GAMMA CONVERGENCE AND APPLICATIONS TO PHASE TRANSITIONS

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1 GAMMA CONVERGENCE AND APPLICATIONS TO PHASE TRANSITIONS GIOVANNI LEONI. Liquid Liquid Phase Transitions Consider a fluid confined into a container R N. Assume that the total mass of the fluid is m, so that admissible density distributions u : R satisfy the constraint u(x) dx = m. The total energy is given by the functional u W (u(x)) dx, where W : R [0, ) is the energy per unit volume. Assume that W supports two phases a < b, that is, W is a double-well potential, with {t R : W (t) = 0} = {a, b}. Then any density distribution u that renders the body stable in the sense of Gibbs is a minimizer of the following problem { } (P 0 ) min W (u(x)) dx : u(x) dx = m. If L N () = and a < m < b, then given any measurable set E with () L N (E) = m a b a, the function u = bχ E +aχ \E is a solution of problem (P 0 ). Here L N stands for the N-dimensional Lebesgue measure. This lack of uniqueness is due the fact that interfaces between the two phases a and b are not penalized by the total energy. The physically preferred solutions should be the ones that arise as limiting cases of a theory that penalizes interfacial energy, so it is expected that these solutions should minimize the surface area of E. In the van der Walls Cahn Hilliard theory of phase transitions [CH958], [Ro979], [VdW893], the energy depends not only on the density u but also on its gradient, precisely, (2) G ε (u) := W (u(x)) dx + ε 2 u(x) 2 dx. Note that the gradient term penalizes rapid changes of the density u, and thus it plays the role of an interfacial energy. Stable density distributions u are now solutions of the minimization problem { } (P ε ) min W (u(x)) dx + ε 2 u(x) 2 dx, where the minimum is taken over all smooth functions u satisfying u(x) dx = m. In 983 Gurtin [Gu985] conjectured that the limits, as ε 0, of solutions u ε of (P ε ) are solutions u 0 of (P 0 ) with minimal surface area, that is, if u 0 = aχ E0 + bχ \E0, then (3) surface area of E 0 surface area of E

2 for every measurable set with L N (E) = m a. Moreover, he also conjectured that b a (4) G ε (u ε ) ε surface area of E 0. Using results of Modica and Mortola [MM977], this conjecture was proved independently for N 2 by Modica [Mo987] and by Sternberg [St988] in the setting of Γ-convergence. The one-dimensional case N = had been studied by Carr, Gurtin, and Slemrod in [CGS984]... Γ-Convergence. The notion of gamma convergence was introduced by De Giorgi in [DG975] (see also [Br2002], [DM993]). Definition.. Let (Y, d) be a metric space and consider a sequence {F n } of functions F n : Y [, ]. We say that {F n } Γ-converges to a function F : Y [, ] if the following properties hold: (i) (Liminf Inequality) For every y Y and every sequence {y n } Y such that y n y, (5) F(y) lim inf n + F n(y n ). (ii) (Limsup Inequality) For every y Y there exists {y n } Y such that y n y and (6) lim sup F n (y n ) F(y). The function F is called the Γ-limit of the sequence {F n }. Exercise.2. Let (Y, d) be a metric space and consider a sequence {F n } of functions F n : Y (, ). Assume that there exists min F n (z) = F n (y n ), z Y that {F n } Γ-converges to F, and that y n y for some y Y. Prove that there exists min z Y F (z) and that In applications the main challenges are F (y) = min F (z) = lim min F n (y). z Y y Y Finding the appropriate rescaling of F n and an appropriate metric d. These usually follow by studying equibounded sequences and by a compactness argument. Identifying the Γ-limit F. Proving (i) and (ii). In [MM977] Modica and Mortola studied the Γ-convergence of the sequence of functionals R ε sin2 (πu (x)) dx + ε u (x) 2 dx N R N with respect to the convergence in L ( R N). 2

3 Exercise.3. Take Y = L 2 () and assume that W (t) = t a b t. Study the Γ-convergence of the family of functionals (W (u) + ε G ε (u) := 2 u 2 ) dx if u W,2 () and u dx = m, otherwise in L 2 (). Exercise.4. Take Y = L 2 () and assume that W (t) = t a b t. Study the Γ-convergence of the family of functionals (W (u) + ε G ε (u) := 2 u 2 ) dx if u W,2 () and u dx = m, otherwise in L 2 () with respect to weak convergence in L 2 () Compactness. In view of (4), for ε > 0 we consider the rescaled functional defined by (7) F ε (u) := F ε : W,2 () [0, ] ( ) W (u) + ε u 2 dx, ε where the double well potential W : R [0, ) satisfies the following hypotheses: (H ) W is continuous, W (t) = 0 if and only if t {a, b} for some a, b R with a < b. (H 2 ) There exist L > 0 and T > 0 such that for all t R with t T. W (t) L t. Definition.5. Let R N be an open set. We define the space of functions of bounded variation BV () as the space of all functions u L () whose distributional first-order partial derivatives are finite signed Radon measures; that is, for all i =,..., N there exists a finite signed measure λ i : B () R such that (8) u φ dx = φ dλ i x i for all φ Cc (). The measure λ i is called the weak, or distributional, partial derivative of u with respect to x i and is denoted D i u. See [AFP2000], [Le2009] for more information about functions of bounded variation. 2 Which means that in the definition of Γ-convegence, you should replace y n y with u n u in L 2 (). 3

4 Theorem.6 (Compactness). Let R N be an open set with finite measure. Assume that the double-well potential W satisfies conditions (H ) and (H 2 ). Let ε n 0 + and let {u n } W,2 () be such that (9) M := sup F εn (u n ) <. n Then there exist a subsequence {u nk } of {u n } and u BV (; {a, b}) such that u nk u for in L (). Proof. We begin by showing that {u n } is bounded in L () and equi-integrable. By (9) and (H 2 ), (0) n dx { u n T } u W (u n (x)) dx M L { u n T } L ε n. Since has finite measure, L u n dx = L { u n T } { u n T } u n dx + L { u n <T } u n dx W (u n ) dx + LT L N () Mε n + LT L N (). Thus, {u n } is bounded in L (). We claim that {u n } is equi-integrable. Indeed, let γ > 0 be fixed and find N ε N so large that M ε L n γ for all n N 2 ε. Then, by (0), () u n dx 2 γ { u n T } for all n N ε, while if E is measurable, (2) u n dx T L N (E) 2 γ, E { u n <T } provided that L N (E) γ. It follows from () and (2) that for every measurable set E 2T with L N (E) γ, 2T u n dx = u n dx + u n dx 2 γ + 2 γ E E { u n >T } E { u n <T } for all n N ε. Finally, since the finite family {u,..., u Nε } is equi-integrable (exercise), there exists δ > 0 such that u n dx γ E for all n N ε and for every measurable set E with L N (E) δ. It suffices to take δ := min { δ, γ}. This implies that {u 2T n } is equi-integrable. Since has finite measure, in view of the Vitali s convergence theorem and of the Egoroff convergence theorem, to obtain strong convergence of a subsequence, it suffices to prove pointwise convergence of a subsequence. 4

5 For K > 0 define (3) W (t) := min {W (t), K}, t R and (4) f (t) := 2 t a W (s) ds, t R. Since 0 W W, for every n N, we have (5) F εn (u n ) 2 W (u n (x)) u n (x) dx = (f u n ) (x) dx. Note that the function f is Lipschitz continuous and we are using the chain rule in W, () (see Theorem 6.8) Then by (9), (6) sup n (f u n ) dx M. Moreover, since Lip f 2 K, and f (a) = 0, f (u n (x)) = f (u n (x)) f (a) 2 K u n (x) a for L N a.e. x and for all n N. Since {u n } is bounded in L (), it follows that the sequence {f u n } is bounded in L (). By the Rellich Kondrachov theorem, there exist a subsequence of {u n } (not relabeled) and a function w BV () such that w n := f u n w in L loc (). By taking a further subsequence, if necessary, without loss of generality, we may assume that w n (x) w (x) and that W (u n (x)) 0 for L N a.e. x. Since the function W (t) > 0 for all t a, b, it follows from (4) that the function f is strictly increasing and continuous. Thus, its inverse f is continuous and u nk (x) = f (w k (x)) f (w (x)) =: u (x) for L N a.e. x. It follows by (H ) and the fact that W (u nk (x)) 0 for L N a.e. x, that u (x) {a, b} for L N a.e. x. In turn, w (x) {f (a), f (b)} = {0, f (b)} for L N a.e. x, and so we may write (7) w = f (b) χ E for a set E. Since w BV () and has finite measure, we have that χ E BV (). Hence, (8) u = bχ E + a ( χ E ) belongs to BV (). Remark.7. Theorem.6 was proved by Modica [Mo987] and by Sternberg [St988] under the stronger assumption that c t p W (t) c t p 5

6 for all t T and for some c > 0 and p 2. The weaker hypothesis (H 2 ) is due to Fonseca and Tartar [FT989]..3. Liminf Inequality. In view of the previous theorem, the metric convergence in the definition of Γ-convergence should be L (). Thus, we extend F ε to L () by setting ( ) W (u) + ε u 2 dx if u W (9) F ε (u) :=,2 () and u dx = m, ε otherwise in L (). Let ε n 0 +. Under appropriate hypotheses on W and, we will show that the sequence of functionals {F εn } Γ-converges to the functional { cw P (E, ) if u BV (; {a, b}) and u dx = m, (20) F (u) := otherwise in L (), where (2) c W := 2 and E := {x : u (x) = b}. For u BV () we set b a W (t) dt Du := (D u,..., D N u). Thus, if u BV (), then Du M b ( ; R N ), and since M b ( ; R N ) may be identified with the dual of C 0 ( ; R N ), we have that Du () := Du Mb (;R N ) { N ( = sup Φ i dd i u : Φ C ) } 0 ; R N, Φ C0 (;R N ) <. i= Definition.8. Let R N defined by { N V (u, ) := sup be an open set and let u L loc (). The variation of u in is i= Φ i x i u dx : Φ C c ( ; R N ), Φ C0 (;R N ) } Exercise.9. Let R N be an open set and let u L loc (). Prove the following. ( (i) If the distributional gradient Du of u belongs to M ) b ; R N, then Du () = V (u, ). ( (ii) If V (u, ) <, then the distributional gradient Du of u belongs to M ) b ; R N. In particular, if u L (), then u belongs( to BV () if and only if V (u, ) <. Hint: Use the Riesz representation theorem in C ) 0 ; R N. 6.

7 (iii) If {u n } L loc () is a sequence of functions converging to u in L loc (), then V (u, ) lim inf V (u n, ). The previous example shows that characteristic functions of smooth sets belong to BV (). More generally, we have the following. Definition.0. Let E R N be a Lebesgue measurable set and let R N be an open set. The perimeter of E in, denoted P (E, ), is the variation of χ E in, that is, P (E, ) := V (χ E, ) { N = sup i= E Φ i x i dx : Φ C c The set E is said to have finite perimeter in if P (E, ) <. If = R N, we write P (E) := P ( E, R N). ( ; R N ), Φ C0 (;R N ) } Remark.. In view of Exercise.9, if R N is an open set and E R N is a Lebesgue measurable set with L N (E ) <, then χ E belongs to BV () if and only if P (E, ) <. We are now ready to study the Γ-convergence of the sequence of functionals (9). Theorem.2 (Liminf inequality). Let R N be an open set with finite measure. Assume that the double-well potential W satisfies conditions (H ) and (H 2 ). Let ε n 0 +. Let ε n 0 + and let {u n } L () be such that u n u in L (). Then (22) lim inf F ε n (u n ) F(u), where F n and F are the functionals defined in (9) and (20), respectively. Proof. Consider a sequence {u n } L () such that u n u in L () for some u L (). If lim inf F ε n (u n ) =, then there is nothing to prove, thus we assume that (23) lim inf F ε n (u n ) <. Let {ε nk } be a subsequence of {ε n } such that lim inf F ε n (u n ) = lim k F εnk (u nk ) <. Then F εnk (u nk ) < for all k sufficiently large. Hence, u nk W,2 () for all k sufficiently large. By Theorem.6, u BV (; {a, b}). Finally, by extracting a further subsequence, not relabelled, we can assume that u n (x) u (x) for L N a.e. x. Hence, in what follow, without loss of generality, we will assume that (23) holds, that {u n } W,2 (), that u BV (; {a, b}), that lim inf n + F εn (u n ) is actually a limit, and that {u n } converges to u in L () and pointwise L N a.e. in. 7.

8 Step : We begin by truncating the sequence {u n }. Consider the Lipschitz function b if t b, h (t) := t if a < t < b, a if t a. Note that 0 if t > b, h (t) = if a < t < b, 0 if t < a. By the chain rule in Sobolev spaces, the functions v n := h u n are still in W,2 () and for L N a.e. x, { un (x) if a < u (h u n ) (x) = n (x) < b, 0 otherwise, where we used the fact that u n (x) = 0 for L N a.e. x such that u n (x) = a or u n (x) = b. Hence, v n u n. Moreover, since W 0 and W (a) = W (b) = 0, if a < u n (x) < b, we have that W (v n (x)) = W (u n (x)), otherwise W (v n (x)) = 0 W (u n (x)). Hence, ( ε W (v n) + ε v n 2 ) dx ( ε W (u n) + ε u n 2 ) dx. Finally, since u BV (; {a, b}), we have that h u = u and so using the fact that Lip h, h (u n (x)) u (x) = h (u n (x)) h (u (x)) u n (x) u (x), which shows that v n u in L () and v n (x) v (x) for L N a.e. x. Step 2: Consider the function W and f defined in (3) and (4), respectively, where K := max t [a,b] W (t). Note that W (t) = W (t) for all t [a, b]. As in the proof of Theorem.6 we have that F εn (v n ) 2 W (v n (x)) v n (x) dx = (f v n ) (x) dx, that f v n f u in L () and (f v n ) (x) (f u) (x) for L N a.e. x. By the lower semicontinuity of the seminorms in BV, we have that lim inf F ε n (v n ) lim inf (f v n ) (x) dx = lim inf D (f v n) () Write D (f u) (). u = bχ E + a ( χ E ), 8

9 for a set E, so that f u = f (b) χ E + f (a) ( χ E ) = f (b) χ E. Hence, D (f u) () = D (f (b) χ E ) () = f (b) D (χ E ) () = c W P (E, ). This concludes the proof..4. Limsup Inequality. To prove the limsup inequality, we will need stronger assumptions on the set. Theorem.3 (Limsup inequality). Let R N be an open bounded set with Lipschitz boundary. Assume that the double-well potential W satisfies condition (H ). Then for every u L () there exists a sequence {u n } L () be such that u n u in L (). Then (24) lim sup F εn (u n ) F(u), where F n and F are the functionals defined in (9) and (20), respectively. Proof. If F(u) =, then we can take u n := u for all n. Thus, assume that F(u) <, so that u BV (; {a, b}) and u dx = m. Write u = bχ E + a ( χ E ). Step : Sketch of the proof for N = : Assume first that N = and that = ( l, l) and that the function u takes the form { a if t < 0, (25) g 0 (t) := b if t 0. We would like to approximate g 0 with a Lipschitz function g ε such that g ε ( l) = a, g ε (l) = b and minimizing the one dimensional functional l ( ) ε W (g) + ε g 2 dt. l The Euler Lagrange equation of this functional is 2ε 2 g = W (g). To see this, assume that g is a minimizer and take h C 2 c ( l, l). and consider the function g + sh. Then F εn (g + sh) F εn (g) for all s and so the function s F εn (g + sh) 9

10 has a minimum at s = 0. In turn, 0 = df ε n l ( ) (g + sh) ds = s=0 l ε W (g + sh)h + ε2 (g + sh ) h dt l ( ) l ( ) = ε W (g)h + ε2g h dt = ε W (g) ε2g h dt l for all h C 2 c ( l, l). A density argument gives 2ε 2 g = W (g). Multiplying by g and integrating gives l ε 2 g 2 = c ε + W (g). The constant c ε cannot be zero. Indeed, if c ε = 0 and if g (t 0 ) = a or g (t 0 ) = b, then since W (a) = W (b) = 0, then g would be a constant. On the other hand we need g to go from a to b as fast as possible. We take c ε = ε. Hence, we define ϕ ε (z) := z a ε ε + W (s) ds if a z b. Since ϕ ε is strictly increasing, it has an inverse ϕ ε ϕ ε (0) = a, ϕ ε (ϕ ε (b)) = b and dϕ ε dt (t) = ϕ ε (ϕ ε (t)) = which is what we wanted. Finally, since W 0, ϕ ε (b) = b a ε ε + W (s) ds ε + W (ϕ ε ε b a s=0 : [0, ϕ ε (b)] [a, b]. Moreover, (t)), ε ε ds = (b a) ε /2. Extend ϕ ε (t) to be a for t < 0 and b for t > ϕ ε (b). The function ϕ ε has all the desired properties, except the mass constraint. Step 2: Assume that N 2 and that that E is an open set with E a nonempty compact hypersurface of class C 2 and that E meets the boundary of transversally, that is, H N ( E ) = 0. Here H N is the (N )-dimensional Hausdorff measure. Let s rewrite u as follows u (x) = g 0 (d E (x)), where g 0 is the function (25) and d E is the signed distance of E, that is, { dist (x, E) if x E, d E (x) := dist (x, E) if x R N \ E. The idea is now to consider the function v ε (x) = ϕ ε (d E (x)). 0

11 The problem is that v ε does not satisfy the mass constraint v ε (x) dx = m. To solve this problem, observe that for every t R, ϕ ε (t) g 0 (t), while g 0 (t) ϕ ε (t + ϕ ε (b)). Hence, ϕ ε (d E (x)) dx g 0 (d E (x)) dx = u (x) dx = m, ϕ ε (d E (x) + ϕ ε (b)) dx g 0 (d E (x)) dx = u (x) dx = m. By the continuity of the function s [0, ϕ ε (b)] ϕ ε (d E (x) + s) dx and the intermediate value theorem, we may find s ε [0, ϕ ε (b)] such that ϕ ε (d E (x) + s ε ) dx = m. Hence, we can now define g ε (t) := ϕ ε (t + s ε ) and (26) u ε (x) := g ε (d E (x)). The function d E is a Lipschitz function with d E (x) = for L N a.e. x R N (see Propositions 5.2 and 5.4). Moreover, using the fact that H N ( E ) = 0 and that E is of class C 2, we have that (see Lemma 5.8), lim r 0 HN ({x : d E (x) = r}) = H N ( E). Hence, by the coarea formula for Lipschitz functions (see Theorem.4)), we have ( ) ε W (u ε (x)) + ε u ε (x) 2 dx ( ) = ε W (g ε (d E (x))) + ε g ε (d E (x)) 2 dx ϕε(b) sε ( ) (27) = s ε ε W (g ε (r)) + ε g ε (r) 2 H N ({x : d E (x) = r}) dr ( ϕε(b) sup H N ({x : d E (x) = t}) t ϕ ε(b) sup H N ({x : d E (x) = t}) t ϕ ε(b) sup H N ({x : d E (x) = t}) t ϕ ε(b) = sup H N ({x : d E (x) = t}) t ϕ ε(b) 0 ϕε(b) 0 ϕε(b) 0 b a dϕ ε ε W (ϕ ε (t)) + ε dt ( ε + W (ϕ ε (t)) + ε ε 2 ε + W (ϕ ε 2 ε + W (s) ds. (t)) dϕ ε dt 2) (t) dt dϕ ε dt (t) dt 2) (t) dt

12 In turn, ( ) b lim sup ε 0 + ε W (u ε) + ε u ε 2 dx H N ( E) 2 W (s) ds. a Finally, it remains to show that u ε u in L (). Again by the coarea formula for Lipschitz functions and the fact that d E (x) =, u ε (x) u (x) dx = g ε (d E (x)) g 0 (d E (x)) d E (x) dx = ϕε(b) sε s ε Cϕ ε (b) g ε (r) g 0 (r) H N ({x : d E (x) = r}) dr sup t ϕ ε(b) C (b a) ε /2 H N ({x : d E (x) = t}) sup t ϕ ε(b) 0 H N ( E) = 0. H N ({x : d E (x) = t}) Step 3: To remove the regularity assumption on the set E, by Lemma.5 there exists a sequence of open sets E k with E k a nonempty compact hypersurface of class C 2 and H N ( E k ) = 0 such that χ Ek χ E in L (), P (E k, ) P (E, ) and L N (E k ) = L N (E) for all k. By Step 2, for each fixed k we can find a sequence {u n,k } W,2 () such that u n,k u k := bχ Ek +a ( χ Ek ) in L () and In turn, lim lim sup k ( ) W (u n,k ) + ε u n,k 2 dx = c W H N ( E k ) ε n lim ( ) W (u n,k ) + ε u n,k 2 ε n b a 2 W (s) ds. dx c W lim sup H N ( E k ) k = c W P (E, ). A diagonalization argument (see Proposition 6.9) yields a sequence {u n,kn } such that u n,kn u in L () and ( ) lim sup W (u n,kn ) + ε u n,kn 2 dx c W P (E, ). ε n Theorem.4 (Coarea Formula for Lipschitz Functions). Let R N be an open set, let ψ : R be a Lipschitz function and let h : R R be a Borel function and assume that h ψ is integrable. Then h (ψ (x)) ψ (x) dx = h (r) H N ({x : ψ (x) = r}) dr. R 2

13 Lemma.5. Assume that R N is a bounded open set with Lipschitz boundary and that E R N is a set of finite perimeter. Then there exists a sequence of open sets E n with E n a nonempty compact hypersurface of class C 2 and H N ( E n ) = 0 such that χ En χ E in L (), P (E n, ) P (E, ) and L N (E n ) = L N (E) for all n. Proof. Extend χ E outside to a function w BV ( R N), with 0 w, such that Dw ( ) = 0. Let w n := w ϕ n, where ϕ n are standard mollifiers. Then w n C ( R N), w n w in L ( R N) and w n dx Dw ( R N) R N w n dx Dw () = P (E, ). Consider the open sets E n,t := { x R N : w n (x) > t }. By Sard s theorem, for all n and all but L a.e. t R, we have that E n,t is a C manifold of dimension N. Since H N ( ) <, and for every fixed n the sets { E n,t } t are disjoint, we have that H N ( E n,t ) = 0 for all but countably many t. Moreover, for t (0, ), using the definition of E n,t, we have that w n χ E dx w n χ E dx + w n χ E dx = (E n,t\e) (E n,t\e) w n dx + (E\E n,t) (E\E n,t) w n dx tl N ( (E n,t \ E)) + ( t) L N ( (E \ E n,t )). Letting n and since w n χ E in L (), we conclude that L N ( (E n,t \ E)) 0 and L N ( (E \ E n,t )) 0, so that χ En,t χ E as n for every t (0, ). In turn, by the lower semicontinuity of the total variation, (28) lim inf P (E n,t, ) P (E, ) for every t (0, ). On the other hand, by the coarea formula, the fact that 0 w, and Fatou s lemma. P (E, ) = Dw () = lim w n dx = lim = lim 0 0 P ({x : w n (x) > t}, ) dt P (E n,t, ) dt 3 0 lim inf P (E n,t, ) dt.

14 Hence, 0 ( ) lim inf P (E n,t, ) P (E, ) dt 0. It follows by (28), that lim inf P (E n,t, ) = P (E, ) for L a.e. t (0, ). In conclusion, we have shown that for L a.e. t (0, ), E n,t is a C manifold of dimension N for all n with H N ( E n,t ) = 0, and χ En,t χ E in L () and lim inf P (E n,t, ) = P (E, ). We choose one such t and set E n := E n,t. Next, we want to modify E n in such a way that L N (E n ) = L N (E) for all n. The argument below is due to Ryan Murray. For a set of finite perimeter, it can be shown that the total variation measure Dχ E coincides with the measure H N of the essential boundary E of E, which is given by complement of the set of points x R N such that the limit L N (E B (x, r)) lim r 0 L N (B (x, r)) exists and is either 0 or. It can be shown that for H N a.e. x E, there exists L N (E B (x, r)) lim r 0 L N (B (x, r)) = 2. Let x and x 2 E be two such points and consider the sets Then χ Dk χ E in L () and Dχ Dk () = H N ( D k ) D k := (E B (x, /k)) \ B (x 2, /k). H N ( E) + H N ( B (x, /k)) + H N ( B (x 2, /k)) H N ( E) = Dχ D (). Hence, lim k P (D k,, ) = P (E, ). Using the fact that x and x 2 are points of density for E, 2 for k large we have that (29) (30) L N (E B (x, /k)) > 4 LN (B (x, /k)), L N (E B (x 2, /k)) < 3 4 LN (B (x 2, /k)). Now we approximate D k as before to get smooth sets D k,n. For fixed k, for all n large we have that P (D k,, ) P (D k,n, ) k, χ Dk χ Dk,n dx k, 4

15 and by properties of mollifiers, ( ( ) ) N 4 (3) B x, 5 k D k,n, \ D k,n B ( x, ( ) ) N 4. 5 k Assume that L N (D k,n ) > L N (E). Let A k,n := D k,n \ B (x, r k,n ), where r k,n is chosen so that L N (B (x, r k,n )) = L N (D k,n ) L N (E) > 0. We claim that r k,n < ( 4 ) N. Indeed, by (29), 5 k Hence, for n large enough, L N (D k ) = L N (E) + L N (B (x, /k) \ E) L N (E B (x 2, /k)) = L N (E) + L N (B (x, /k)) L N (B (x, /k) E) L N (E B (x 2, /k)) < L N (E) + L N (B (x, /k)) 4 LN (B (x, /k)) = L N (E) LN (B (x, /k)). L N (D k,n ) L N (E) < 3 4 LN (B (x, /k)). This shows that r k,n ( ) 3 N < ( 4 N 4 k 5). Hence, in view of (3), the set A k k,n is still smooth. In the case L N (D k,n ) < L N (E), we will consider instead and use (30). A k,n := D k,n B (x 2, r k,n ) Corollary.6 (Gurtin s Conjectures). Let R N be an open bounded set with Lipschitz boundary. Assume that the double-well potential W satisfies conditions (H ) and (H 2 ). Then Gurtin s conjectures hold. Proof. Let v ε W,2 () be a solution of (P ε ) (why does this exist? Exercise). Then v ε is also a minimizer of the functional F ε defined in (9). Fix any u BV (; {a, b}) with u dx = m and let u ε be the function defined in (26). By the minimality of v ε and by (27), F ε (v ε ) F ε (u ε ) M. It follows by the compactness theorem that up to a subsequence v ε converges in L () to a function v BV (; {a, b}) with v dx = m. It follows by Exercise.2 that v is a minimizer of the functional F defined in (20), and so, writing v = bχ E0 + a ( χ E0 ), we have that c W P (E 0, ) c W P (E, ) 5

16 for all functions u BV (; {a, b}) and u dx = m, where E := {x : u (x) = b}. Moreover, again by Exercise.2, lim F ε (v ε ) = c W P (E 0, ) ε 0 + and so G ε (v ε ) εc W P (E 0, ), where G ε is the functional defined in (2). 2. Variants 2.. Phase Transitions for Second Order Materials. Let s consider the functional ( ) (32) ε W (u) qε u 2 + ε 3 2 u 2 dx, where q R. The case q = 0 was studied by Fonseca and Mantegazza [FMa2000], the case q < 0 by Hilhorst, Peletier, and Schätzle [HPS2002], while the case q > 0 small by Chermisi, Dal Maso, Fonseca, and G. L. [CDMFL20] and by Cicalese, Spadaro, and Zeppieri [CSZ20] for N =. When N =, ε =, and (33) W (t) = ( t 2 ) 2, 2 the functional reduces to ( ( u 2 ) ) 2 q u 2 + u 2 dx. I 2 In this case the Euler Lagrange equation of the functional is given by (34) u (iv) + qu + u 3 u = 0. When q < 0 this is the stationary solution of the extended Fisher Kolmogorov equation v t = γ 4 v x + 2 v 4 x + v 2 v3, γ > 0, which was introduced by Coullet, Elphick, and Repaux [CER987] and by Dee and van Saarloos [DvS988] to study pattern formation in bistable systems, while for q > 0 it is the stationary solution of the Swift Hohenberg equation ) 2 v ( t = + 2 v + αv v 3, α > 0, x 2 which was introduced in [SH977] to study Rayleigh-Bénard convection. The equation (34) has been studied by several authors in both cases q > 0 and q < 0 (see, e.g., [MPT998], [PT997], [SvdB2002] and the references therein). Our motivation in [CDMFL20] was a nonlocal variational model introduced by Andelman, Kawasaki, Kawakatsu, and Taniguchi [KAKT993], [TKAK994], (see also [LA987], [SA995]) for the shape deformation of unilamellar membranes undergoing an inplane phase separation. A simplified local version of this model (see [SA995]) leads to the study of (32). 6

17 The model (32) in the one-dimensional case was independently proposed by Coleman, Marcus, and Mizel in [CMM992] (see also [LM989]) in connection with the study of periodic or quasiperiodic layered structures. In the case q < 0, compactness follows from what we have done before. The difficult case is q 0. Note that if q > 0 is very large, then the functional may not be bounded from below. We will study here the case in which q is very small. For simplicity, we take N = and W takes the form (33) and refer to [CDMFL20] for the case N and more general wells. The main result behind compactness is the following nonlinear interpolation result. The proof is taken from [CSZ20] (see also [CDMFL20] for an alternative proof). Classical interpolation results are due to Gagliardo [Ga959] and Nirenberg [Ni966]. Theorem 2.. There exists q 0 > 0 such that d q 0 u 2 dx for all c < d and all u W 2,2 (c, d). c (d c) 2 d c d W (u) dx + (d c) 2 u 2 dx Proof. By rescaling and translating, we can assume that (c, d) = (0, ). By the mean value theorem there exists x 0 (0, ) such that u (x 0 ) = and so, by the fundamental theorem of calculus, In turn, It follows by Hölder s inequality, u (x) = u (x 0 ) + u (x) u (x 0 ) (35) u (x) u (x 0 ) 2 and so Upon integration over x, we get 0 0 u dx x x 0 u dt. 0 u (x) 2 2 u (x 0 ) u dt. u 2 dt, u (x) 2 dx 2 u (x 0 ) u 2 dt. 0 c u 2 dt. To conclude the proof, it remains to show that there exists a constant l > 0 such that l u (x 0 ) 2 0 W (u) dx u 2 dx.

18 There are two cases. If u (x 0 ) 2 4 then there is nothing to prove. Thus, assume that Then by (35), u 2 dx, u 2 dx < u (x 0 ) 2. (36) u (x) u (x 0 ) 2 < 4 u (x 0 ) 2. This implies that (37) 2 u (x 0 ) < u (x) < 3 2 u (x 0 ) for all x (0, ). Therefore u is strictly monotone. Hence, it does not vanish in one of the two intervals ( ( 0, 2) and, ), say u > 0 in (, ) (the other cases are analogous). By the classical 2 2 interpolation inequality applied to the function u, we get 2 u 2 dx C 2 (u ) 2 dx + C 2 u 2 dx. Now, since u > 0 in (, ), (u ) 2 (u + ) 2 (u ) 2, and so we obtain 2 In turn, from (36), 2 u 2 dx C u (x 0 ) 2 C which is what we wanted to prove. 0 0 W (u) dx + C W (u) dx + C 0 From the previous theorem, we obtain the following result. 0 u 2 dx. u 2 dx, Corollary 2.2. For every open interval I and every q < q 0 there exists ε 0 = ε 0 (I, q) > 0 such that for 0 < ε < ε 0, qε 2 u 2 dx W (u) dx + ε 4 u 2 dx for all u W 2,2 loc (I). I I Proof. Assume 0 < q < q 0. Consider the function v (y) := u (εy) for y I/ε := {z R : εz I}. Let n ε be the integer part of length (I) and divide I into n ε ε open intervals I k,ε of length εn ε length (I) and apply the previous theorem to the function v in each interval I k,ε to get q 0 ε I 2 u (εy) 2 ε 2 n 2 ε dy k,ε length 2 W (u (εy)) dy + length 2 (I) ε 4 u (I) I k,ε ε 2 n 2 ε I (εy) 2 dy. k,ε 8 I

19 Summing over k and changing variables, we get qε 2 u (x) 2 dx q ε 2 n 2 ε I q 0 length 2 W (u (x)) dx + q length 2 (I) ε 4 u (x) 2 dx. (I) I q 0 ε 2 n 2 ε I The result now follows by observing that q q 0 < and εn ε length (I) as ε 0 +. Using the previous corollary, we can prove compactness for q < q 0. Theorem 2.3 (Compactness). Let I R be an interval and let q < q 0. Let ε n 0 + and let {u n } W,2 () be such that ( ) sup W (u n ) qε n u n ε n 2 + ε 3 n u n 2 dx <. n I Then there exist a subsequence {u nk } of {u n } and u BV (I; {, }) such that u nk u for in L (I). Proof. Let δ > 0 be so small that q+δ < q δ 0 and write ( W (u n ) qε n u ε n 2 + ε 3 n u n 2 = ( δ) W (u n ) q + δ ) n ε n δ ε n u n 2 + ε 3 n u n 2 dx ( ) + δ W (u n ) + ε n u ε n 2 + ε 3 n u n 2. n By the previous theorem, for all n large enough ( W (u n ) q + δ ) ε n δ ε n u n 2 + ε 3 n u n 2 dx 0 I and so ( ) ( ) δ W (u n ) + ε n u I ε n 2 dx W (u n ) qε n u n I ε n 2 + ε 3 n u n 2 dx n for all n sufficiently large. We can now apply Theorem The Vectorial Case d. This was studied by Sternberg [St988] in the case when the wells are two closed curves in R 2, by Fonseca and Tartar [FT989] in the case of two wells in R d, by Baldo [Ba990] in the case of multiple wells, and by Ambrosio [Am990] who considered the case in which the set of zeros of W is a compact set (see also [St99]). Let s describe the case of two wells. For ε > 0 consider the functional defined by F ε : W,2 ( ; R d) [0, ] F ε (u) := ( ε W (u) + ε u 2 ) dx, 9

20 where the double well potential W : R d [0, ) satisfies the following hypotheses: (H ) W is continuous, W (z) = 0 if and only if z {a, b} for some a, b R d with a b. (H 2 ) There exist L > 0 and T > 0 such that for all z R d with z T. W (z) L z. The analogue of Theorem.6 is the following compactness theorem. There are several proofs of this result: the one in [FT989] uses Young measures, while the one [Ba990] does not. We present here another proof, due to Massimiliano Morini, that makes use of Theorem.6. Theorem 2.4 (Compactness). Let R N be an open set with finite measure. Assume that the double-well potential W satisfies conditions (H ) and (H 2 ). W (,2 ; R d) be such that Let ε n 0 + and let {u n } M := sup F εn (u n ) <. n Then there exist a subsequence {u nk } of {u n } and u BV (; {a, b}) such that u nk u for in L ( ; R d). First proof. Step : Assume that a b. For every t 0 define V (t) := min W (z). z =t Then V is upper semicontinuous, V (t) > 0 for t a, b, V ( a ) = V ( b ) = 0, and V (t) Lt for t T. For every u W (,2 ; R d) define ( ) H ε (u) := V ( u ) + ε u 2 dx F ε (u). ε Then by (9), sup n H εn (u n ) sup F εn (u n ; ) <, n and so by the compactness in the scalar case d =, there exist a subsequence {u nk } and w BV () such that w k := Φ 2 u nk w in L loc (), where and Hence, Φ 2 (t) := 2 t 0 V (s) ds, t R V (z) := min {V (z), K}, z R. u nk v := Φ 2 w in L (). 20

21 By taking a further subsequence, if necessary, without loss of generality, we may assume that u nk (x) v (x) and that W (u nk (x)) 0 for L N a.e. x. This implies that v BV (; {a, b}). Define { a if v (x) = a, u (x) := b if v (x) = b. We claim that u nk u in L ( ; R d). To see this, fix x such that u nk (x) v (x) and W (u nk (x)) 0. Then, by (H ), necessarily, u nk (x) u (x). Step 2: If a = b, let e i be a vector of the canonical basis of R d such that a e i b e i. Then a + e i b + e i. It suffices to apply the previous step with W replaced by and u n by u n + e i. Ŵ (z) := W (z e i ), z R d, The second proof is adapted from [Ba990] and [FT989]. Second proof. For K > 0 define (38) W (z) := min {W (z), K}, z R d and consider the geodesic distance in R d given by { (39) d (v, w) := 2 inf W (g (t)) g (t) dt : g piecewise C curve, g ( ) = v, g () = w } for v, w R d. We claim that the function f (z) := d (a, z), z R d, is Lipschitz. Indeed, let v, w R d and let γ be a piecewise C curve joining a with v. Then f (w) 2 W ds + 2 W ds 2 W ds + 2 K v w, γ [v,w] where [v, w] is the segment of endpoints v, w. Taking the infimum over all curves γ, we get f (w) f (v) + 2 K v w, which shows that f is Lipschitz continuous. Next we prove that (40) (f u) (x) d x 2 W (u (x)) u (x) dx for all u W,2 ( ; R d). Assume first that u C ( ; R d). Then the function f u is locally Lipschitz, and thus, by Rademacher s theorem, it is differentiable L N a.e. Fix a point x 0 such 2 γ

22 that f u is differentiable at x 0. Let i {,..., N}, let h > 0 and let γ be a piecewise C curve joining a with u (x 0 ). Then f (u (x 0 + he i )) 2 W ds + 2 W ds = 2 W ds γ [u(x 0 ),u(x 0 +he i )] γ + 2 u (x 0 + he i ) u (x 0 ) W (su (x 0 ) + ( s) u (x 0 + he i )) ds. 0 Taking the infimum over all curves γ and applying the mean value theorem to the last integral, we get f (u (x 0 + he i )) f (u (x 0 )) 2 u (x 0 + he i ) u (x 0 ) W (θu (x 0 ) + ( θ) u (x 0 + he i )) for some θ [0, ]. Dividing by h and letting h 0 + yields (f u) (x 0 ) 2 u x i (x 0 ) x i W (u (x 0 )). By inverting the roles of u (x 0 + he i ) and u (x 0 ) we get (4) (f u) (x 0 ) x i 2 u (x 0 ) x i W (u (x 0 )). In turn, (f u) (x 0 ) = 2 (f u) (x 0 ) x i= i ( 2 u 2 (x 0 ) x i= i W (u (x 0 ))) = 2 W (u (x 0 )) u (x 0 ). This proves the claim for u C ( ; R d). In the general case, u W,2 ( ; R d), we can use the Meyers Serrin theorem to approximate u W,2 ( ; R d) with a sequence of functions u k W,2 ( ; R d) C ( ; R d) converging to u in W,2 ( ; R d). By selecting a subsequence, we may assume that {u k } and { u k } converge to u and u pointwise L N a.e. and that (42) u k (x) 2 + u k (x) 2 h (x) for L N a.e. x and all k and for some integrable function h. Since f is Lipschitz, it follows that {f u k } converges to f u in L 2 () and pointwise L N a.e. Moreover, by (4) applied to u k and the fact that W is bounded, we have that {f u k } is bounded in W,2 (). Hence, it converges weakly to f u in W,2 (). By (40) applied to u k, (f u k ) (x) d x 2 W (uk (x)) u k (x) dx 22

23 for all k. Letting k, and using the lower semicontinuity of the L 2 norm on the left-hand side and Lebesgue dominated convergence theorem (which can be applied by (40)) on the right-hand side, we conclude that (40) holds for u. Using (40) in place of (5), we can conclude as in the proof of Theorem.6 that if {u n } W,2 ( ; R d) is such that sup n F εn (u n ) <, then there exist a subsequence, not relabeled, and a function w BV () such that w n := f u n w in L loc (). By selecting a further subsequence, we may assume that {w n } converges to w pointwise L N a.e. It remains to show that {u n } converges to u pointwise L N a.e. Define and E := {x : w (x) > 0} u := bχ E + aχ \E. Let x E be such that w n (x) w (x) and W (u n (x)) 0. Consider a subsequence { {u nk (x)}. } By (H ) and the fact that W (u n (x)) 0, there exists a further subsequence u nki (x) of {u nk (x)} such that either u nki (x) a or u nki (x) b. We claim that the case u nki (x) a cannot happen. Indeed, if this were the case, then by the continuity of f, w ki (x) = f (u nk (x)) f (a) = 0, which contradicts the fact that w n (x) w (x) > 0. This shows that u nki (x) b, and by the arbitrariness of the subsequence, that u n (x) b. Similarly, we can show that if w (x) = 0, w n (x) w (x), and W (u n (x)) 0, then u n (x) a. This proves that {u n } converges pointwise to u. A Γ-convergence result similar to the one given in Theorems.2 and.3 holds. The main changes are the constant c W in (2) and the fact that, in addition to (H ) and (H 2 ), W is also assumed to be locally Lipschitz and quadratic near the wells, precisely: (H 3 ) W is Lipschitz on compact sets and there exist l > 0 and δ > 0 such that l z a 2 W (z) l z a 2 for all z R d with z a δ, l z b 2 W (z) l z b 2 for all z R d with z b δ. As in the scalar case we extend F ε to L ( ; R d) by setting ( ) W (u) + ε u 2 F ε (u) := ε dx if u W (,2 ; R d) and u dx = m, otherwise in L ( ; R d). 23

24 Let ε n 0 +. Under appropriate hypotheses on W and, we will show that the sequence of functionals {F εn } Γ-converges to the functional { cw P (E, ) if u BV (; {a, b}) and u dx = m, F (u) := otherwise in L ( ; R d), where c W := 2 inf and E := {x : u (x) = b}. { W (g (t)) g (t) dt : g piecewise C curve, g ( ) = a, g () = b }, Theorem 2.5 (Liminf inequality). Let R N be an open set with finite measure. Assume that the double-well potential W satisfies conditions (H ), and (H 2 ), and (H 3 ). Let ε n 0 +. Let ε n 0 + and let {u n } L ( ; R d) be such that u n u in L (). Then lim inf F ε n (u n ) F(u). Proof. We begin by proving that (43) c W = f (b) = d (a, b) if the constant K in (38) is chosen large enough. To see this, not that since W W, we have that f (b) c W. To prove the converse inequality, let h (r) := W (z). By (H 2 ) there exists r > r 0 := a b 2 such that r min z a+b 2 =r r 0 h (r) dr > c W. Take K := max a+b z 2 r W (z). Let now g be a piecewise C curve with g ( ) = a and g () = b. If g (t) a+b 2 r for all t [, ], then W (g (t)) = W (g (t)) for all t [, ]. Hence, W (g (t)) g (t) dt = W (g (t)) g (t) dt c W. On the other hand, if there exists a t 0 [, ] such that g (t 0 ) a+b > r 2, then since g ( ) a+b 2 = r0 < r, we have that there exists a first time t such that g (t ) a+b 2 = r. In 24

25 turn, W (g (t)) = W (g (t)) for all t [, t ], and so by the area formula for Lipschitz functions W (g (t)) g (t) dt t W (g (t)) g (t) dt = t W (g (t)) g (t) dt = W (g (t)) dy = R d t [,t ] g ({y}) R d t [,t ] g ({y}) W (y) dy r r 0 h (r) dr > c W. This shows that f (b) c W. The remaining of the proof is very similar to the one of Theorem.2. It can be shown that { c W = inf ( W (g (t)) + g (t) 2) dt : g piecewise C curve, g ( ) = a, g ( ) = b }, { R ( = inf W (g (t)) + g (t) 2) dt : R R > 0, g piecewise C curve, g ( R) = a, g (R) = b } The Anisotropic Case. In the scalar case d = the family of functionals (7) was generalized to f(x, u, ε u) dx ε by Bouchitté [Bo990], Owen [Ow988], and Owen and Sternberg [OS99], while the vectorial case d was considered by Barroso and Fonseca [BF994], who studied the functional ( ) ε W (u) + εφ2 (x, u) dx, u W (,2 ; R d). Of particular importance in the case d = is the functional ( ) H ε (u) := ε W (u) + εφ2 ( u) dx R N defined for u W (,2 R N) and u(x) dx = m, where Φ : R N R is convex and positively R N homogeneous of degree one. We extend H ε to L ( R N) by setting H ε (u) := + if u / W (,2 R N) or if the constraint u(x) dx = m is not satisfied. R N By adapting the arguments developed in [BF994], [Bo990], [OS99], it can be shown the Γ-limit of {H ε }, is given by (44) H 0 (u) := c W P Φ (E) 25

26 if u = bχ E + aχ \E, with E a set of finite perimeter satisfying (), while H 0 (u) := otherwise. Here, c W is the constant given in 2 and P Φ is the Φ-perimeter, defined for every E R N with finite perimeter by (45) P Φ (E) := Φ (ν E ) dh N, E where E is the reduced boundary of E, ν E is the measure theoretic outer unit normal of E. It was established by Fonseca [Fo99] and Fonseca and Müller [FMu99] (see also the work of Taylor [Ta97], [Ta975], [Ta978]) that the minimum of the problem { (46) min P Φ (E) : E set of finite perimeter, L N (E) = m a } b a is uniquely attained (up to a translation) by an appropriate rescaling E 0 of the Wulff set (47) B Φ := {x R n : Φ (x) }, where Φ is the polar function of Φ. More precisely E 0 := rb Φ (up to a translation), where r > 0 is chosen so that L N (E Φ ) = m a. A key ingredient in the proof is the Brunn-Minkowski b a inequality (see [Gar2002]) (48) (L N (A)) /N + (L N (B)) /N (L N (A + B)) /N, which holds for all Lebesgue measurable sets A, B R N such that A + B is also Lebesgue measurable. Functionals of the type (44) describe the surface energy of crystals and, since the fundamental work of Herring [He95], they play a central role in many fields of physics, chemistry and materials science. If the dimension of the crystals is sufficiently small, then the leading morphological mechanism is driven by the minimization of surface energy Solid-Solid Phase Transitions. The corresponding problem for gradient vector fields, where in place of F ε we introduce ( ) (49) I ε (u) := ε W ( u) + ε 2 u 2 dx, u W ( 2,2 ; R d), arises naturally in the study of elastic solid-to-solid phase transitions [BJ987] and [KM994]. Here u : R d stands for the deformation. One of the main differences with the Modica Mortola functional is that in the case of gradients, some geometrical compatibility conditions must exist between the wells. Indeed, the following Hadamard s compatibility condition holds. Theorem 2.6. Let R N be an open connected set, let u W, ( ; R d) satisfy u (x) = χ E (x) A + ( χ E (x)) B for L N a.e. x, where E is a measurable set with 0 < L N (E) < L N (), and A, B R d N. Then there exist ν S N, a, u 0 R d, with a u 0 = 0, θ W, () with θ (x) = χ E (x) ν for L N a.e. x, such that (50) A B = a ν 26

27 and (5) u (x) = u 0 + Bx + θ (x) a for L N a.e. x. Proof. Let z (x) := u (x) Bx, and let C := A B. Then z (x) = χ E (x) C for L N a.e. x, and since χ E is not constant because E > 0, we may find ϕ Cc () such that ϕ dx 0. (Exercise). Define We have 0 = E ν := ϕ dx E = C ij ν k C ik ν j, ϕ dx E ϕ dx. E ( zi ϕ z ) i ϕ dx x j x k x k x j and so the vectors C i := (C i,..., C in ) are paralell to ν, i.e., there exists a = (a,..., a d ) R d such that C i = a i ν, and this proves (50). Note that if b R d is orthogonal to a, then (z (x) b) = χ E (x) (a b) ν = 0, therefore for any fixed x 0 the function z z (x 0 ) is parallel to a, hence there exists θ W, () such that z (x) z (x 0 ) = θ (x) a for L N a.e. x. Therefore, (52) where u (x) = z (x 0 ) + Bx + θ (x) a = u 0 + Bx + θ (x) a, u 0 := z (x 0 ) (z (x 0) a) a 2 a, θ (x) := θ (x) + (z (x 0) a) a 2. Moreover, in view of (50) we have and in turn by (52) u (x) = B + χ E (x) a ν, u (x) = B + a θ (x). We conclude that θ (x) = χ E (x) ν for L N a.e. x. 27

28 In the case of elastic solid-to-solid phase transitions, A, B represent two variants of martensite. The Γ-convergence for the family {I ε } in the case of two wells satisfying (50) was studied by Conti, Fonseca, and G.L. in [CFL2002]. This result was extended by Conti and Schweizer [CS2006] in dimension N = 2, who took into account frame-indifference, and assumed that {W = 0} = SO(N)A SO(N)B, where SO(N) is the set of rotations in R N, and by Chermisi and Conti [CC200], who considered the case of multiple wells in any dimension and under frame-indifference. More recently, Zwicknagl [Z203] has studied several variants of a classical model of Kohn and Müller [KM994] for the fine scale structure of twinning near an austenite-twinned-martensite interface Nonlocal Functionals. In [ABII998] (see also [ABI998]) Alberti and Bellettini studied the Γ-convergence of the family of nonlocal functionals defined by (53) J ε (u) := (u (x)) dx + ε W J ε (x y) (u (x) u (y)) 2 dxdy 4ε for u L (), where J ε (x y) := ( x ) ε J, N ε with J : R N [0, ) an even integrable function satisfying J (x) x dx <, and W satisfies R N (H ) and (H 2 ) with a = and b =. In equilibrium statistical mechanics functionals of the form (53) arise as free energies of continuum limits of Ising spin systems on lattices. In this context, u plays the role of a macroscopic magnetization density and J is a ferromagnetic Kac potential (see [ABCP996]). It was proved in [ABII998] that a compactness result similar to that of Theorem.6 holds, and that the family {J ε } Γ-converges in L () to the functionalis given by J 0 (u) := Φ (ν E ) dh N, E if u = χ E χ \E, with E a set of finite perimeter, while J 0 (u) := otherwise, for an appropriate function Φ. We recall that E is the reduced boundary of E, ν E is the measure theoretic outer unit normal of E. Note that the kernel J is assumed to be integrable and thus it excludes the classical seminorm for fractional Sobolev spaces W s,p, 0 < s <, introduced by Gagliardo [Ga957] to characterize traces of functions in the Sobolev space W,p (), p >, (see also [DNPV202] and [Le2009]). This case was considered for s = and N = by Alberti, Bouchitté, and Seppecher in [ABS994] 2 (see also [ABS998]), who studied the functional K ε (u) := λ ε I W (u) dx + ε I I u (x) u (y) x y 2 dxdy, where ελ ε l (0, ) and I R is a bounded interval. Under hypotheses (H ) and (H 2 ) (with quadratic growth at infinity in place of linear growth), they proved that the family of functionals K ε (extended to L (I) by setting + ) Γ-converges to 2l (b a) 2 P (E, I), where E := {x I : u (x) = b}, if u BV (I; {a, b}) and to otherwise. The case s = and N is 2 28

29 contained in [ABS998], althought it is not stated explicitly. Garroni and Palatucci [GP2006] extended the results in [ABS994] to the functional p W (u) dx + ε p 2 u (x) u (y) ε x y dxdy, I p > 2. More recently Savin and Valdinoci [SV202] considered the functional X ε (u) : = W (u) dx + u (x) u (y) 2 2 ε2s dxdy + ε 2s u (x) u (y) x y N+2s x y I I R N \ p dxdy, where s (0, ) and dimension N 2 and found the Γ-limit of the rescaled functionals ε 2s X ε (u) if 0 < s < 2, ε log ε X ε (u) if s = 2, ε X ε (u) if 2 < s < Higher Order Γ-Convergence. A sequence of functionals F ε : Y (, ], defined on an arbitrary metric space Y, has the asymptotic development of order k and we write F ε Γ = F (0) + εf () + + ε k F (k) + o ( ε k) if F ε Γ F (0) and (54) F (i) ε := F ε (i ) inf X F (i ) ε Γ F (i) for i =,..., k, where F ε (0) := F ε (see [AB993], [ABO996]). The second order asymptotic development for the Modica Mortola functional 7 is still an open problem. In [AB993] Anzellotti and Baldo considered the case N = under the assumption that {t R : W (t) = 0} = [a, c] [d, b] and the constraint u(x) dx = m is replaced by boundary condition u = g on, while in the N- dimensional setting Anzellotti, Baldo, and Orlandi [ABO996] studied the second order asymptotic development for 7 in the case in which W is the one-well potential W (t) = t 2 and again with the boundary condition u = g on. More recently Dal Maso, Fonseca, and G. L. [DMFL203] have proved that the second order asymptotic development of 7 with the constraint u(x) dx = m is zero when W is of class C but not C 2 near the wells and under the additional assumption that u = a on, which forces the phase {x : u (x) = b} of minimizers u to stay away from the boundary of. 3. Related Problems and Other Applications of Γ-Convergence 3.. Allen Cahn Equation Cahn Hiliard Equation Dimension Reduction. 29

30 3.4. Discrete to Continuum Dislocations Ginsburgh Landau Homogenization. 4. The Distance Function The material in this section is taken from [DZ200], [Fe959], [Fo984], [KP98]. Given a nonempty set E R N, let where f (x) := dist (x, E), x R N, (55) dist (x, E) := inf { x y : y E}. Proposition 4.. Let E R N be a nonempty set. Then f is Lipschitz continuous with Lipschitz constant less than or equal to one. Proof. If y E, by (55) we have Hence, dist (x, E) x y = x z + z y x z + z y. dist (x, E) x z z y for all y E, which shows that dist (x, E) x z is a lower bound for the set { z y : y E}. Since the infimum is the greatest of all lower bounds, it follows that Hence, dist (x, E) x z inf { z y : y E} = f (z). (56) f (x) f (z) + z x. By interchanging the role of x, z, we have that or, equivalently, which together with (56) gives or, equivalently, f (z) f (x) + z x = f (x) + x z, x z f (x) f (z), x z f (x) f (z) x z, f (x) f (z) x z. 30

31 Since f is Lipschitz continuous, it follows by Rademacher s theorem (see, e.g., Theorem.49 in [Le2009]) that f is differentiable for L N a.e. x R N. For x R N define the set Π E (x) := { y E : f (x) = x y }. Proposition 4.2. Let E R N be a nonempty set. Then for every x R N the set Π E (x) is nonempty and compact. Moroever, if x R N \ E, then Π E (x) E, while if x E, then Π E (x) = {x}. Proof. Let s prove that Π E (x) is nonempty. By (55) for every n N there exists y n E such that (57) dist (x, E) x y n < dist (x, E) + n. Since E is nonempty, dist (x, E) <, and so the sequence {y n } is bounded. It follows by the Weierstrass theorem that there exist a subsequence of {y n }, not relabeled, such that y n y for some y R N. Since y n E, we have that y E and, by letting n in (57), we obtain that dist (x, E) = x y. This shows that y Π E (x). The remaining of the proof is left as an exercise. Exercise 4.3. Let E R N be a nonempty set. Prove that for every x R N, dist (x, E) = dist ( x, E ). Proposition 4.4. Let E R N be a nonempty set. Let v B (0, ) be a direction and let x R N. Then there exists + f 2 (x) v := lim t 0 + f 2 (x + tv) f 2 (x) t Proof. Let y Π E (x) and y t Π E (x + tv). Then, by (55), and so f 2 (x + tv) f 2 (x) t = 2 min [(x y) v]. y Π E (x) x + tv y t = f (x + tv) x + tv y It follows that f 2 (x + tv) f 2 (x) lim sup t 0 + t for all p Π E (x), which implies that lim sup t 0 + = x + tv y t 2 x y 2 t x + tv y 2 x y 2 = t + 2 (x y) v. t f 2 (x + tv) f 2 (x) 2 t 3 2 (x y) v min y Π E (x) [(x y) v].

32 On the other hand, let t n 0 + be such that lim inf t 0 + f 2 (x + tv) f 2 (x) t and let y n Π E (x + t n v). Then by Proposition 4., = lim f 2 (x + t n v) f 2 (x) t n x y n x + t n v y n + t n v = f (x + t n v) + t n f (x) + 2t n. It follows that the sequence {y n } is bounded, and so, up to a subsequence, not relabeled, y n y 0 E. In turn, since f is continuous, f (x) = lim f (x + t n v) = lim x + t n v y n = x y 0. This implies that y 0 Π E (x). Hence, by (55), and so Letting n we get lim inf t 0 + which concludes the proof. x y 0 = f (x) x y n, f 2 (x + t n v) f 2 (x) = x + t nv y n 2 x y 0 2 t n t n x + t nv y n 2 x y n 2 t n = t n + 2 (x y n ) v. f 2 (x + tv) f 2 (x) t 2 (x y 0 ) v 2 min y Π E (x) [(x y) v], Let F be the set of points x R N for which Π E (x) is a singleton and define the function and define p (x) := y, where Π E (x) = {y}. Proposition 4.5. Let E R N be a nonempty set. Then the function p : F R N is continuous. Proof. Let x F. If P is not continuous at x, then we can find ε > 0 and a sequence {x n } F converging to x such that (58) p (x) p (x n ) ε for all n. Then x p (x n ) x n p (x n ) + x n x = f (x n ) + x n x f (x) + 2 x n x. 32

33 It follows that the sequence {p (x n )} is bounded, and so, up to a subsequence, not relabeled, p (x n ) y E. In turn, since f is continuous, f (x) = lim f (x n ) = lim x n p (x n ) = x y. This implies that y = p (x). This contradicts (58) and completes the proof. Exercise 4.6. Let g : R N R be Lipschitz on compact sets and let x R N. Assume that there exist + g(x) for every direction v and b R N such that v + g (x) = b v v for all v. Prove that g is differentiable at x. Proposition 4.7. Let E R N be a nonempty set. Then f 2 is differentiable at x R N if and only if x belongs to F, with (59) f 2 (x) = 2 (p (x) x). Moreover, the partial derivatives of f 2 are continuous in F. Proof. Assume x belongs to F. Then by Proposition 4.4 and the fact that Π E (x) = {p (x)}, + f 2 (x) = 2 (x p (x)) v v for every direction v. It follows by Exercise 4.6 that f 2 is differentiable at x, with f 2 (x) = 2 (p (x) x). Moreover, its partial derivatives are continuous by Proposition 4.5. Conversely, assume that f 2 is differentiable at x. Then by standard properties of differentiation and by Proposition 4.4, f 2 (x) v = + f 2 (x) v = 2 min (x y) v y Π E (x) for all v. We claim that this implies that Π E (x) is a singleton. If x E, then Π E (x) = {x} and so there is nothing to prove. Thus, assume that x R N \E and let y 0 Π E (x). Then x y 0 0. Let v 0 := x y 0 x y 0. Then for every y Π E (x), with y y 0, Hence, x y 0 = (x y 0 ) v 0 < (x y) v 0. f 2 (x) v 0 = 2 min (x y) v 0 = x y 0 = f (x). y Π E (x) Since f (x) > 0, we have that f = f 2 is differentiable at x with (60) f (x) = f (x) f 2 (x). 33