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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction The stickiness property of nonlocal Limits Boundary behavior minimal surfaces of nonlocal minimal surfaces Enrico Valdinoci University of Western

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

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The stickiness property of nonlocal minimal surfaces

Enrico Valdinoci

University of Western Australia

2020 Fields Medal Symposium

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

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Outline

Nonlocal minimal surfaces Energy functional dealing with “pointwise interactions” between a given set and its complement Main idea: the “surface tension” is the byproduct of long-range interactions Implications: nonlocal phase transitions and nonlocal capillarity theories New effects due to the long-range interactions Contributions from “far-away” can have a significant influence

  • n the local structures of these new objects

STICKINESS Differently from classical minimal surfaces, the nonlocal minimal surfaces have the strong tendency to “stick at the boundary”

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The fractional perimeter functional

Given s ∈ (0, 1) and a bounded open set Ω ⊂ Rn with C1,γ-boundary, the s-perimeter of a (measurable) set E ⊆ Rn in Ω is defined as Pers(E; Ω) := L(E ∩ Ω, (CE) ∩ Ω) + L(E ∩ Ω, (CE) ∩ (CΩ)) + L(E ∩ (CΩ), (CE) ∩ Ω), where CE = Rn \ E denotes the complement of E, and L(A, B) denotes the following nonlocal interaction term L(A, B) :=

  • A
  • B

1 |x − y|n+s dx dy ∀ A, B ⊆ Rn, This notion of s-perimeter and the corresponding minimization problem were introduced in [Caffarelli-Roquejoffre-Savin, 2010].

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Side 1. Perimeter 4. Approximate Perimeter 4 √ 2. Error 4( √ 2 − 1).

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Error in each pixel O(ǫ2−s). Number of pixels O(ǫ−1) Error O(ǫ1−s).

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[Caffarelli-Roquejoffre-Savin, 2010]

1) Existence theorem: there exists E s-minimizer for Pers in Ω with E \ Ω = E0 \ Ω. 2) Maximum principle: E s-minimizer and (∂E) \ Ω ⊂ {|xn| a} ⇒ ∂E ⊂ {|xn| a}. 3) If ∂E is an hyperplane, then E is s-minimizer. 4) If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 2. 5) If E is s-minimizer and 0 ∈ ∂E, then

  • Rn

χE(y) − χEc(y) |y|n+s dy = 0.

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[Caffarelli-Roquejoffre-Savin, 2010]

1) Existence theorem: there exists E s-minimizer for Pers in Ω with E \ Ω = E0 \ Ω. 2) Maximum principle: E s-minimizer and (∂E) \ Ω ⊂ {|xn| a} ⇒ ∂E ⊂ {|xn| a}. 3) If ∂E is an hyperplane, then E is s-minimizer. 4) If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 2. 5) If E is s-minimizer and 0 ∈ ∂E, then

  • Rn

χE(y) − χEc(y) |y|n+s dy = 0.

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[Caffarelli-Roquejoffre-Savin, 2010]

1) Existence theorem: there exists E s-minimizer for Pers in Ω with E \ Ω = E0 \ Ω. 2) Maximum principle: E s-minimizer and (∂E) \ Ω ⊂ {|xn| a} ⇒ ∂E ⊂ {|xn| a}. 3) If ∂E is an hyperplane, then E is s-minimizer. 4) If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 2. 5) If E is s-minimizer and 0 ∈ ∂E, then

  • Rn

χE(y) − χEc(y) |y|n+s dy = 0.

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[Caffarelli-Roquejoffre-Savin, 2010]

1) Existence theorem: there exists E s-minimizer for Pers in Ω with E \ Ω = E0 \ Ω. 2) Maximum principle: E s-minimizer and (∂E) \ Ω ⊂ {|xn| a} ⇒ ∂E ⊂ {|xn| a}. 3) If ∂E is an hyperplane, then E is s-minimizer. 4) If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 2. 5) If E is s-minimizer and 0 ∈ ∂E, then

  • Rn

χE(y) − χEc(y) |y|n+s dy = 0.

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[Caffarelli-Roquejoffre-Savin, 2010]

1) Existence theorem: there exists E s-minimizer for Pers in Ω with E \ Ω = E0 \ Ω. 2) Maximum principle: E s-minimizer and (∂E) \ Ω ⊂ {|xn| a} ⇒ ∂E ⊂ {|xn| a}. 3) If ∂E is an hyperplane, then E is s-minimizer. 4) If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 2. 5) If E is s-minimizer and 0 ∈ ∂E, then

  • Rn

χE(y) − χEc(y) |y|n+s dy = 0.

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Regularity in dimension 2

[Savin-Valdinoci, 2013]: Regularity of cones in dimension 2. If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n−3.

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Regularity in dimension 2

[Savin-Valdinoci, 2013]: Regularity of cones in dimension 2. If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n−3.

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Regularity for graphs in dimension 3

[Figalli-Valdinoci, 2013]: Bernstein-type result:

◮ E is s-minimal in Rn+1 and ∂E is a global graph, ◮ s-minimal surfaces are smooth in Rn

⇒ ∂E is hyperplane. Regularity of minimal graph in dimension 3.

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Regularity for graphs in dimension 3

[Figalli-Valdinoci, 2013]: Bernstein-type result:

◮ E is s-minimal in Rn+1 and ∂E is a global graph, ◮ s-minimal surfaces are smooth in Rn

⇒ ∂E is hyperplane. Regularity of minimal graph in dimension 3.

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Limit as s → 1

[Bourgain-Brezis-Mironescu, 2001], [Dávila, 2002], [Ponce, 2004], [Caffarelli-Valdinoci, 2011], [Ambrosio-De Philippis-Martinazzi, 2011], [Lombardini, 2018]: (1 − s)Pers → Per, as s ր 1 (up to normalizing multiplicative constants). ⇓ [Caffarelli-Valdinoci, 2013]: s close to 1: nonlocal minimal surfaces are as regular as classical minimal surfaces. (If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 8.)

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Limit as s → 1

[Bourgain-Brezis-Mironescu, 2001], [Dávila, 2002], [Ponce, 2004], [Caffarelli-Valdinoci, 2011], [Ambrosio-De Philippis-Martinazzi, 2011], [Lombardini, 2018]: (1 − s)Pers → Per, as s ր 1 (up to normalizing multiplicative constants). ⇓ [Caffarelli-Valdinoci, 2013]: s close to 1: nonlocal minimal surfaces are as regular as classical minimal surfaces. (If E is s-minimizer in B1, then ∂E is C1,α in B1/2 except in a closed set Σ, with Hausdorff dimension less or equal than n − 8.)

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Limit as s → 0

[Maz’ya-Shaposhnikova, 2002] and [Dipierro-Figalli-Palatucci-Valdinoci, 2013]: If there exists the limit α(E) := lim

sց0 s

  • E∩(CB1)

1 |y|n+s dy, then lim

sց0 s Pers(E, Ω) =

  • ωn−1 − α(E)

|E ∩ Ω| ωn−1 + α(E) |Ω \ E| ωn−1 .

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Stickiness to half-balls

For any δ > 0, Kδ :=

  • B1+δ \ B1
  • ∩ {xn < 0}.

We define Eδ to be the set minimizing the s-perimeter among all the sets E such that E \ B1 = Kδ.

K

δ

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Stickiness to half-balls

There exists δo > 0 such that for any δ ∈ (0, δo] we have that Eδ = Kδ.

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Stickiness to the sides of a box

Given a large M > 1 we consider the s-minimal set EM in (−1, 1) × R with datum outside (−1, 1) × R given by the jump JM := J−

M ∪ J+ M, where

J−

M := (−∞, −1] × (−∞, −M)

and J+

M := [1, +∞) × (−∞, M).

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Stickiness to the sides of a box

There exist Mo > 0 and Co ≥ C′

  • > 0, depending on s, such

that if M ≥ Mo then [−1, 1) × [CoM

1+s 2+s , M] ⊆ Ec

M

and (−1, 1] × [−M, −CoM

1+s 2+s ] ⊆ EM.

Also, the exponent β := 1+s

2+s above is optimal.

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Stickiness to the sides of a box

EM M −M 1 −1 M

β

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Stickiness as s → 0+

We consider a sector in R2 outside B1, i.e. Σ := {(x, y) ∈ R2 \ B1 s.t. x > 0 and y > 0}. Let Es be the s-minimizer of the s-perimeter among all the sets E such that E \ B1 = Σ. Then, there exists so > 0 such that for any s ∈ (0, so] we have that Es = Σ.

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Stickiness as s → 0+

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Instability of the flat fractional minimal surfaces

Fix ǫ0 > 0 arbitrarily small. Then, there exists δ0 > 0, possibly depending on ǫ0, such that for any δ ∈ (0, δ0] the following statement holds true. Assume that F ⊃ H ∪ F− ∪ F+, where H := R × (−∞, 0), F− := (−3, −2) × [0, δ) and F+ := (2, 3) × [0, δ). Let E be the s-minimal set in (−1, 1) × R among all the sets that coincide with F outside (−1, 1) × R. Then E ⊇ (−1, 1) × (−∞, δ

2+ǫ0 1−s ]. 40 / 91

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Instability of the flat fractional minimal surfaces

β := 2+ǫ0

1−s

δ δβ 1 −1

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A useful barrier

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The usual suspects

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The leading role of infinity [Bucur-Lombardini-Valdinoci, 2019]

Let Ω ⊂ Rn be bounded, connected and smooth. Let E0 ⊂ Rn be such that Br(x0) \ Ω ⊂ E0 for some x0 ∈ ∂Ω and r > 0, and α(E0) < α(halfplane). Then, there exists s0 ∈ (0, 1) such that if s ∈ (0, s0) and E is the nonlocal minimal set in Ω with external datum E0, we have E ∩ Ω = ∅.

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The leading role of infinity [Bucur-Lombardini-Valdinoci, 2019]

Let Ω ⊂ Rn be bounded, connected and smooth. Let E0 ⊂ Rn be such that Br(x0) \ Ω ⊂ E0 for some x0 ∈ ∂Ω and r > 0, and α(E0) < α(halfplane). Then, there exists s0 ∈ (0, 1) such that if s ∈ (0, s0) and E is the nonlocal minimal set in Ω with external datum E0, we have E ∩ Ω = ∅.

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Three further questions [Dipierro-Savin-Valdinoci, 2020]

  • 1. How regular are the nonlocal minimal surfaces coming from

inside the domain?

  • 2. Is the Euler-Lagrange equation satisfied up to the boundary?
  • 3. How typical is the stickiness phenomenon?

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Three further questions [Dipierro-Savin-Valdinoci, 2020]

  • 1. How regular are the nonlocal minimal surfaces coming from

inside the domain?

  • 2. Is the Euler-Lagrange equation satisfied up to the boundary?
  • 3. How typical is the stickiness phenomenon?

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Three further questions [Dipierro-Savin-Valdinoci, 2020]

  • 1. How regular are the nonlocal minimal surfaces coming from

inside the domain?

  • 2. Is the Euler-Lagrange equation satisfied up to the boundary?
  • 3. How typical is the stickiness phenomenon?

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Regularity coming from inside

“Continuity implies differentiability” Consider a nonlocal minimal graph in (0, 1), with a smooth external graph u0. There is a dichotomy:

◮ either

lim

xր0 u0(x) = lim xց0 u(x)

and lim

xց0 |u′(x)| = +∞, ◮ or

lim

xր0 u0(x) = lim xց0 u(x)

and u is C1, 1+s

2 at 0. 65 / 91

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Regularity coming from inside

“Continuity implies differentiability” Consider a nonlocal minimal graph in (0, 1), with a smooth external graph u0. There is a dichotomy:

◮ either

lim

xր0 u0(x) = lim xց0 u(x)

and lim

xց0 |u′(x)| = +∞, ◮ or

lim

xր0 u0(x) = lim xց0 u(x)

and u is C1, 1+s

2 at 0. 66 / 91

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Regularity coming from inside

“Continuity implies differentiability” Consider a nonlocal minimal graph in (0, 1), with a smooth external graph u0. There is a dichotomy:

◮ either

lim

xր0 u0(x) = lim xց0 u(x)

and lim

xց0 |u′(x)| = +∞, ◮ or

lim

xր0 u0(x) = lim xց0 u(x)

and u is C1, 1+s

2 at 0. 67 / 91

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Regularity coming from inside

“Continuity implies differentiability” Consider a nonlocal minimal graph in (0, 1), with a smooth external graph u0. There is a dichotomy:

◮ either

lim

xր0 u0(x) = lim xց0 u(x)

and lim

xց0 |u′(x)| = +∞, ◮ or

lim

xր0 u0(x) = lim xց0 u(x)

and u is C1, 1+s

2 at 0. 68 / 91

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  • f nonlocal minimal

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Some remarks

This dichotomy is a purely nonlinear effect, since the boundary behavior of linear equation is of Hölder type [Serra-Ros Oton].

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

Stickiness + dichotomy = butterfly effect An arbitrarily small perturbation of the flat data produce a boundary discontinuity, which entails an infinite derivative at the boundary. An arbitrarily small perturbation of the flat data produce an infinite derivative at the boundary.

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

Stickiness + dichotomy = butterfly effect An arbitrarily small perturbation of the flat data produce a boundary discontinuity, which entails an infinite derivative at the boundary. An arbitrarily small perturbation of the flat data produce an infinite derivative at the boundary.

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

Stickiness + dichotomy = butterfly effect An arbitrarily small perturbation of the flat data produce a boundary discontinuity, which entails an infinite derivative at the boundary. An arbitrarily small perturbation of the flat data produce an infinite derivative at the boundary.

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

As a curve, the nonlocal minimal graph turns out to be always C1, 1+s

2 :

it is either the graph of a C1, 1+s

2 -function (when it is continuous

at the boundary!), or it is discontinuous and sticks vertically detaching in a C1, 1+s

2 fashion [De Silva-Savin] (then the inverse

function is a C1, 1+s

2 function). 73 / 91

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

As a curve, the nonlocal minimal graph turns out to be always C1, 1+s

2 :

it is either the graph of a C1, 1+s

2 -function (when it is continuous

at the boundary!), or it is discontinuous and sticks vertically detaching in a C1, 1+s

2 fashion [De Silva-Savin] (then the inverse

function is a C1, 1+s

2 function). 74 / 91

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

The nonlocal mean curvature can be written in the form +∞

−∞

F u(x + y) − u(x) |y|

  • dy

|y|1+s . And this is a “C1,s operator”. But 1+s

2

> s, therefore we can “pass the equation to the limit”...

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

The nonlocal mean curvature can be written in the form +∞

−∞

F u(x + y) − u(x) |y|

  • dy

|y|1+s . And this is a “C1,s operator”. But 1+s

2

> s, therefore we can “pass the equation to the limit”...

76 / 91

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Some remarks

The nonlocal mean curvature can be written in the form +∞

−∞

F u(x + y) − u(x) |y|

  • dy

|y|1+s . And this is a “C1,s operator”. But 1+s

2

> s, therefore we can “pass the equation to the limit”...

77 / 91

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Boundary Euler-Lagrange equations

If u is a nonlocal minimal graph in (0, 1) with smooth datum

  • utside, then

+∞

−∞

F u(x + y) − u(x) |y|

  • dy

|y|1+s = 0 for all x ∈ [0, 1].

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With this, we can take any configuration, add an arbitrarily small bump and use the unperturbed configuration as a barrier. At touching points the additional bump produces an extra-mass violating the Euler-Lagrange equation. Notice that now also touching at the boundary can be taken into account!

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

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With this, we can take any configuration, add an arbitrarily small bump and use the unperturbed configuration as a barrier. At touching points the additional bump produces an extra-mass violating the Euler-Lagrange equation. Notice that now also touching at the boundary can be taken into account!

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

With this, we can take any configuration, add an arbitrarily small bump and use the unperturbed configuration as a barrier. At touching points the additional bump produces an extra-mass violating the Euler-Lagrange equation. Notice that now also touching at the boundary can be taken into account!

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Stickiness is generic

Let ϕ ∈ C∞

0 ([−2, −1], [0, 1]), with ϕ ≡ 0.

Let u(t) be the nonlocal minimal graph in (0, 1) with external datum u(t)

0 := u0 + tϕ.

Suppose that lim

xր0 u0(x) = lim xց0 u(x).

Then lim

xր0 u(t) 0 (x) < lim xց0 u(t)(x).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Stickiness is generic

Let ϕ ∈ C∞

0 ([−2, −1], [0, 1]), with ϕ ≡ 0.

Let u(t) be the nonlocal minimal graph in (0, 1) with external datum u(t)

0 := u0 + tϕ.

Suppose that lim

xր0 u0(x) = lim xց0 u(x).

Then lim

xր0 u(t) 0 (x) < lim xց0 u(t)(x).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Stickiness is generic

Let ϕ ∈ C∞

0 ([−2, −1], [0, 1]), with ϕ ≡ 0.

Let u(t) be the nonlocal minimal graph in (0, 1) with external datum u(t)

0 := u0 + tϕ.

Suppose that lim

xր0 u0(x) = lim xց0 u(x).

Then lim

xր0 u(t) 0 (x) < lim xց0 u(t)(x).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Stickiness is generic

Let ϕ ∈ C∞

0 ([−2, −1], [0, 1]), with ϕ ≡ 0.

Let u(t) be the nonlocal minimal graph in (0, 1) with external datum u(t)

0 := u0 + tϕ.

Suppose that lim

xր0 u0(x) = lim xց0 u(x).

Then lim

xր0 u(t) 0 (x) < lim xց0 u(t)(x).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Proof of dichotomy

Think about the usual suspects (discontinuous, Lispchitz, Hölder, smooth). Blow-up. The “worst” cases to understand are the Hölder and the smooth (the Lispchitz produces non-minimal corners). The smooth case produces flat objects: use a boundary improvement of flatness (combined with a boundary monotonicity formula) to deduce smoothness of the initial minimizer (for this, use new barrier to go beyond the linear theory!). The Hölder case produces vertical angles: rule them out by proving that close-to-vertical nonlocal minimal graphs are indeed vertical (for this, slide balls).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Proof of dichotomy

Think about the usual suspects (discontinuous, Lispchitz, Hölder, smooth). Blow-up. The “worst” cases to understand are the Hölder and the smooth (the Lispchitz produces non-minimal corners). The smooth case produces flat objects: use a boundary improvement of flatness (combined with a boundary monotonicity formula) to deduce smoothness of the initial minimizer (for this, use new barrier to go beyond the linear theory!). The Hölder case produces vertical angles: rule them out by proving that close-to-vertical nonlocal minimal graphs are indeed vertical (for this, slide balls).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Proof of dichotomy

Think about the usual suspects (discontinuous, Lispchitz, Hölder, smooth). Blow-up. The “worst” cases to understand are the Hölder and the smooth (the Lispchitz produces non-minimal corners). The smooth case produces flat objects: use a boundary improvement of flatness (combined with a boundary monotonicity formula) to deduce smoothness of the initial minimizer (for this, use new barrier to go beyond the linear theory!). The Hölder case produces vertical angles: rule them out by proving that close-to-vertical nonlocal minimal graphs are indeed vertical (for this, slide balls).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Proof of dichotomy

Think about the usual suspects (discontinuous, Lispchitz, Hölder, smooth). Blow-up. The “worst” cases to understand are the Hölder and the smooth (the Lispchitz produces non-minimal corners). The smooth case produces flat objects: use a boundary improvement of flatness (combined with a boundary monotonicity formula) to deduce smoothness of the initial minimizer (for this, use new barrier to go beyond the linear theory!). The Hölder case produces vertical angles: rule them out by proving that close-to-vertical nonlocal minimal graphs are indeed vertical (for this, slide balls).

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The stickiness property of nonlocal minimal surfaces Enrico Valdinoci Introduction Limits Boundary behavior

  • f nonlocal minimal

surfaces

Proof of dichotomy

Think about the usual suspects (discontinuous, Lispchitz, Hölder, smooth). Blow-up. The “worst” cases to understand are the Hölder and the smooth (the Lispchitz produces non-minimal corners). The smooth case produces flat objects: use a boundary improvement of flatness (combined with a boundary monotonicity formula) to deduce smoothness of the initial minimizer (for this, use new barrier to go beyond the linear theory!). The Hölder case produces vertical angles: rule them out by proving that close-to-vertical nonlocal minimal graphs are indeed vertical (for this, slide balls).

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Thank you very much for your attention!

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