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12 1 z #(JN )≤c V (s) ds −1 , z z independent of N . We can suppose that JN = {ki : i = 1, . . , L}, with L independent of N , and that ki /N → ti ∈ [a, b]. Let S = {t1 , . . , tL }. For every η > 0 we have k IN ⊂ S + [−η, η] 50 Examples of approximation z for all N > 1/η and k ∈ JN . Hence |uj |2 dt ≤ lim inf j lim inf j ψ(z) I\(S+[−η,η]) z k∈JN ≤ lim inf j z k∈JN ≤ lim inf j k IN k IN ψ(z)|uj |2 dt ψ(vj )|uj |2 dt ψ(vj )|uj |2 dt I ≤ lim inf j Gεj (uj , vj , I) . We have that u ∈ H 1 (I \ (S + [−η, η])), and that |uj |2 dt ≤ lim inf j Gεj (uj , vj , I) .

Then there exists a family (uε ) converging to u in L1 (I) such that lim supε→0+ Fε (uε ) ≤ F (u).

In this case, we can suppose that Ω = (−1, 1) and S(u) = {0}. Let z = |u(0+) − u(0−)|, and let yz ∈ [0, 1] be such that ϑj (z) = ψj (yz )jz + cV (yz ). With fixed η > 0, let T > 0 and v ∈ H 1 (0, T ) be such that T (V (v) + |v |2 ) dt ≤ 0 1 (cV (yz ) + η), 2 v(0) = yz , v(T ) = 1. Choose ξε = o(ε), and set  if t < −(ξε + εT ) u(t + ξε + εT )    if −ξε + εT < t < −ξε  u(0−)   u(0+) − u(0−) uε (t) = u(0−) + (t + ξε ) if |t| ≤ ξε + ε T  2ξε    if ξε < t < ξε + εT   u(0+) u(t − ξε − εT ) if t > ξε + εT and Elliptic approximations  yz      |t| − ξε vε (t) = v  ε     1 55 if |t| ≤ ξε if ξε < |t| < ξε + εT if |t| ≥ ξε + εT .

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