(Non)local phase transitions and minimal perimeter interfaces · 2012. 10. 3. · Uniform...

(Non)local phasetransitions and

minimal perimeterinterfaces

Enrico Valdinoci

Outline

The fractionalLaplacian operator

Back to theLaplacian caseThe Allen-Cahn equation

A conjecture of De Giorgi

Some research linesfor (−∆)s

The symmetry problem for(−∆)s

Density estimates for thelevel sets

Γ-convergence for (−∆)s

Few words on the nonlocalperimeter

Uniform regularityof s-minimalsurfacesFurther research needed

(Non)local phase transitions and minimalperimeter interfaces

Enrico Valdinoci

Dipartimento di MatematicaUniversità di Milano

[email protected]

ADMAT 2012

PDEs for multiphase advanced materials

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Enrico Valdinoci

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Outline of the talk

The fractional Laplacian operator

Back to the Laplacian caseThe Allen-Cahn equationA conjecture of De Giorgi

Some research lines for (−∆)s

The symmetry problem for (−∆)s

Density estimates for the level setsΓ-convergence for (−∆)s

Few words on the nonlocal perimeter

Uniform regularity of s-minimal surfacesFurther research needed

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(−∆)su = F−1(|ξ|2s(Fu)

),

where s ∈ (0, 1) and F is the Fourier transform.This definition is consistent with the case s = 1:

−∆u = F−1(|ξ|2(Fu)).

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(−∆)su = F−1(|ξ|2s(Fu)

),


−∆u = F−1(|ξ|2(Fu)).

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(−∆)su = F−1(|ξ|2s(Fu)

),


−∆u = F−1(|ξ|2(Fu)).

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An equivalent definition may be given by integrating against asingular kernel, which suitably averages a second-orderincremental quotient:

−(−∆)su(x) =

∫Rn

u(x + y) + u(x− y)− 2u(x)|y|n+2s dy.

Up to a factor 2, this is the same as defining the operator as anintegral in the principal value sense

−(−∆)su(x) = limε→0+

∫Rn\Bε

u(x + y)− u(x)|y|n+2s dy.

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−(−∆)su(x) =

∫Rn

u(x + y) + u(x− y)− 2u(x)|y|n+2s dy.


−(−∆)su(x) = limε→0+

∫Rn\Bε


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−(−∆)su(x) =

∫Rn

u(x + y) + u(x− y)− 2u(x)|y|n+2s dy.


−(−∆)su(x) = limε→0+

∫Rn\Bε


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Motivation: the fractional Laplacian naturally surfaces inprobability, water waves, and lower dimensional obstacleproblems (among others). In statistical mechanics it is a way totake into account long-range particle interactions.

Difficulty: The operator is nonlocal, hence one needs toestimate also the contribution coming from far. Also,integrating is usually harder than differentiating.

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Motivation: the fractional Laplacian naturally surfaces inprobability, water waves, and lower dimensional obstacleproblems (among others). In statistical mechanics it is a way totake into account long-range particle interactions.

Difficulty: The operator is nonlocal, hence one needs toestimate also the contribution coming from far. Also,integrating is usually harder than differentiating.

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Goal: Understand the geometric properties of the solutions of

(−∆)su = u− u3.

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When s = 1, the equation

−∆u = u− u3

is named after Allen-Cahn (or Ginzburg-Landau, orModica-Mortola...) and it is a model for phase transitions.The pure phases correspond to u ∼ +1 and u ∼ −1.The set in which u ∼ 0 is the interface which separates the purephases.

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−∆u = u− u3


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−∆u = u− u3


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Let u be a smooth bounded solution of

−∆u = u− u3

in Rn, with∂xnu > 0.

Is it true that u depends only on one Euclidean variable?

I.e. ∃uo : R → R and ω ∈ Sn−1 such that u(x) = uo(ω · x)?

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−∆u = u− u3




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−∆u = u− u3




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The answer is YES when n ≤ 3 and NO when n ≥ 9.The answer is also YES when n ≤ 8 and

limxn→±∞

u(x′, xn) = ±1.

The answer is also YES for any n if

limxn→±∞

u(x′, xn) = ±1

uniformly.

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limxn→±∞

u(x′, xn) = ±1.


limxn→±∞

u(x′, xn) = ±1

uniformly.

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limxn→±∞

u(x′, xn) = ±1.


limxn→±∞

u(x′, xn) = ±1

uniformly.

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This is due to the work of many outstanding mathematicians(Ambrosio, Barlow, Bass, Berestycki, Cabré, Caffarelli, DelPino, Farina, Ghoussub, Gui, Hamel, Kowalczyk, Modica,Monneau, Nirenberg, Savin, Wei, etc.)

The problem is still open in dimension 4 ≤ n ≤ 8 if the extraassumptions are dropped.

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Enrico Valdinoci

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This is due to the work of many outstanding mathematicians(Ambrosio, Barlow, Bass, Berestycki, Cabré, Caffarelli, DelPino, Farina, Ghoussub, Gui, Hamel, Kowalczyk, Modica,Monneau, Nirenberg, Savin, Wei, etc.)

The problem is still open in dimension 4 ≤ n ≤ 8 if the extraassumptions are dropped.

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One can ask a similar question for the fractional Laplacian:

Let s ∈ (0, 1) and u be a smooth bounded solution of

(−∆)su = u− u3



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One can ask a similar question for the fractional Laplacian:

Let s ∈ (0, 1) and u be a smooth bounded solution of

(−∆)su = u− u3



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In this case, Cabré and Solà-Morales proved that the answer isYES when n = 2 and s = 1/2.

Also YES when n = 2 and any s ∈ (0, 1) (Cabré, Sire and V.)

and when n = 3 and s ∈ [1/2, 1) (Cabré and Cinti).

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Also YES for any n and any s ∈ (0, 1) if

limxn→±∞

u(x′, xn) = ±1

uniformly (Farina and V., Cabré and Sire).

The problem is open for n ≥ 4, and even for n = 3 ands ∈ (0, 1/2) (and no counterexamples are known in anydimension).

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Also YES for any n and any s ∈ (0, 1) if

limxn→±∞

u(x′, xn) = ±1

uniformly (Farina and V., Cabré and Sire).

The problem is open for n ≥ 4, and even for n = 3 ands ∈ (0, 1/2) (and no counterexamples are known in anydimension).

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If one cannot prove symmetry, it is still good to have someinformation on the measure of the level sets (i.e., on theprobability of finding some phase in a given region).

For the case of the Laplacian, these density estimates wereobtained by Caffarelli and Cordoba.

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If one cannot prove symmetry, it is still good to have someinformation on the measure of the level sets (i.e., on theprobability of finding some phase in a given region).

For the case of the Laplacian, these density estimates wereobtained by Caffarelli and Cordoba.

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For the fractional Laplacian, here is a density estimate (Savinand V.):

If uε minimizes Fε in Br and u(0) = 0 then

|uε > 1/2 ∩ Br| ≥ c rn

provided that ε ≤ cr.

Here, Fε is the (rescaled) associated energy functional.

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For the fractional Laplacian, here is a density estimate (Savinand V.):

If uε minimizes Fε in Br and u(0) = 0 then

|uε > 1/2 ∩ Br| ≥ c rn

provided that ε ≤ cr.

Here, Fε is the (rescaled) associated energy functional.

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The scaling of the energy functional Fε is chosen to satisfy thefollowing asymptotics:

If s ∈ [1/2, 1), the functional Fε Γ-converges to the perimeterfunctional.

If s ∈ (0, 1/2), the functional Fε Γ-converges to the nonlocalperimeter functional (as introduced by Caffarelli, Roquejoffreand Savin).A minimizer uε converges a.e. to a step function χE − χRn\E,and the level sets of uε converge to ∂E locally uniformly.

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For Γ-convergence of nonlocal functionals related with phasetransitions, see also Alberti, Bellettini, Bouchitté, Garroni,González, Seppecher, etc.

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For any s ∈ (0, 1/2) the s-perimeter of a set E inside a givendomain Ω is defined by

Pers(E, Ω) :=

ZE∩Ω

Z(CE)∩Ω

1|x − y|n+2s dy dx

+

ZE∩Ω

Z(CE)∩(CΩ)

1|x − y|n+2s dy dx

+

ZE∩(CΩ)

Z(CE)∩Ω

1|x − y|n+2s dy dx,

where C means the complement (see Caffarelli, Roquejoffreand Savin).

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Then (see Caffarelli and V.), if E is a smooth set,

lims→(1/2)−

s(1− 2s)Pers(E, Br) = Per(E, Br)

for a dense set of r’s.Also, if Ek are minimal for Persk and sk → (1/2)−, then Ek

converges to some set E which is minimal for Per.

Results of these type may be given in the Γ-convergence sense(Ambrosio, De Philippis and Martinazzi).

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lims→(1/2)−





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lims→(1/2)−





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(Caffarelli and V.) There exists ε? > 0, such that if

∂E ∩ B1 ⊆ |xn| ≤ ε?

then ∂E is a C1,α-graph in the en-direction.

Differently from Caffarelli, Roquejoffre and Savin, here ε?

does not depend on s.

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(Caffarelli and V.) There exists ε? > 0, such that if

∂E ∩ B1 ⊆ |xn| ≤ ε?

then ∂E is a C1,α-graph in the en-direction.

Differently from Caffarelli, Roquejoffre and Savin, here ε?

does not depend on s.

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There exists ε ∈ (0, 1/2) such that, if

12− ε < s <

12,

then we have:I if n ≤ 7, any s-minimal cone is a hyperplane and any

s-minimal surface is C1,α;I in any dimension, any s-minimal surface is C1,α possibly

outside a closed set Σ, with Hd(Σ) = 0 for any d > n− 8.

Open problem: find ε! (no estimate available!)

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Enrico Valdinoci

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There exists ε ∈ (0, 1/2) such that, if

12− ε < s <

12,

then we have:I if n ≤ 7, any s-minimal cone is a hyperplane and any

s-minimal surface is C1,α;I in any dimension, any s-minimal surface is C1,α possibly

outside a closed set Σ, with Hd(Σ) = 0 for any d > n− 8.

Open problem: find ε! (no estimate available!)

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The only full-regularity result available is in dimension 2,where ε = 1/2:

(Savin and V.) Let s ∈ (0, 1/2). If n = 2 any s-minimal cone isa straight line and any s-minimal surface is C1,α.

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Further regularity:

(Barrios Barrera, Figalli and V.) C1,α =⇒ C∞.

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It would be desirable to better understand the behavior ofnonlocal minimal perimeter sets and to exploit their rigid (?)geometry in order to obtain information on the level sets of u...

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(Non)local phase transitions and minimal perimeter interfaces · 2012. 10. 3. · Uniform...

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