Partial Regularity for Minimizers of Degenerate Polyconvex Energies

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20 L. Esposito, G. Mingione / Partial regularity for minimizers Taking into account (51)–(57), using strong convergences stated in (54) we have C M τ µ ≤ lim sup λ −p m U(x m , τr m ) m ∫ ≤ C(p) − | Dw − (Dw) τ | 2 dz B τ ≤ C(p)C(M)τ 2 ≤ C(p)C(M)τ µ contradicting (32) if we choose C M such that C M > C(p)C(M). 7. Decay estimate in the second case In this section we end the proof of Theorem 1 showing the decay estimate in the second case. Also this time the proof will be based on the fact that the convergences stated in (35) are actually strong; we remind the reader that this fact will be proved subsequently. Up to a (not relabelled) subsequence we may suppose this time (recall (39)) where l ∈ R + , Ā ∈ R nN , | Ā |= 1. By (58) and the very definition of ∧ i A m we have also that: We divide the Euler equation (37) by λ p−1 m ∫ λ 1−p m B 1 DF 1 (A m + λ m Dv m )Dφ dz ∫ + B 1 k∑ i=2 λ −1 m A m → lĀ, (58) | ∧ i A m | ≤ c|A m | i ≤ cλ i m ≤ cλ m . (59) to get λ 1−p m [DF i (∧ i (A m + λ m Dv m )) − DF i (∧ i A m )]· · [∧ i−1 (A m + λ m Dv m ) ˜⊙Dφ] dz = 0. (60) Now we preliminarily show that the terms indexed with i ≥ 2 in (60) are converging to 0. In order to do this we jump back to (44); using this formula the general i-term in (60) can be controlled by ∫ cλ 2−p m [| ∧ i A m | p−2 + | B 1 [| i∑ ∧ i−j A m ⊙ λ j m ∧ j Dv m | p−2 ] j=1 i∑ ∑i−1 ∧ i−j A m ⊙ λ j−1 m ∧ j Dv m |][| ∧ i−1−j A m ⊙ λ j m ∧ j Dv m |] dz j=1 ≤ cλ m ∫ B 1 [(|A m |λ −1 m ) p−2 + ( i∑ j=1 j=0 λ j−1 m | ∧ j Dv m |) p−2 ][1 + ( i∑ j=1 λm j−1 | ∧ j Dv m |) 2 ] dz
≤ cλ m ∫B 1 [1 + ( by (35) and (58)–(59). L. Esposito, G. Mingione / Partial regularity for minimizers 21 i∑ j=1 λ j−1 m | ∧ j Dv m |) p ] dz → 0 , As in the previous section we claim that the weak convergences in (35) are strong, that is, always up to a (not relabelled) subsequence: v m → v strongly in W 1,p loc (B 1; R N ), λ i−1 m ∧ i Dv m → 0 strongly in L p loc (B 1; ∧ i H N n ), i ≥ 2 . (61) Remark 7.1. In the follwing we do not need explicitely the convergence in (35) 6 to be strong. This is also because the strong convergence of the terms in (35) 6 follows from the one in (61) 2 . Indeed, assuming (61) 2 and using (59) together with Hölder inequality, we have: ∫ B τ (∫ ) 2 | ∧ i A m | p−2 λ 2i−p m | ∧ i Dv m | 2 dz ≤ c λm p(i−1) | ∧ i Dv m | p p dz → 0 , B τ an estimate that will be useful in the sequel. Using (H6), (58) and (61) 1 , we have that ∫ λ 1−p m |DF 1 (A m + λ m Dv m ) − DF 1 (λ m lĀ + λ mDv)| dz → 0 . (62) B 1 To prove this, we argue as follows: by Egorov theorem, fixed ɛ, δ > 0 it is possible to determine A ⊂ B 1−ɛ such that |B 1 \ A| ≤ δ and Dv m → Dv uniformly in A; then the previous integral can be controlled by: ∫ cλ 2−p m (B 1−ɛ \A)∪(B 1 \B 1−ɛ ) ∫ + ∫ ≤ c A ∫ 1 0 |D 2 F 1 (A m + λ m Dv m + τ(λ m (Dv − Dv m ) +λ m lĀ − A m))| · |[(Dv − Dv m ) + lĀ − λ−1 m A m ]| dτ dz λ 1−p m |DF 1 (A m + λ m Dv m ) − DF 1 (λ m lĀ + λ mDv)| dz (B 1−ɛ \A)∪(B 1 \B 1−ɛ ) ∫ + A (1 + |Dv| p−1 + |Dv m | p−1 ) dz λ 1−p m |DF 1 (A m + λ m Dv m ) − DF 1 (λ m lĀ + λ mDv)| dz . The second integral clearly converges to zero as m → +∞, by (H6), since the uniform convergence on A. The first one can be made arbitrarily small, letting ɛ, δ → 0, since the sequence {|Dv m | p−1 } m∈N is equiintegrable, by (35) 2 . By using a similar argument, (H6), (59) and the previous estimates, we infer: ∫ | lĀ + Dv |p−2 < (lĀ + Dv), Dφ > dz = 0 (63) B 1
Page 1 and 2: Journal of Convex Analysis Volume 8
Page 3 and 4: with p > 2. L. Esposito, G. Mingion
Page 5 and 6: L. Esposito, G. Mingione / Partial
Page 9 and 10: 4. Preliminary results and notation
Page 15 and 16: +λ m ∫ B 1 L. Esposito, G. Mingi
Page 17 and 18: y (35), (38) and (41). L. Esposito,
Page 19: L. Esposito, G. Mingione / Partial
Page 33 and 34: Letting m → ∞, by (59) we have

Partial Regularity for Minimizers of Degenerate Polyconvex Energies

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