AbstractWe consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (x)  \log x\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$
d
i
v
Λ
exp
(


x

)

log

x

I
D
u
=
d
i
v
F
+
g
in
B
.
Here B denotes the unit ball of $$\mathbb {R}^n$$
R
n
, for $$n > 2$$
n
>
2
, centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$
W
1
,
2
(
B
,
R
n
×
n
)
, g is a vector in $$L^2(B, \mathbb {R}^n)$$
L
2
(
B
,
R
n
)
and $$\Lambda $$
Λ
is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$
u
∈
W
0
1
,
2
(
B
,
R
n
)
to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$
Λ
is not large enough.