Partial Regularity for Minimizers of Degenerate Polyconvex Energies

34 L. Esposito, G. Mingione / Partial regularity for minimizers
≤ ˜c(
i∑
j=1
·[
i∑
j=1
λ j−1
m ∧ i−j A m ⊙ ∧ j Dφ s ] 2 dz (102)
λ j−1
m | ∧ i−j A m |) 2 → 0,
by (35) 7 and (59). As in the first case and roughly estimating | Dv m −Dv |≤| Dv m −Dφ s |
+ | Dv − Dφ s | we find that the first integral in (80) controls the following quantity:
∫
∫
|Dv m − Dv| p dz − c |Dφ s − Dv| p dz .
B t B t
So collecting (97)–(102) and letting first m → ∞ and then s → ∞, we finally obtain by
the triangle inequality
∫
lim sup | Dv m − Dv | p dz
m B t
k∑
∫
i∑
+ lim sup [(| ∧ i A m | λ −1
m ) p−2 + | λm j−1 ∧ i−j A m ⊙ ∧ j Dv m | p−2 ] ·
m
B t
·[
i∑
j=1
i=2
λ j−1
m
∧ i−j A m ⊙ ∧ j Dv m ] 2 dz
j=1
∫
∫
≤ cδ[1 + | Dv | 2 dH n−1 + | Dv | p dH n−1 ]
∂B t ∂B t
∫
∫
+c | Dv | 2 dz + c | Dv | p dz.
B t −B t−δ B t −B t−δ
Letting δ → 0 and arguing by induction as done for the first case, we can finally prove
the strong convergences as stated in (61). This completes the proof of theorem 5.2.
10. Proof of the Main Theorem
In this section we finally prove the Main Theorem stated in section 5. The proof rests on
a standard iteration argument involving U(x, r), essentially based on Theorem 5.2.
Lemma 10.1. Let 0 < α < 1 and M > 0, then there exists 0 < τ < 1 and ɛ > 0, both
2
depending on α and M, such that if:
then
B(x, r) ⊂ Ω, | (Du) x,r |≤ M,
U(x, r) ≤ ɛ,
for each l ∈ N and µ has been introduced in Theorem 4.1.
U(x, τ l r) ≤ (τ l ) µα U(x, r), (103)
Proof. Just follow Lemma 6.1 from [21], iterating Theorem 5.2 and adapting the proof
to the different structure of U(x, r) and the different statement of Theorem 5.2 which
involves the exponent µ rather than the exponent 2.