Partial Regularity for Minimizers of Degenerate Polyconvex Energies

L. Esposito, G. Mingione / Partial regularity for minimizers 23
Preliminary construction. We use a sequence of comparison functions, firstly introduced
in [21]. For t ∈ ( 1, 1) and δ ∈ (0, 1) we define z 2 4 m ≡ zm
t,δ as follows, let x ∈ B 1 and
x = rω be its polar decomposition, i.e. r =| x |; ω = x | x | −1 ; we put
with φ ∈ C ∞ (B 1 ; R N ).
⎧
φ(rω)
r < t − δ
⎪⎨
φ([t − δ + 2(r − (t − δ))]ω) t − δ ≤ r ≤ t − δ 2
z m (x) =
t−r
r−(t−δ/2)
φ(tω) + v
δ/2 δ/2 m (tω) t − δ 2
⎪⎩
≤ r ≤ t
v m (rω) t ≤ r ≤ 1
We derive some estimates for ∧ k Dz m that will also show that z m ∈ ∧ k W 1,p (B 1 ; R N ), for
a.e. t ∈ (1/2, 1).
Let (τ 1 , .., τ n−1 , ν) be an orthonormal basis where ν is a radial vector. Then we have on
B t − B t−
δ
2
D τi z m = t − r
δ/2
t
r D τ i
φ(tω) +
r − (t − δ/2)
δ/2
D ν z m (rω) = 2δ −1 (v m (tω) − φ(tω)).
t
r D τ i
v m (tω),
We have, keeping into account that t/r ≤ 2, (t−r)(δ/2) −1 ≤ 1 and (r−(t−δ/2))(δ/2) −1 ≤
1,
| D τi z m (rω) |≤ c(| Dφ | + | Dv m |)(tω),
| D ν z m (rω) |≤ cδ −1 | v m (tω) − φ(tω) | .
When 2 ≤ i ≤ k a straightforward computation gives (look at section 2):
and
| 〈∧ i Dz m (rω), τ j1 ∧ ... ∧ τ ji 〉 |≤ c(|| Dφ || i +
| 〈∧ i Dz m (rω), ν ∧ τ j1 ∧ ... ∧ τ ji−1 〉 |
i∑
|| Dφ || i−j | ∧ j Dv m (tω) |),
∑i−1
≤ c(δ −1 | v m (tω) − φ(tω) |)(|| Dφ || i−1 + || Dφ || i−j−1 | ∧ j Dv m (tω)) |),
where || Dφ || stands for || Dφ || L ∞ (B t ) . In this way we finally have
j=1
j=1
(67)
| ∧ i Dz m (rω) |≤ c(|| Dφ || +δ −1 | v m (tω) − φ(tω) |) · (68)
∑i−1
·(|| Dφ || i−1 + || Dφ || i−j−1 | ∧ j Dv m (tω) |) + c | ∧ i Dv m (tω) | .
j=1
Note that all the functions involved in the left hand side of (68) are evaluated at a generic
x ∈ B t − B t−δ/2 , x = rω, while in the right hand side x ∈ ∂B t i.e. x = tω and t is fixed.