(Non)local phase transitions and minimal surfaces

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1 Dipartimento di Matematica Italian-Japanese workshop Cortona, June 2011

2 Outline of the talk

3 Outline of the talk

4 ( ) s u = 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) ).

5 ( ) s u = 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) ).

6 ( ) s u = 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) ).

7 An equivalent definition may be given by integrating against a singular kernel, which suitably averages a second-order incremental quotient: ( ) s u(x + y) + u(x y) 2u(x) u(x) = R n y n+2s dy. Up to a factor 2, this is the same as defining the operator as an integral in the principal value sense ( ) s u(x + y) u(x) u(x) = lim ɛ 0 + R n \B ɛ y n+2s dy.

8 An equivalent definition may be given by integrating against a singular kernel, which suitably averages a second-order incremental quotient: ( ) s u(x + y) + u(x y) 2u(x) u(x) = R n y n+2s dy. Up to a factor 2, this is the same as defining the operator as an integral in the principal value sense ( ) s u(x + y) u(x) u(x) = lim ɛ 0 + R n \B ɛ y n+2s dy.

9 An equivalent definition may be given by integrating against a singular kernel, which suitably averages a second-order incremental quotient: ( ) s u(x + y) + u(x y) 2u(x) u(x) = R n y n+2s dy. Up to a factor 2, this is the same as defining the operator as an integral in the principal value sense ( ) s u(x + y) u(x) u(x) = lim ɛ 0 + R n \B ɛ y n+2s dy.

10 Motivation: the fractional Laplacian naturally surfaces in probability, water waves, and lower dimensional obstacle problems (among others). In statistical mechanics it is a way to take into account long-range particle interactions. Difficulty: The operator is nonlocal, hence one needs to estimate also the contribution coming from far. Also, integrating is usually harder than differentiating.

11 Motivation: the fractional Laplacian naturally surfaces in probability, water waves, and lower dimensional obstacle problems (among others). In statistical mechanics it is a way to take into account long-range particle interactions. Difficulty: The operator is nonlocal, hence one needs to estimate also the contribution coming from far. Also, integrating is usually harder than differentiating.

12 Goal: Understand the geometric properties of the solutions of ( ) s u + u u 3 = 0.

13 The Allen-Cahn equation When s = 1, the equation u + u u 3 = 0 is named after Allen-Cahn (or Ginzburg-Landau, or Modica-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 pure phases.

14 The Allen-Cahn equation When s = 1, the equation u + u u 3 = 0 is named after Allen-Cahn (or Ginzburg-Landau, or Modica-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 pure phases.

15 The Allen-Cahn equation When s = 1, the equation u + u u 3 = 0 is named after Allen-Cahn (or Ginzburg-Landau, or Modica-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 pure phases.

16 A conjecture of De Giorgi Let u be a smooth bounded solution of u + u u 3 = 0 in R n, with xn u > 0. Is it true that u depends only on one Euclidean variable? I.e. u o : R R and ω S n 1 such that u(x) = u o (ω x)?

17 A conjecture of De Giorgi Let u be a smooth bounded solution of u + u u 3 = 0 in R n, with xn u > 0. Is it true that u depends only on one Euclidean variable? I.e. u o : R R and ω S n 1 such that u(x) = u o (ω x)?

18 A conjecture of De Giorgi Let u be a smooth bounded solution of u + u u 3 = 0 in R n, with xn u > 0. Is it true that u depends only on one Euclidean variable? I.e. u o : R R and ω S n 1 such that u(x) = u o (ω x)?

19 A conjecture of De Giorgi The answer is YES when n 3 and NO when n 9. The answer is also YES when n 8 and lim x u(x, x n ) = ±1. n ± The answer is also YES for any n if uniformly. lim x u(x, x n ) = ±1 n ±

20 A conjecture of De Giorgi The answer is YES when n 3 and NO when n 9. The answer is also YES when n 8 and lim x u(x, x n ) = ±1. n ± The answer is also YES for any n if uniformly. lim x u(x, x n ) = ±1 n ±

21 A conjecture of De Giorgi The answer is YES when n 3 and NO when n 9. The answer is also YES when n 8 and lim x u(x, x n ) = ±1. n ± The answer is also YES for any n if uniformly. lim x u(x, x n ) = ±1 n ±

22 A conjecture of De Giorgi This is due to the work of many outstanding mathematicians (Ambrosio, Barlow, Bass, Berestycki, Cabré, Caffarelli, Del Pino, Farina, Ghoussub, Gui, Hamel, Kowalczyk, Modica, Monneau, Nirenberg, Savin, Wei, etc.) The problem is still open in dimension 4 n 8 if the extra assumptions are dropped.

23 A conjecture of De Giorgi This is due to the work of many outstanding mathematicians (Ambrosio, Barlow, Bass, Berestycki, Cabré, Caffarelli, Del Pino, Farina, Ghoussub, Gui, Hamel, Kowalczyk, Modica, Monneau, Nirenberg, Savin, Wei, etc.) The problem is still open in dimension 4 n 8 if the extra assumptions are dropped.

24 The symmetry problem for ( ) s One can ask a similar question for the fractional Laplacian: Let s (0, 1) and u be a smooth bounded solution of ( ) s u + u u 3 = 0 in R n, with xn u > 0. Is it true that u depends only on one Euclidean variable?

25 The symmetry problem for ( ) s One can ask a similar question for the fractional Laplacian: Let s (0, 1) and u be a smooth bounded solution of ( ) s u + u u 3 = 0 in R n, with xn u > 0. Is it true that u depends only on one Euclidean variable?

26 The symmetry problem for ( ) s In this case, Cabré and Solà-Morales proved that the answer is YES 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).

27 The symmetry problem for ( ) s In this case, Cabré and Solà-Morales proved that the answer is YES 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).

28 The symmetry problem for ( ) s In this case, Cabré and Solà-Morales proved that the answer is YES 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).

29 The symmetry problem for ( ) s Also YES for any n and any s (0, 1) if lim x u(x, x n ) = ±1 n ± uniformly (Farina and V., Cabré and Sire). The problem is open for n 4, and even for n = 3 and s (0, 1/2) (and no counterexamples are known in any dimension).

30 The symmetry problem for ( ) s Also YES for any n and any s (0, 1) if lim x u(x, x n ) = ±1 n ± uniformly (Farina and V., Cabré and Sire). The problem is open for n 4, and even for n = 3 and s (0, 1/2) (and no counterexamples are known in any dimension).

31 Density estimates for the level sets If one cannot prove symmetry, it is still good to have some information on the measure of the level sets (i.e., on the probability of finding some phase in a given region). For the case of the Laplacian, these density estimates were obtained by Caffarelli and Cordoba.

32 Density estimates for the level sets If one cannot prove symmetry, it is still good to have some information on the measure of the level sets (i.e., on the probability of finding some phase in a given region). For the case of the Laplacian, these density estimates were obtained by Caffarelli and Cordoba.

33 Density estimates for the level sets For the fractional Laplacian, here is a density estimate (Savin and V.): If u ɛ minimizes F ɛ in B r and u(0) = 0 then {u ɛ > 1/2} B r c r n provided that ɛ cr. Here, F ɛ is the (rescaled) associated energy functional.

34 Density estimates for the level sets For the fractional Laplacian, here is a density estimate (Savin and V.): If u ɛ minimizes F ɛ in B r and u(0) = 0 then {u ɛ > 1/2} B r c r n provided that ɛ cr. Here, F ɛ is the (rescaled) associated energy functional.

35 Γ-convergence for ( ) s The scaling of the energy functional F ɛ is chosen to satisfy the following asymptotics: If s [1/2, 1), the functional F ɛ Γ-converges to the perimeter functional. If s (0, 1/2), the functional F ɛ Γ-converges to the nonlocal perimeter functional (as introduced by Caffarelli, Roquejoffre and Savin). A minimizer u ɛ converges a.e. to a step function χ E χ R n \E, and the level sets of u ɛ converge to E locally uniformly.

36 Γ-convergence for ( ) s The scaling of the energy functional F ɛ is chosen to satisfy the following asymptotics: If s [1/2, 1), the functional F ɛ Γ-converges to the perimeter functional. If s (0, 1/2), the functional F ɛ Γ-converges to the nonlocal perimeter functional (as introduced by Caffarelli, Roquejoffre and Savin). A minimizer u ɛ converges a.e. to a step function χ E χ R n \E, and the level sets of u ɛ converge to E locally uniformly.

37 Γ-convergence for ( ) s The scaling of the energy functional F ɛ is chosen to satisfy the following asymptotics: If s [1/2, 1), the functional F ɛ Γ-converges to the perimeter functional. If s (0, 1/2), the functional F ɛ Γ-converges to the nonlocal perimeter functional (as introduced by Caffarelli, Roquejoffre and Savin). A minimizer u ɛ converges a.e. to a step function χ E χ R n \E, and the level sets of u ɛ converge to E locally uniformly.

38 Γ-convergence for ( ) s The scaling of the energy functional F ɛ is chosen to satisfy the following asymptotics: If s [1/2, 1), the functional F ɛ Γ-converges to the perimeter functional. If s (0, 1/2), the functional F ɛ Γ-converges to the nonlocal perimeter functional (as introduced by Caffarelli, Roquejoffre and Savin). A minimizer u ɛ converges a.e. to a step function χ E χ R n \E, and the level sets of u ɛ converge to E locally uniformly.

39 Γ-convergence for ( ) s For Γ-convergence of nonlocal functionals related with phase transitions, see also Alberti, Bellettini, Bouchitté, Garroni, González, Seppecher, etc.

40 Few words on the nonlocal perimeter For any s (0, 1/2) the s-perimeter of a set E inside a given domain Ω is defined by Z Z 1 Per s(e, Ω) := dy dx E Ω (CE) Ω x y n+2s Z Z 1 + dy dx E Ω (CE) (CΩ) x y n+2s Z Z 1 + dy dx, x y n+2s E (CΩ) (CE) Ω where C means the complement (see Caffarelli, Roquejoffre and Savin).

41 Few words on the nonlocal perimeter Then (see Caffarelli and V.), if E is a smooth set, lim s(1 2s)Per s(e, B r ) = Per(E, B r ) s (1/2) for a dense set of r s. Also, if E k are minimal for Per sk and s k (1/2), then E k 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).

42 Few words on the nonlocal perimeter Then (see Caffarelli and V.), if E is a smooth set, lim s(1 2s)Per s(e, B r ) = Per(E, B r ) s (1/2) for a dense set of r s. Also, if E k are minimal for Per sk and s k (1/2), then E k 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).

43 Few words on the nonlocal perimeter Then (see Caffarelli and V.), if E is a smooth set, lim s(1 2s)Per s(e, B r ) = Per(E, B r ) s (1/2) for a dense set of r s. Also, if E k are minimal for Per sk and s k (1/2), then E k 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).

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

Sharp energy estimates and 1D symmetry for nonlinear equations involving fractional Laplacians

$Sharp energy estimates and 1D symmetry for nonlinear equations involving fractional Laplacians$ Sharp energy estimates and 1D symmetry for nonlinear equations involving fractional Laplacians Eleonora Cinti Università degli Studi di Bologna - Universitat Politécnica de Catalunya (joint work with Xavier