Does smoothness of Ω imply smoothness of dωdσ ?
Q2 (free boundary): Does smoothness of dωdσ imply smoothness of Ω?
(some a priori topology)
Lots of work: Badger, Bortz, David, E., Feneuil, Hofmann, Kenig, Martell,Mayboroda, Tolsa, Toro, Uriarte-Tuero, Zhao...and that’s just the peoplehere at PCMI! Many more also!
Connections to the Dirichlet problem, probability, potential theory...
4
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Relationship between Geometry and H.M.
Two big questions (ω is H.M., σ = Hn−1|∂Ω):
Q1 (direct): Does smoothness of Ω imply smoothness of dωdσ ?
Q2 (free boundary): Does smoothness of dωdσ imply smoothness of Ω?
(some a priori topology)
Lots of work: Badger, Bortz, David, E., Feneuil, Hofmann, Kenig, Martell,Mayboroda, Tolsa, Toro, Uriarte-Tuero, Zhao...and that’s just the peoplehere at PCMI! Many more also!
Connections to the Dirichlet problem, probability, potential theory...
4
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Relationship between Geometry and H.M.
Two big questions (ω is H.M., σ = Hn−1|∂Ω):
Q1 (direct): Does smoothness of Ω imply smoothness of dωdσ ?
Q2 (free boundary): Does smoothness of dωdσ imply smoothness of Ω?
(some a priori topology)
Lots of work: Badger, Bortz, David, E., Feneuil, Hofmann, Kenig, Martell,Mayboroda, Tolsa, Toro, Uriarte-Tuero, Zhao...and that’s just the peoplehere at PCMI! Many more also!
Connections to the Dirichlet problem, probability, potential theory...
4
![Page 13: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/13.jpg)
Relationship between Geometry and H.M.
Two big questions (ω is H.M., σ = Hn−1|∂Ω):
Q1 (direct): Does smoothness of Ω imply smoothness of dωdσ ?
Q2 (free boundary): Does smoothness of dωdσ imply smoothness of Ω?
(some a priori topology)
Lots of work: Badger, Bortz, David, E., Feneuil, Hofmann, Kenig, Martell,Mayboroda, Tolsa, Toro, Uriarte-Tuero, Zhao...and that’s just the peoplehere at PCMI! Many more also!
Connections to the Dirichlet problem, probability, potential theory...
4
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Two Important Theorems
Rn\E ≡ Ω, ω-harmonic measure of Ω, σ = Hn−1|E .E is (n − 1)-Ahlfors regular: σ(B(Q, r)) ' rn−1 for all Q ∈ E , r > 0.
Theorem (Hofmann-Martell & Azzam-Mourgoglou-Tolsa)
E is (n − 1)-uniformly rectifiable (+ weak topological assumption) if andonly if ω is (quantitatively) absolutely continuous (weak-A∞) w.r.t σ.
Theorem (Kenig-Toro)
Some topological assumptions on Ω. Then osc(n) and oscdωdσ control eachother.In particular, ω = σ iff Ω is a half-space.
A few seconds on quantitative regularity (Ahlfors regular, A∞, uniformlyrectifiable etc...)
Takeaway: Geometry is characterized by solutions of Laplacian incomplement!
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Two Important Theorems
Rn\E ≡ Ω, ω-harmonic measure of Ω, σ = Hn−1|E .E is (n − 1)-Ahlfors regular: σ(B(Q, r)) ' rn−1 for all Q ∈ E , r > 0.
Theorem (Hofmann-Martell & Azzam-Mourgoglou-Tolsa)
E is (n − 1)-uniformly rectifiable (+ weak topological assumption) if andonly if ω is (quantitatively) absolutely continuous (weak-A∞) w.r.t σ.
Theorem (Kenig-Toro)
Some topological assumptions on Ω. Then osc(n) and oscdωdσ control eachother.In particular, ω = σ iff Ω is a half-space.
A few seconds on quantitative regularity (Ahlfors regular, A∞, uniformlyrectifiable etc...)
Takeaway: Geometry is characterized by solutions of Laplacian incomplement!
5
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Two Important Theorems
Rn\E ≡ Ω, ω-harmonic measure of Ω, σ = Hn−1|E .E is (n − 1)-Ahlfors regular: σ(B(Q, r)) ' rn−1 for all Q ∈ E , r > 0.
Theorem (Hofmann-Martell & Azzam-Mourgoglou-Tolsa)
E is (n − 1)-uniformly rectifiable (+ weak topological assumption) if andonly if ω is (quantitatively) absolutely continuous (weak-A∞) w.r.t σ.
Theorem (Kenig-Toro)
Some topological assumptions on Ω. Then osc(n) and oscdωdσ control eachother.In particular, ω = σ iff Ω is a half-space.
A few seconds on quantitative regularity (Ahlfors regular, A∞, uniformlyrectifiable etc...)
Takeaway: Geometry is characterized by solutions of Laplacian incomplement!
5
![Page 17: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/17.jpg)
Two Important Theorems
Rn\E ≡ Ω, ω-harmonic measure of Ω, σ = Hn−1|E .E is (n − 1)-Ahlfors regular: σ(B(Q, r)) ' rn−1 for all Q ∈ E , r > 0.
Theorem (Hofmann-Martell & Azzam-Mourgoglou-Tolsa)
E is (n − 1)-uniformly rectifiable (+ weak topological assumption) if andonly if ω is (quantitatively) absolutely continuous (weak-A∞) w.r.t σ.
Theorem (Kenig-Toro)
Some topological assumptions on Ω. Then osc(n) and oscdωdσ control eachother.In particular, ω = σ iff Ω is a half-space.
A few seconds on quantitative regularity (Ahlfors regular, A∞, uniformlyrectifiable etc...)
Takeaway: Geometry is characterized by solutions of Laplacian incomplement!
5
![Page 18: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/18.jpg)
Two Important Theorems
Rn\E ≡ Ω, ω-harmonic measure of Ω, σ = Hn−1|E .E is (n − 1)-Ahlfors regular: σ(B(Q, r)) ' rn−1 for all Q ∈ E , r > 0.
Theorem (Hofmann-Martell & Azzam-Mourgoglou-Tolsa)
E is (n − 1)-uniformly rectifiable (+ weak topological assumption) if andonly if ω is (quantitatively) absolutely continuous (weak-A∞) w.r.t σ.
Theorem (Kenig-Toro)
Some topological assumptions on Ω. Then osc(n) and oscdωdσ control eachother.In particular, ω = σ iff Ω is a half-space.
A few seconds on quantitative regularity (Ahlfors regular, A∞, uniformlyrectifiable etc...)
Takeaway: Geometry is characterized by solutions of Laplacian incomplement!
5
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Higher Co-dimension
Would like to characterize geometry of higher-co-dimension sets!
Problem: Elliptic PDE don’t see sets of co-dim > 2! (removable!)
Need degenerate elliptic PDE. Could try p-Laplacian (but it’s non-linear!Yuck!).
E ⊂ Rn a d-Ahlfors regular set. David-Feneuil-Mayboroda: solutions to
Lu = −div(
A(x)
dist(x ,E )n−d−1∇u)
= 0,
satisfy all the usual elliptic estimates (A an elliptic matrix).
Question: Geometry of E characterized by ωL vs σ?
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Higher Co-dimension
Would like to characterize geometry of higher-co-dimension sets!
Problem: Elliptic PDE don’t see sets of co-dim > 2! (removable!)
Need degenerate elliptic PDE. Could try p-Laplacian (but it’s non-linear!Yuck!).
E ⊂ Rn a d-Ahlfors regular set. David-Feneuil-Mayboroda: solutions to
Lu = −div(
A(x)
dist(x ,E )n−d−1∇u)
= 0,
satisfy all the usual elliptic estimates (A an elliptic matrix).
Question: Geometry of E characterized by ωL vs σ?
6
![Page 21: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/21.jpg)
Higher Co-dimension
Would like to characterize geometry of higher-co-dimension sets!
Problem: Elliptic PDE don’t see sets of co-dim > 2! (removable!)
Need degenerate elliptic PDE. Could try p-Laplacian (but it’s non-linear!Yuck!).
E ⊂ Rn a d-Ahlfors regular set. David-Feneuil-Mayboroda: solutions to
Lu = −div(
A(x)
dist(x ,E )n−d−1∇u)
= 0,
satisfy all the usual elliptic estimates (A an elliptic matrix).
Question: Geometry of E characterized by ωL vs σ?
6
![Page 22: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/22.jpg)
Higher Co-dimension
Would like to characterize geometry of higher-co-dimension sets!
Problem: Elliptic PDE don’t see sets of co-dim > 2! (removable!)
Need degenerate elliptic PDE. Could try p-Laplacian (but it’s non-linear!Yuck!).
E ⊂ Rn a d-Ahlfors regular set. David-Feneuil-Mayboroda: solutions to
Lu = −div(
A(x)
dist(x ,E )n−d−1∇u)
= 0,
satisfy all the usual elliptic estimates (A an elliptic matrix).
Question: Geometry of E characterized by ωL vs σ?
6
![Page 23: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/23.jpg)
Higher Co-dimension
Would like to characterize geometry of higher-co-dimension sets!
Problem: Elliptic PDE don’t see sets of co-dim > 2! (removable!)
Need degenerate elliptic PDE. Could try p-Laplacian (but it’s non-linear!Yuck!).
E ⊂ Rn a d-Ahlfors regular set. David-Feneuil-Mayboroda: solutions to
Lu = −div(
A(x)
dist(x ,E )n−d−1∇u)
= 0,
satisfy all the usual elliptic estimates (A an elliptic matrix).
Question: Geometry of E characterized by ωL vs σ?
6
![Page 24: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/24.jpg)
Regularized Distance I: The Direct Result
Problem: x 7→ dist(x ,E ) is not a nice function. Hard to talk about ωL.
David-Feneuil-Mayboroda: family of smoothed out distances, Dα(x).Dα(x) ' dist(x ,E ).
Lαu = −div(
1
Dα(x)n−d−1∇u).
Theorem (David-Feneuil-Mayboroda (for Lipschitz graphs),David-Mayboroda (in progress))
Let E be a d-uniformly rectifiable set, then ωα ∈ A∞(dσ).
This is great! We have that nice geometry implies nice behavior of the(degenerate) elliptic measure (analogue of Hofmann-Martell for higherco-dimension).
Note: no topological assumptions needed. That is because E is so thin!
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Regularized Distance I: The Direct Result
Problem: x 7→ dist(x ,E ) is not a nice function. Hard to talk about ωL.
David-Feneuil-Mayboroda: family of smoothed out distances, Dα(x).Dα(x) ' dist(x ,E ).
Lαu = −div(
1
Dα(x)n−d−1∇u).
Theorem (David-Feneuil-Mayboroda (for Lipschitz graphs),David-Mayboroda (in progress))
Let E be a d-uniformly rectifiable set, then ωα ∈ A∞(dσ).
This is great! We have that nice geometry implies nice behavior of the(degenerate) elliptic measure (analogue of Hofmann-Martell for higherco-dimension).
Note: no topological assumptions needed. That is because E is so thin!
7
![Page 26: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/26.jpg)
Regularized Distance I: The Direct Result
Problem: x 7→ dist(x ,E ) is not a nice function. Hard to talk about ωL.
David-Feneuil-Mayboroda: family of smoothed out distances, Dα(x).Dα(x) ' dist(x ,E ).
Lαu = −div(
1
Dα(x)n−d−1∇u).
Theorem (David-Feneuil-Mayboroda (for Lipschitz graphs),David-Mayboroda (in progress))
Let E be a d-uniformly rectifiable set, then ωα ∈ A∞(dσ).
This is great! We have that nice geometry implies nice behavior of the(degenerate) elliptic measure (analogue of Hofmann-Martell for higherco-dimension).
Note: no topological assumptions needed. That is because E is so thin!
7
![Page 27: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/27.jpg)
Regularized Distance I: The Direct Result
Problem: x 7→ dist(x ,E ) is not a nice function. Hard to talk about ωL.
David-Feneuil-Mayboroda: family of smoothed out distances, Dα(x).Dα(x) ' dist(x ,E ).
Lαu = −div(
1
Dα(x)n−d−1∇u).
Theorem (David-Feneuil-Mayboroda (for Lipschitz graphs),David-Mayboroda (in progress))
Let E be a d-uniformly rectifiable set, then ωα ∈ A∞(dσ).
This is great! We have that nice geometry implies nice behavior of the(degenerate) elliptic measure (analogue of Hofmann-Martell for higherco-dimension).
Note: no topological assumptions needed. That is because E is so thin!
7
![Page 28: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/28.jpg)
Regularized Distance I: The Direct Result
Problem: x 7→ dist(x ,E ) is not a nice function. Hard to talk about ωL.
David-Feneuil-Mayboroda: family of smoothed out distances, Dα(x).Dα(x) ' dist(x ,E ).
Lαu = −div(
1
Dα(x)n−d−1∇u).
Theorem (David-Feneuil-Mayboroda (for Lipschitz graphs),David-Mayboroda (in progress))
Let E be a d-uniformly rectifiable set, then ωα ∈ A∞(dσ).
This is great! We have that nice geometry implies nice behavior of the(degenerate) elliptic measure (analogue of Hofmann-Martell for higherco-dimension).
Note: no topological assumptions needed. That is because E is so thin!
7
![Page 29: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/29.jpg)
Regularized Distance II: Who is Dα?
E ⊂ Rn is d-Ahlfors regular. σ = Hd |E and α > 0. Define
Dα(x) ≡(ˆ
E
1
|x − y |d+αdσ
)−1/α
.
Why Dα?
• Ahlfors Regular ⇒ Dα <∞.
• α > 0 ensures Dα(x) 'α,AR dist(x ,E ).
• Dα sees whole geometry of E (non-local!) and is smooth in Rn\E .
• ∇Dα “sees” flatness of E (How....?)
8
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Regularized Distance II: Who is Dα?
E ⊂ Rn is d-Ahlfors regular. σ = Hd |E and α > 0. Define
Dα(x) ≡(ˆ
E
1
|x − y |d+αdσ
)−1/α
.
Why Dα?
• Ahlfors Regular ⇒ Dα <∞.
• α > 0 ensures Dα(x) 'α,AR dist(x ,E ).
• Dα sees whole geometry of E (non-local!) and is smooth in Rn\E .
• ∇Dα “sees” flatness of E (How....?)
8
![Page 31: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/31.jpg)
Regularized Distance II: Who is Dα?
E ⊂ Rn is d-Ahlfors regular. σ = Hd |E and α > 0. Define
Dα(x) ≡(ˆ
E
1
|x − y |d+αdσ
)−1/α
.
Why Dα?
• Ahlfors Regular ⇒ Dα <∞.
• α > 0 ensures Dα(x) 'α,AR dist(x ,E ).
• Dα sees whole geometry of E (non-local!) and is smooth in Rn\E .
• ∇Dα “sees” flatness of E (How....?)
8
![Page 32: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/32.jpg)
Regularized Distance II: Who is Dα?
E ⊂ Rn is d-Ahlfors regular. σ = Hd |E and α > 0. Define
Dα(x) ≡(ˆ
E
1
|x − y |d+αdσ
)−1/α
.
Why Dα?
• Ahlfors Regular ⇒ Dα <∞.
• α > 0 ensures Dα(x) 'α,AR dist(x ,E ).
• Dα sees whole geometry of E (non-local!) and is smooth in Rn\E .
• ∇Dα “sees” flatness of E (How....?)
8
![Page 33: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/33.jpg)
Regularized Distance II: Who is Dα?
E ⊂ Rn is d-Ahlfors regular. σ = Hd |E and α > 0. Define
Dα(x) ≡(ˆ
E
1
|x − y |d+αdσ
)−1/α
.
Why Dα?
• Ahlfors Regular ⇒ Dα <∞.
• α > 0 ensures Dα(x) 'α,AR dist(x ,E ).
• Dα sees whole geometry of E (non-local!) and is smooth in Rn\E .
• ∇Dα “sees” flatness of E (How....?)
8
![Page 34: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/34.jpg)
Regularized Distance II: Who is Dα?
E ⊂ Rn is d-Ahlfors regular. σ = Hd |E and α > 0. Define
Dα(x) ≡(ˆ
E
1
|x − y |d+αdσ
)−1/α
.
Why Dα?
• Ahlfors Regular ⇒ Dα <∞.
• α > 0 ensures Dα(x) 'α,AR dist(x ,E ).
• Dα sees whole geometry of E (non-local!) and is smooth in Rn\E .
• ∇Dα “sees” flatness of E (How....?)
8
![Page 35: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/35.jpg)
Oscillation of |∇Dα| and the flatness of E
David-E.-Mayboroda: “|∇Dα| doesn’t oscillate too much near E iff E isregular.”
LetFα(x) = dist(x ,E )|∇|∇Dα|2|.
LetΓη(Q) = x ∈ Rn\E | dist(x ,E ) ≤ (1 + η)|x − Q|.
Theorem (David-E.-Mayboroda, in preparation)
E is uniformly rectifiable if and only if F 2α(x)δ(x)−n+d is a Carleson
measure on Rn\E .E is rectifiable if and only if limQ←x∈Γη(Q) |∇Dα(x)| exists for σ-a.e.Q ∈ E .
Carleson Measure:´B(Q,R) F
2δ−n+ddx ≤ CRd for all Q ∈ E .
9
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Oscillation of |∇Dα| and the flatness of E
David-E.-Mayboroda: “|∇Dα| doesn’t oscillate too much near E iff E isregular.”
LetFα(x) = dist(x ,E )|∇|∇Dα|2|.
LetΓη(Q) = x ∈ Rn\E | dist(x ,E ) ≤ (1 + η)|x − Q|.
Theorem (David-E.-Mayboroda, in preparation)
E is uniformly rectifiable if and only if F 2α(x)δ(x)−n+d is a Carleson
measure on Rn\E .E is rectifiable if and only if limQ←x∈Γη(Q) |∇Dα(x)| exists for σ-a.e.Q ∈ E .
Carleson Measure:´B(Q,R) F
2δ−n+ddx ≤ CRd for all Q ∈ E .
9
![Page 37: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/37.jpg)
Oscillation of |∇Dα| and the flatness of E
David-E.-Mayboroda: “|∇Dα| doesn’t oscillate too much near E iff E isregular.”
LetFα(x) = dist(x ,E )|∇|∇Dα|2|.
LetΓη(Q) = x ∈ Rn\E | dist(x ,E ) ≤ (1 + η)|x − Q|.
Theorem (David-E.-Mayboroda, in preparation)
E is uniformly rectifiable if and only if F 2α(x)δ(x)−n+d is a Carleson
measure on Rn\E .E is rectifiable if and only if limQ←x∈Γη(Q) |∇Dα(x)| exists for σ-a.e.Q ∈ E .
Carleson Measure:´B(Q,R) F
2δ−n+ddx ≤ CRd for all Q ∈ E .
9
![Page 38: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/38.jpg)
Oscillation of |∇Dα| and the flatness of E
David-E.-Mayboroda: “|∇Dα| doesn’t oscillate too much near E iff E isregular.”
LetFα(x) = dist(x ,E )|∇|∇Dα|2|.
LetΓη(Q) = x ∈ Rn\E | dist(x ,E ) ≤ (1 + η)|x − Q|.
Theorem (David-E.-Mayboroda, in preparation)
E is uniformly rectifiable if and only if F 2α(x)δ(x)−n+d is a Carleson
measure on Rn\E .E is rectifiable if and only if limQ←x∈Γη(Q) |∇Dα(x)| exists for σ-a.e.Q ∈ E .
Carleson Measure:´B(Q,R) F
2δ−n+ddx ≤ CRd for all Q ∈ E .
9
![Page 39: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/39.jpg)
Oscillation of |∇Dα| and the flatness of E
David-E.-Mayboroda: “|∇Dα| doesn’t oscillate too much near E iff E isregular.”
LetFα(x) = dist(x ,E )|∇|∇Dα|2|.
LetΓη(Q) = x ∈ Rn\E | dist(x ,E ) ≤ (1 + η)|x − Q|.
Theorem (David-E.-Mayboroda, in preparation)
E is uniformly rectifiable if and only if F 2α(x)δ(x)−n+d is a Carleson
measure on Rn\E .E is rectifiable if and only if limQ←x∈Γη(Q) |∇Dα(x)| exists for σ-a.e.Q ∈ E .
Carleson Measure:´B(Q,R) F
2δ−n+ddx ≤ CRd for all Q ∈ E .
9
![Page 40: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/40.jpg)
Connection with other work
Carleson Measure/Non-tangential limits, important concepts in HarmonicAnalysis.
Carleson Measure: Quantitative, scale-invariant measure of smallness.Intimately connected to uniform rectifiability. E.g. David-Semmes, usualsquare function estimates: closely related to our work.
Non-tangential limits ! principle values for SIOs. Work of Tolsa relatingprinciple values to rectifiability.
∇Dα = − 1
αD1+αα
(ˆE
x − y
|x − y |d+α+1dσ
)
10
![Page 41: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/41.jpg)
Connection with other work
Carleson Measure/Non-tangential limits, important concepts in HarmonicAnalysis.
Carleson Measure: Quantitative, scale-invariant measure of smallness.Intimately connected to uniform rectifiability. E.g. David-Semmes, usualsquare function estimates: closely related to our work.
Non-tangential limits ! principle values for SIOs. Work of Tolsa relatingprinciple values to rectifiability.
∇Dα = − 1
αD1+αα
(ˆE
x − y
|x − y |d+α+1dσ
)
10
![Page 42: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/42.jpg)
Connection with other work
Carleson Measure/Non-tangential limits, important concepts in HarmonicAnalysis.
Carleson Measure: Quantitative, scale-invariant measure of smallness.Intimately connected to uniform rectifiability. E.g. David-Semmes, usualsquare function estimates: closely related to our work.
Non-tangential limits ! principle values for SIOs. Work of Tolsa relatingprinciple values to rectifiability.
∇Dα = − 1
αD1+αα
(ˆE
x − y
|x − y |d+α+1dσ
)
10
![Page 43: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/43.jpg)
Connection with other work
Carleson Measure/Non-tangential limits, important concepts in HarmonicAnalysis.
Carleson Measure: Quantitative, scale-invariant measure of smallness.Intimately connected to uniform rectifiability. E.g. David-Semmes, usualsquare function estimates: closely related to our work.
Non-tangential limits ! principle values for SIOs. Work of Tolsa relatingprinciple values to rectifiability.
∇Dα = − 1
αD1+αα
(ˆE
x − y
|x − y |d+α+1dσ
)
10
![Page 44: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/44.jpg)
Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
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Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
![Page 46: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/46.jpg)
Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
![Page 47: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/47.jpg)
Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
![Page 48: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/48.jpg)
Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
![Page 49: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/49.jpg)
Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
![Page 50: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/50.jpg)
Intuition/Proof: |∇D| = c ⇒ flat
Key Step: |∇Dα| is constant if and only if E is an affine space.
⇒ is interesting. Essentially two steps: if dist(x ,E ) is C 1 then E isconvex. If it is C 2 (and E is d-Ahlfors regular with d < n) then E is affine.
Proof Sketches:
• UR ⇒ Carleson Condition: compare |∇Dα| to |∇Dα| of bestapproximation. Use α numbers.
• Carleson ⇒ UR: use a compactness argument. Limit to a plane, useβ numbers.
• NT limits ⇒ Rectifiability: blow-up! Get a plane at almost everypoint.
• Rectifiability ⇒ NT Limits: blow-up! Get a constant at almost everypoint.
11
![Page 51: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/51.jpg)
What about ωα?
We got distracted: we know E uniformly rectifiable implies ωα ∈ A∞(dσ).
If ωα ∈ A∞(dσ) then can we conclude regularity on E?
General Free Boundary Problem:
Does the oscillation of dωαdσ control the regularity of E?
NO!!!!
12
![Page 52: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/52.jpg)
What about ωα?
We got distracted: we know E uniformly rectifiable implies ωα ∈ A∞(dσ).
If ωα ∈ A∞(dσ) then can we conclude regularity on E?
General Free Boundary Problem:
Does the oscillation of dωαdσ control the regularity of E?
NO!!!!
12
![Page 53: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/53.jpg)
What about ωα?
We got distracted: we know E uniformly rectifiable implies ωα ∈ A∞(dσ).
If ωα ∈ A∞(dσ) then can we conclude regularity on E?
General Free Boundary Problem:
Does the oscillation of dωαdσ control the regularity of E?
NO!!!!
12
![Page 54: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/54.jpg)
What about ωα?
We got distracted: we know E uniformly rectifiable implies ωα ∈ A∞(dσ).
If ωα ∈ A∞(dσ) then can we conclude regularity on E?
General Free Boundary Problem:
Does the oscillation of dωαdσ control the regularity of E?
NO!!!!
12
![Page 55: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/55.jpg)
Magic α!
Let E ⊂ Rn be d-Ahlfors regular. And α = n − d − 2 > 0.
Theorem (David-E.-Mayboroda, in preparation)
For any E as above and α = n − d − 2, C−1σ ≤ ωα ≤ Cσ.
If E is rectifiable then ωα ≡ cσ.
NOTE: E could be a fractal! d could be a non-integer dimension!!!
Recall the work of Kenig-Toro:
Theorem (Kenig-Toro)
Ω ⊂ Rn with mild topological assumptions. ∂Ω is (n − 1)-Ahlfors regular.Then ω = σ if and only if Ω is a half-space.More generally, regularity of dω
dσ controls regularity of ∂Ω.
Takeaway: For magic α, dωαdσ doesn’t control the regularity of E , and fails
to do so in the most spectacular way possible!
13
![Page 56: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/56.jpg)
Magic α!
Let E ⊂ Rn be d-Ahlfors regular. And α = n − d − 2 > 0.
Theorem (David-E.-Mayboroda, in preparation)
For any E as above and α = n − d − 2, C−1σ ≤ ωα ≤ Cσ.If E is rectifiable then ωα ≡ cσ.
NOTE: E could be a fractal! d could be a non-integer dimension!!!
Recall the work of Kenig-Toro:
Theorem (Kenig-Toro)
Ω ⊂ Rn with mild topological assumptions. ∂Ω is (n − 1)-Ahlfors regular.Then ω = σ if and only if Ω is a half-space.More generally, regularity of dω
dσ controls regularity of ∂Ω.
Takeaway: For magic α, dωαdσ doesn’t control the regularity of E , and fails
to do so in the most spectacular way possible!
13
![Page 57: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/57.jpg)
Magic α!
Let E ⊂ Rn be d-Ahlfors regular. And α = n − d − 2 > 0.
Theorem (David-E.-Mayboroda, in preparation)
For any E as above and α = n − d − 2, C−1σ ≤ ωα ≤ Cσ.If E is rectifiable then ωα ≡ cσ.
NOTE: E could be a fractal! d could be a non-integer dimension!!!
Recall the work of Kenig-Toro:
Theorem (Kenig-Toro)
Ω ⊂ Rn with mild topological assumptions. ∂Ω is (n − 1)-Ahlfors regular.Then ω = σ if and only if Ω is a half-space.More generally, regularity of dω
dσ controls regularity of ∂Ω.
Takeaway: For magic α, dωαdσ doesn’t control the regularity of E , and fails
to do so in the most spectacular way possible!
13
![Page 58: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/58.jpg)
Magic α!
Let E ⊂ Rn be d-Ahlfors regular. And α = n − d − 2 > 0.
Theorem (David-E.-Mayboroda, in preparation)
For any E as above and α = n − d − 2, C−1σ ≤ ωα ≤ Cσ.If E is rectifiable then ωα ≡ cσ.
NOTE: E could be a fractal! d could be a non-integer dimension!!!
Recall the work of Kenig-Toro:
Theorem (Kenig-Toro)
Ω ⊂ Rn with mild topological assumptions. ∂Ω is (n − 1)-Ahlfors regular.Then ω = σ if and only if Ω is a half-space.More generally, regularity of dω
dσ controls regularity of ∂Ω.
Takeaway: For magic α, dωαdσ doesn’t control the regularity of E , and fails
to do so in the most spectacular way possible!
13
![Page 59: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/59.jpg)
Magic α!
Let E ⊂ Rn be d-Ahlfors regular. And α = n − d − 2 > 0.
Theorem (David-E.-Mayboroda, in preparation)
For any E as above and α = n − d − 2, C−1σ ≤ ωα ≤ Cσ.If E is rectifiable then ωα ≡ cσ.
NOTE: E could be a fractal! d could be a non-integer dimension!!!
Recall the work of Kenig-Toro:
Theorem (Kenig-Toro)
Ω ⊂ Rn with mild topological assumptions. ∂Ω is (n − 1)-Ahlfors regular.Then ω = σ if and only if Ω is a half-space.More generally, regularity of dω
dσ controls regularity of ∂Ω.
Takeaway: For magic α, dωαdσ doesn’t control the regularity of E , and fails
to do so in the most spectacular way possible!
13
![Page 60: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/60.jpg)
What’s up with “magic α”?
Can compute: see that for α = n − d − 2 we have
LαDα = −div(
1
Dn−d−1α
∇Dα)
= 0.
“The distance is a solution to the equation”
Dα is “Green function with pole at infinity”: |∇Dα| on E gives dωαdσ .
Dα ' dist(x ,E )⇒ ωα ' σ
When α is magic Dα(x) =(´
E1
|x−y |n−2 dσ)−1/α
. Note: 1|x |n−2 is harmonic!
14
![Page 61: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/61.jpg)
What’s up with “magic α”?
Can compute: see that for α = n − d − 2 we have
LαDα = −div(
1
Dn−d−1α
∇Dα)
= 0.
“The distance is a solution to the equation”
Dα is “Green function with pole at infinity”: |∇Dα| on E gives dωαdσ .
Dα ' dist(x ,E )⇒ ωα ' σ
When α is magic Dα(x) =(´
E1
|x−y |n−2 dσ)−1/α
. Note: 1|x |n−2 is harmonic!
14
![Page 62: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/62.jpg)
What’s up with “magic α”?
Can compute: see that for α = n − d − 2 we have
LαDα = −div(
1
Dn−d−1α
∇Dα)
= 0.
“The distance is a solution to the equation”
Dα is “Green function with pole at infinity”: |∇Dα| on E gives dωαdσ .
Dα ' dist(x ,E )⇒ ωα ' σ
When α is magic Dα(x) =(´
E1
|x−y |n−2 dσ)−1/α
. Note: 1|x |n−2 is harmonic!
14
![Page 63: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/63.jpg)
What’s up with “magic α”?
Can compute: see that for α = n − d − 2 we have
LαDα = −div(
1
Dn−d−1α
∇Dα)
= 0.
“The distance is a solution to the equation”
Dα is “Green function with pole at infinity”: |∇Dα| on E gives dωαdσ .
Dα ' dist(x ,E )⇒ ωα ' σ
When α is magic Dα(x) =(´
E1
|x−y |n−2 dσ)−1/α
. Note: 1|x |n−2 is harmonic!
14
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Open Questions about the Magic α
1 Why is magic α magic?• Dα satisfies an equation but what is really going on?
2 Is this emblematic or pathological?• Is any other β magic?• Can we prove the converse for ωβ with β not magic?
3 What does α 7→ Dα look like?• The power − 1
α makes this question harder.
15
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Open Questions about the Magic α
1 Why is magic α magic?• Dα satisfies an equation but what is really going on?
2 Is this emblematic or pathological?• Is any other β magic?
• Can we prove the converse for ωβ with β not magic?
3 What does α 7→ Dα look like?• The power − 1
α makes this question harder.
15
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Open Questions about the Magic α
1 Why is magic α magic?• Dα satisfies an equation but what is really going on?
2 Is this emblematic or pathological?• Is any other β magic?• Can we prove the converse for ωβ with β not magic?
3 What does α 7→ Dα look like?• The power − 1
α makes this question harder.
15
![Page 67: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/67.jpg)
Open Questions about the Magic α
1 Why is magic α magic?• Dα satisfies an equation but what is really going on?
2 Is this emblematic or pathological?• Is any other β magic?• Can we prove the converse for ωβ with β not magic?
3 What does α 7→ Dα look like?• The power − 1
α makes this question harder.
15
![Page 68: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/68.jpg)
Wild Speculation Part I: Energy
Minimizers to (nice) energies often self-improve regularity (flat impliessmooth). Ex: minimal surfaces, obstacle problems, etc etc.
No energy for harmonic measure in general. But underlying Kenig-Torotype results: secret energy
ˆ|∇u|2 + χu>0.
Perhaps no (nice) energy in co-dim > 1 case (operator too dependent onset). Or perhaps energy is not nice for magic α (loss of coercivity)?
Question: Does this phenomenon exist for other problems with energy?E.g. given a set E , can you come up with an obstacle type problem such
that E is the contact set of the minimizer? Coefficients will be nasty.
16
![Page 69: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/69.jpg)
Wild Speculation Part I: Energy
Minimizers to (nice) energies often self-improve regularity (flat impliessmooth). Ex: minimal surfaces, obstacle problems, etc etc.
No energy for harmonic measure in general. But underlying Kenig-Torotype results: secret energy
ˆ|∇u|2 + χu>0.
Perhaps no (nice) energy in co-dim > 1 case (operator too dependent onset). Or perhaps energy is not nice for magic α (loss of coercivity)?
Question: Does this phenomenon exist for other problems with energy?E.g. given a set E , can you come up with an obstacle type problem such
that E is the contact set of the minimizer? Coefficients will be nasty.
16
![Page 70: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/70.jpg)
Wild Speculation Part I: Energy
Minimizers to (nice) energies often self-improve regularity (flat impliessmooth). Ex: minimal surfaces, obstacle problems, etc etc.
No energy for harmonic measure in general. But underlying Kenig-Torotype results: secret energy
ˆ|∇u|2 + χu>0.
Perhaps no (nice) energy in co-dim > 1 case (operator too dependent onset). Or perhaps energy is not nice for magic α (loss of coercivity)?
Question: Does this phenomenon exist for other problems with energy?E.g. given a set E , can you come up with an obstacle type problem such
that E is the contact set of the minimizer? Coefficients will be nasty.
16
![Page 71: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/71.jpg)
Wild Speculation Part I: Energy
Minimizers to (nice) energies often self-improve regularity (flat impliessmooth). Ex: minimal surfaces, obstacle problems, etc etc.
No energy for harmonic measure in general. But underlying Kenig-Torotype results: secret energy
ˆ|∇u|2 + χu>0.
Perhaps no (nice) energy in co-dim > 1 case (operator too dependent onset). Or perhaps energy is not nice for magic α (loss of coercivity)?
Question: Does this phenomenon exist for other problems with energy?E.g. given a set E , can you come up with an obstacle type problem such
that E is the contact set of the minimizer? Coefficients will be nasty.
16
![Page 72: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/72.jpg)
Wild Speculation Part II: co-dimension 1
Let E ⊂ Rn−1 be an (n − 1)-Ahlfors regular set.
Question: Under what conditions on E can you find an elliptic operator, Lsuch that ωL ' Hn−1|E?
“Opposite” question: “what operators, L, “detect” rectifiability inco-dimension 1?” a matter of great interest.
Nothing known for either question except in the “perturbative” regime(e.g. t-independent coefficients, operator close to identity in a Carlesonsense etc).
Explicit Question: Does there exist an operator in R2 on the exterior ofthe four-corner Cantor set, C , such that ωL ' σ on C?
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![Page 73: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/73.jpg)
Wild Speculation Part II: co-dimension 1
Let E ⊂ Rn−1 be an (n − 1)-Ahlfors regular set.
Question: Under what conditions on E can you find an elliptic operator, Lsuch that ωL ' Hn−1|E?
“Opposite” question: “what operators, L, “detect” rectifiability inco-dimension 1?” a matter of great interest.
Nothing known for either question except in the “perturbative” regime(e.g. t-independent coefficients, operator close to identity in a Carlesonsense etc).
Explicit Question: Does there exist an operator in R2 on the exterior ofthe four-corner Cantor set, C , such that ωL ' σ on C?
17
![Page 74: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/74.jpg)
Wild Speculation Part II: co-dimension 1
Let E ⊂ Rn−1 be an (n − 1)-Ahlfors regular set.
Question: Under what conditions on E can you find an elliptic operator, Lsuch that ωL ' Hn−1|E?
“Opposite” question: “what operators, L, “detect” rectifiability inco-dimension 1?” a matter of great interest.
Nothing known for either question except in the “perturbative” regime(e.g. t-independent coefficients, operator close to identity in a Carlesonsense etc).
Explicit Question: Does there exist an operator in R2 on the exterior ofthe four-corner Cantor set, C , such that ωL ' σ on C?
17
![Page 75: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/75.jpg)
Wild Speculation Part II: co-dimension 1
Let E ⊂ Rn−1 be an (n − 1)-Ahlfors regular set.
Question: Under what conditions on E can you find an elliptic operator, Lsuch that ωL ' Hn−1|E?
“Opposite” question: “what operators, L, “detect” rectifiability inco-dimension 1?” a matter of great interest.
Nothing known for either question except in the “perturbative” regime(e.g. t-independent coefficients, operator close to identity in a Carlesonsense etc).
Explicit Question: Does there exist an operator in R2 on the exterior ofthe four-corner Cantor set, C , such that ωL ' σ on C?
17
![Page 76: Regularized Distances and Harmonic Measure in Co-Dimension](https://reader033.vdocuments.site/reader033/viewer/2022050306/626f3c66602f632abb109afd/html5/thumbnails/76.jpg)
Thanks!
Thank You For Listening!
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