Abstract
AbstractThroughout this section, we will use the notation
$$\displaystyle W_0(u)=\int _{B_1}|\nabla u|{ }^2\,dx-\int _{\partial B_1}u^2\,d\mathcal {H}^{d-1}\qquad \text{and}\qquad W(u)=W_0(u)+|\{u>0\}\cap B_1|, $$
W
0
(
u
)
=
∫
B
1
|
∇
u
|
2
d
x
−
∫
∂
B
1
u
2
d
ℋ
d
−
1
and
W
(
u
)
=
W
0
(
u
)
+
|
{
u
>
0
}
∩
B
1
|
,
where B1 is the unit ball in
$$\mathbb {R}^d$$
ℝ
d
, d ≥ 2 and u ∈ H1(B1).
Publisher
Springer International Publishing