Mathematical Challenges Motivated by Multi-Phase Materials: Analytical, Stochastic and Discrete Aspects Anogia, Crete, June 21-26, 2009 Symmetry in phase transition Alberto FARINA Universit´ e de Picardie Jules Verne LAMFA, CNRS UMR 6140 Amiens, France
De Giorgi’s conjecture (1978) : Let us consider a solution R N , I u ∈ C 2 (I R) of ∆ u = u 3 − u, (1) such that ∂u | u | ≤ 1 , > 0 ∂x N R N . in the whole I Is it true that all the level sets of u are hyperplanes, at least if N ≤ 8 ? • Equivalently, De Giorgi’s conjecture can be reformulated by saying that the considered solution u is 1D , that is, it depends only on one variable (up to rotations).
The PDE in (1) is the well-known Allen-Cahn equation. It has important physical applications : • study of interfaces in both gasses and solids, [Allen-Cahn (1979), Rowlinson (1979)] • theory of superconductors and superfluids, [Landau (1967), Ginzburg-Pitaevskii (1958)] • cosmology [Gibbons-Townsend (1999)].
(Crude) physical motivation We are given some substance in some container, called, say, Ω, which may exhibit two phases, which we label with “ − 1” and “+1”, and we would like to describe mathematically the pattern and the separation of such phases . The classical Van der Waals/Cahn-Hilliard theory tell us that interface formation is driven by a variational principle, that is the pattern is the outcome of the minimization of an energy of the form : � E o ( u ; Ω) = W ( u ( x )) dx Ω • W is some “double well” function, i.e., such that W ( ± 1) = 0 and W ( r ) > 0 if r � = ± 1 • u ( x ) represents the states of the substance at the point x ∈ Ω. The typical ex. is W ( u ) = ( u 2 − 1) 2 (Allen-Cahn model). 4
One quickly realizes that this cannot be a satisfactory model, since any function attaining only the values − 1 and +1 mini- mizes the energy E o . In particular, the separation between the two phases could be as wild as possible and the energy would not be affected! Since this is not the case in physical applications one has to add to E o , a term that “penalizes” the formation of unnecessary interfaces. This may be accomplished by adding to E o a small gradient term, that is by looking at the energy � ǫ 2 |∇ u ( x ) | 2 + 1 E ǫ ( u ; Ω) = ǫ W ( u ( x )) dx, Ω where ǫ > 0 is a small parameter. Such small gradient term indeed cuts the interfaces as much as possible, in the sense that the minimizers of E ǫ turn out to be smooth functions, taking values between − 1 and +1, and whose level sets approach (in a suitable way) hypersurfaces of least possible area, when ǫ → 0 + [Modica-Mortola (1977), Modica (1987), Caffarelli-Cordoba (1995)].
In this physical interpretation, De Giorgi’s Conjecture states that, at least in low dimension, global phase transitions have a flat interface, or, by a blow-up argument, that the phase, locally, just depends on the distance from the interface. Possible motivation for the conjecture Let u be as in De Giorgi’s conjecture, ǫ > 0 and let u ǫ ( x ) := u ( x/ǫ ) The monotonicity assumption in De Giorgi’s conjecture seems to suggest that : R N − 1 . • the level sets of u (and thus those of u ǫ ) are graphs over I • the phase transition happens in a straight, minimal way.
Thus, when ǫ → 0 + , the level sets of u ǫ are closer and closer to R N − 1 , i.e. a solution of global minimal graphs ϕ ǫ over I � � ∇ ϕ ǫ R N − 1 . − div = 0 in I � 1 + |∇ ϕ ǫ | 2 Since global minimal graphs are flat for N − 1 ≤ 7, due to Bernstein-type Theorems, it follows that the level sets of u ǫ are close to a flat hyperplane. N ≤ 8 is crucial ! Here Now, since elliptic problems are somehow “rigid”, we may suspect that once { u ǫ = c } is close enough to a hyperplane, it is a hyperplane itself. By scaling back, this would give that { u = c } is a hyperplane. Then, the level sets of u would be parallel hyperplanes and thus u would be 1D, as asked by De Giorgi’s conjecture.
In the previous (heuristic) argument several gaps have to be filled : • no minimality condition is explicitly required in De Giorgi’s conjecture, so the results about the asymptotic behavior of min- imizers are not directly applicable. • The monotonicity condition does not assure, in principle, that R N − 1 . the level sets of u are global graphs over I • One would need to proof the rigidity argument.
Available results De Giorgi’s conjecture is settled for • N = 2 [Ghoussoub-Gui (1998)], • N = 3 [Ambrosio-Cabr´ e (2000)]. When • 4 ≤ N ≤ 8, the conjecture is open and no counterexample is available (not even for semilinear equations different from the Allen-Cahn equation). • N ≥ 9, Del Pino, Kowalczyk, Wei have constructed, in a recent work, a solution of (1) satisfying ∂u | u | ≤ 1 , > 0 ∂x N which is not 1D . This implies that the assumption N ≤ 8 in De Giorgi’s conjecture cannot be removed.
Stability R N and assume f ∈ C 1 . Let Ω be a domain of I A solution u ∈ C 2 (Ω) of − ∆ u = f ( u ) in Ω is stable if � |∇ ψ | 2 − f ′ ( u ) ψ 2 ≥ 0 . ∀ ψ ∈ C 1 c (Ω) Q u ( ψ ) := Ω • u local minimizer = ⇒ u stable , • u monotone solution = ⇒ u stable .
The case N = 2 Theorem 1 [F.- Sciunzi - Valdinoci (2008)] Assume N = 2 . R 2 ) , with Let f be locally Lipschitz-continuous, and let u ∈ C 2 (I R 2 ) , be a stable solution of |∇ u | ∈ L ∞ (I R 2 . − ∆ u = f ( u ) I (2) in Then, u is 1 D. Theorem 1 provides different generalizations with respect to the question raised by De Giorgi : • the Allen-Cahn equation is replaced by the the equation (2) with f ∈ C 0 , 1 loc , • the assumption of monotonicity is weakened to stability, • the solution is not necessarily bounded (only a bound on the gradient is needed: for instance linear functions like u ( x ) = x 1 are unbounded monotone solutions of − ∆ u = 0, with bounded gradient).
• Theorem 1 also holds for quasilinear singular and degenerate equations. • Theorem 1 is not true if stability is removed. R 2 there are saddle solutions that are not stable [Dang- In I Fife-Peletier (1992)]. • The assumption |∇ u | ∈ L ∞ is necessary in Theorem 1. The function u ( x, y ) = y − x 2 2 satisfies : R 2 , − ∆ u = 1 in I R 2 , ∂u ∂y = 1 > 0 in I √ 1 + x 2 �∈ L ∞ (I R 2 ) , |∇ u | = and it is not 1D .
Sketch of proof Step 1. Each level set of u is a smooth manifold on the set { ∇ u � = 0 } , hence we can introduce the principal curvatures κ 1 , . . . , κ N − 1 at any point of such manifold and set K 2 := κ 2 1 + . . . + κ 2 N − 1 , ∇ T = tangential gradient along the level set of u . Theorem [Sternberg-Zumbrun (1998)] Under the assump- tions of Theorem 1 we have � � � |∇ u | 2 K 2 + |∇ T |∇ u || 2 � ϕ 2 ≤ 2 |∇ u | 2 |∇ ϕ | 2 (3) I R { ∇ u � =0 } R 2 ) . for any ϕ ∈ C 0 , 1 c (I
Step 2 - Proof of Theorem 1. Given R ≥ 1, we set R ( x ) + 2 ln( R/ | x | ) ϕ R ( x ) := χ B √ χ B R \ B √ R ( x ) . ln R Plug ϕ R inside (3) to obtain C ′ � C � 1 � |∇ u | 2 K 2 + |∇ T |∇ u || 2 � ≤ | x | 2 ≤ (ln R ) 2 log R B √ B R \ B √ R R for appropriate C , C ′ > 0, since |∇ u | is bounded. By taking R arbitrarily large, we have that K = |∇ T |∇ u || ≡ 0.
The case N = 3 Using similar argument we have Theorem 2 [F.- Sciunzi - Valdinoci (2008)] Assume N = 3 . R 3 ) ∩ L ∞ (I R 3 ) be a Let f be locally Lipschitz and let u ∈ C 2 (I solution of R 3 , − ∆ u = f ( u ) in I R 3 , ∂u ∂x 3 > 0 in I Then u is 1D . R 3 , stability ⇒ 1D ? Open question : In I
The case 4 ≤ N ≤ 8 Theorem [Savin (2003, 2008)] Let u be as requested in De Giorgi’s conjecture and let N ≤ 8 . Assume, furthermore that x N → + ∞ u ( x ′ , x N ) = 1 lim and x N →−∞ u ( x ′ , x N ) = − 1 . lim Then, u is 1D . Now we would like to discuss how the above extra conditions may be weakened.
Since u is supposed to be increasing in the N th variable and bounded, we may denote the space variable as x = ( x ′ , x N ) ∈ R N − 1 × I I R and set u ( x ′ ) := x N → + ∞ u ( x ′ , x N ) lim and u ( x ′ ) := x N →−∞ u ( x ′ , x N ) . lim • u and u are the “profiles” of our solution at infinity, R N − 1 . • they satisfy the same equation (1) in I Theorem 3 [F.- Valdinoci (2008)] Let u be as requested in De Giorgi’s conjecture. Suppose that both u and u are 2D . Then, u is identically +1 and u is identically − 1 . Also, if N ≤ 8 , then u is 1D .
When N ≤ 4 the assumptions of Theorem 3 may be weak- ened further on, since it is enough to require only that either u or u is 2D (instead of both). Theorem 4 [F.- Valdinoci (2008)] Assume N ≤ 4 and let u be as requested in De Giorgi’s conjecture. Suppose that either u or u is 2D . Then, u is 1D .
The case in which the level sets are global graphs R N − 1 , • We say that the level set { u = c } is a global graph over I R N − 1 → I whenever there exists Γ : I R such that R N − 1 × I { u = c } = { ( x ′ , x N ) ∈ I R : x N = Γ( x ′ ) } . Theorem 5 [F.- Valdinoci (2008)] Assume N ≤ 8 and let u be as requested in De Giorgi’s conjecture. R N − 1 . Suppose that a level set { u = c } is a global graph over I Then, u is 1D .
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