Allen-Cahn equation in periodic media. Exercises 1

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1 Allen-Cahn equation in periodic media Exercises 1 Annalisa Massaccesi (Universita degli Studi di Milano) Intensive Programme / Summer School Periodic Structures in Applied Mathematics Göttingen, August 18 31, 2013 This project has been funded with support from the European Commission. This publication [communication] reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Grant Agreement Reference Number: D-2012-ERA/MOBIP

2 Notes on Γ-convergence and the Allen-Cahn equation Annalisa Massaccesi August 30, De Giorgi s conjecture on the Allen-Cahn equation 1.1 De Giorgi s conjecture Allen-Cahn equation u R n ] 1,1[ solving In the Ginzburg-Landau model u=f (u). (1.1) u=u 3 u, (1.2) that is, F(u)=(1 u 2 ) 2 /4, but in general we allow any double well potential. De Giorgi s conjecture u R n ] 1,1[ entire solution of the semilinear equation (1.2) with xn u>0, then{u=s} are hyperplanes s ] 1,1[. A serious discussion about AC solutions is beyond the scope, but we can t avoid a few remarks. Remark 1.1.{u=s} are hyperplanes u depends only on 1 variable ν R n and a R such that u(x)=tanh( 1 2 ν,x +a), in fact u(x)=u( ν,x ) and we solve (1.2), which turns out to be a second order nonlinear ODE. 1

3 Remark 1.2. xn u>0 means that{u=s} are locally graphs with respect to x. Any additional information about these graphs is useful for the study of the behaviour of the ratio σ i = x i u xn u. For instance, equi-lipschitz graphs give a bounded σ (Modica-Mortola s result). Remark 1.3. In order to conclude, one often needs a Liouville-type theorem (remember we have entire solutions of (1.2)), in the above mentioned case there s a Liouville-type theorem for non-uniformly elliptic equations in divergence form. Remark 1.4. Thanks to Berestycki-Caffarelli-Nirenberg techniques(symmetry properties of positive solutions of semilinear elliptic equations) DG s conjecture has been established in n=2 (Ghoussoub-Gui) and n=3 (Ambrosio- Cabré). Remark 1.5. From now on the AC equation becomes only the Euler-Lagrange equation of a non-convex energy E(v,Ω) = ω v 2 +F(v). Wrongly enough, we will confuse AC solutions with energy minimizers! 1.2 Asymptotic version of DG s conjecture Asymptotic conjecture u entire solution of AC equation (1.2) with xn u>0 (as above) and n 8, then{u=s} are flat at. The rest of these notes will be spent to understand why De Giorgi put this unusual bound on the dimension: what was in De Giorgi s mind? 2 Connection with the Bernstein problem of area-minimizing graphs 2.1 The Bernstein problem Theorem 2.1 (Bombieri-De Giorgi-Giusti). ψ R m R entire solution of div ψ =0 (2.1) 1+ ψ 2 2

4 is affine if m 7. If m 8, anon-affine entire solution of (2.1) (Simons cone). Remark 2.2. Take n=m+1, in fact we want to know something about {u=s} R n as graphs on x R n 1. The problem is how to pass from AC to Bernstein: (i) blow down; (ii) suitable variational convergence. 2.2 Blow down The blow down goes in the opposite direction of a blow up! Like looking at the solutions from. In formulas u R (x) =u(rx) and something interesting happens as R +. Remark 2.3. u R solves 1 R u R=RF (u R ), (2.2) indeed u R (x)=u(rx) implies u R (x)=r 2 u(rx)=r 2 F (u(rx))=r 2 F (u R (x)). Something even more interesting is happening to the energy E R (v,ω)= 1 2R v 2 +RF(v). 2.3 Modica-Mortola s result Theorem 2.4 (Modica-Mortola). E R (v,ω) Γ-converges to a multiple of the perimeter P(E, Ω). Reminder The perimeter of a set is the total variation of its characteristic function. In the regular cases we have a representation theorem giving Heuristically (i){u R =s} B r {u=s} B Rr ; P(E,Ω)=H n 1 (Ω E). (ii){u R =s} is closer and closer to a minimal graph. Then{u=s} is flat at for m=n

5 2.4 What s going wrong? See Alberti-Ambrosio-Cabré, but (a) We are passing to the limit on a subsequence of radii R, we don t know if the direction ν depends on the subsequence. (b) We would prove that 1 R n 1 BR u 2 ν u 2 R 0, but no info on pointwise convergence. 3 Γ-convergence 3.1 Definition and main aims Remark 3.1. Why a notion of variational convergence? Two directions: (i) given a difficult functional you want to know something about its minimizers knowing the minimizers of approximating functionals(think about numerical analysis...); (ii) vice versa, you re interested in the essential behaviour of a family of problems, like blow-ups, blow down... Ideal statement F ε Γ-converges to F 0 and x ε x 0, with x ε minimizers, then F 0 (x 0 ) F 0 (x) x X. Let s put something in the middle! (Then we will take care of the details...) where F 0 (x 0 ) (1) liminff ε (x ε ) liminf F ε (x ε ) limsupf ε (x ε ) (2) F 0 (x) (3.1) (1) x ε x F 0 (x) liminf F ε (x ε ) (sort of lower semicontinuity for both F ε and x ε ); (2) x x ε x such that limsupf ε (x ε ) F 0 (x) (the existence of a recovery sequence gives a sharp lower bound). 4

6 Remark 3.2. As a first issue, we have to hope for equicoerciveness of F ε : choose x ε such that F ε (x ε ) inff ε +o(1), you need to know that x ε x 0 up to subsequences, in some way. Here you can recognize all the compactness issues of the direct method in Calculus of Variations. Definition 3.3. F ε X R is equi-coercive if t R K t compact such that {F ε t} K t. Equivalently ε j 0 and x j with F εj (x j ) t asubsequence such that F εj (x j) F εj (x j )+o(1) with x j converging somewhere. Remark 3.4. As in classical CalcVar, equicoerciveness and lower semicontinuity are in competition. If the convergence is weaker, then you have a stronger compactness, in spite of lower semicontinuity. Definition 3.5. X separable metric space 1 and F ε,f X R, F ε Γ F iff x X (1) x ε x F(x) liminf F ε (x ε ); (2) x ε x F(x) limsup F ε (x ε ) (actually F(x)=lim F ε (x ε )). Theorem 3.6.(X,d) metric space,(f ε ) equi-coercive sequence of functionals X R, F=Γ lim F ε, then min X F=lim min X F ε. Moreover, if(x ε ) is a precompact sequence such that limf ε (x ε )=liminf F ε, X then every cluster point of(x ε ) is a minimizer for F. Remark 3.7. Γ-convergence gives a choice criterion for minimizers. Γ Remark 3.8. F ε F and G X R continuous (with respect to the distance d on X), then F ε +G F+G. Γ 1 Life is easier! 5

7 Remark 3.9. Mind the dependence on the metric, later we ll see an example (in Remark 3.15). Definition Equivalently to Def. 3.5, we can say the following: F ε X R, let s define Γ liminf Γ limsup F ε (x) = inf{liminf F ε (x ε ) x ε x} F ε (x) = inf{limsupf ε (x ε ) x ε x}, then F=Γ lim F ε iff Γ liminf F ε =Γ limsup F ε =F. Remark Now it is really unavoidable to notice that a Γ-limit is always lower semicontinuous. Indeed, if x (n) x (sequences just for the sake of a nice notation), x (n) ε x (n) such that lim inf F ε (x (n) ε ) limsup F ε (x (n) ε ) F(x (n) ), then, taking a diagonal subsequence x εn x, we get F(x) liminf n F ε n (x εn ) liminf n F(x(n) ). Remark The Γ-limit of a constant sequence is not the functional itself, but its lower semicontinuous envelope, in fact (from Def. 3.10) Γ lim F(x)=inf{liminfF ε (x ε ) x ε x}, which is precisely the relaxation operation (biggest lower semicontinuous functional under F). Remark Pointwise convergence and Γ-convergence are different: if F ε G pointwise and F = Γ lim F ε, then F G. However, uniform convergence to some continuous functional implies Γ-convergence. 3.2 Problems in Γ-convergence Example 3.14 (Homogenization). F ε (u) ={ Ω f ε(x,du) u W 1,p (Ω,R m ) + otherwise with f ε (x,ξ) =f( x ε,ξ). If f is periodic, we have a homogenized functional. 6

8 Remark There are two problems in Γ-convergence, related. (i) The topology and the choice of the convergence. (ii) The correct way to scale the energy. This remark brings back to Modica-Mortola energy As for the scaling problem: take F ε (u)= Ω ε u 2 2 G ε = Ω ε 2 u W(u) ε +W(u). (3.2) instead of (3.2), that is, consider the L 2 convergence. Then where W is the convex hull of W. G ε Γ Ω W, 4 Modica-Mortola s result 4.1 The theorem, with proof in dimension 1 Modica-Mortola s functional (3.2) belongs to those functionals, whose minimizers ask for sharp interfaces, in the limit we will get a partition of Ω in phase domains. Theorem 4.1 (Modica-Mortola). The functional (3.2) Γ-converges with respect to the L 1 norm to F 0 (u)={ c WP({u=1},Ω)=c W H n 1 ( {u=1} Ω) u {0,1} a.e. + otherwise and c W =2 1 0 W(s)ds. Proof. This proof holds for n=1. 7

9 MM s trick: 1 W(u j )+ε j u I ε j 2 2 W(uj ) u j I j 2 so η 1 η W(s)= cη η<<1, (4.1) 1 + >F εj (u j ) W(u j )+ε j u I I ε j 2 {transitions of u j }c η (4.2) j and Ω W(u j )= Ω W(t) {u j =t} dt εc {u j 0,1} j 0 (convergence in measure to some characteristic function). l.s.c.: when u j L 1 u, if u {0,1} then F(u)=+ and F εj (u j ) +, because ε j 0 as j. If u is a characteristic function, then borrowed from (4.2), is enough. rec. seq.: let s solve F εj (u j ) {trans. u j }c η, { v (s)= W(v(s)) v(0)=1/2. If v ε (t) = v( t ε ), then v ε Heaviside function and v ε is a recovery sequence for H because lim supf ε (v ε )=2 v Ω ε(t) 2 dt=2. Analogously for other u characteristic function with bounded variation. 4.2 Slicing A useful technique in Γ-convergence is slicing, because it allows to reduce the dimension. Fix ξ S n 1, we define π ξ = {z R n z,ξ =0} (orthogonal hyperplane) Ω ξ,y = {t R y+tξ Ω} u ξ,y (t) = u(y+tξ). 8

10 (1) We can localize F ε (,A) for every open set A Ω. (2) We choose F ξ,y ε in such a way that F ε (u,a) F ξ ε(u,a) = πξ F ξ,y ε (u ξ,y,a ξ,y )dh n 1 (y). (4.3) (3) Put and F ξ,y (v,i) =Γ liminf F ξ,y ε (v, I) F ξ (u,a) = πξ F ξ,y (u ξ,y,a ξ,y )dh n 1 (y). (4) By Fatou s Lemma we get lim inf F ε (u ε,a) liminf F ξ ε(u ε,a)=liminf πξ F ξ,y ε (u ξ,y,a ξ,y ) liminf 4.3 The proof in higher dimension Slicing: we localize We choose then F ε (u,a)= A W(u) ε F ξ,y ε ((u ε ) ξ,y,a ξ,y ) F ξ (u,a). (4.4) +ε A u 2. Fε ξ,y W(v) (v,i) = I ε +ε v 2, I F ξ ε(u,a)= πξ F ε (u ξ,y,a ξ,y )= A W(u) ε satisfying (4.3). So Γ liminf +ε A u,ξ 2 F ε (u,a), F ξ,y ε (v,i)=c W {jumps of v}= F ξ,y (v,i) and F ξ (u,a)= πξ F ξ,y. 9

11 Moreover by (4.4). Γ liminf F ε (u,a) F ξ (u,a)= πξ c W {jumps of u ξ,y } about the Γ liminf of F ε (u,a): Γ liminff ε (u,a)<+ if and only if u=χ E, with u BV. Then thus Γ lim inf F ξ (u,a)=c W ξ,ν dh n 1 (y), A E F ε (u,a) c W J(u) E sup ξ i,ν dh n 1 c W P({u=1},Ω), i for some dense sequence(ξ i ) S n 1. This covers lower semicontinuity. rec. seq.: if E is smooth u ε (x)=v( d(x) ε ), where d(x) =dist(x,ω E) dist(x,e), so that F ε (u ε )= W(u ε) ε +ε Du ε 2 and lim supf ε (u ε ) c W P({u=1},Ω). For E non-smooth, approximate it! References [1] G. Alberti, L. Ambrosio, X. Cabr: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math., 65 (2001), [2] A. Braides: Γ-convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.

12 [3] A. Braides: A handbook on Γ-convergence. Download from braides/handbook.pdf [4] L. Modica, S. Mortola: Un esempio di Γ -convergenza. Boll. Un. Mat. Ital., 14 (1977),