Let
Ω
⊂
R
m
\Omega \subset {R^m}
be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on
L
2
(
Ω
)
{L^2}(\Omega )
which is the realization of a uniformly elliptic operator A of order
v
v
with suitable coefficients and principal part
a
(
x
,
ξ
)
a(x,\xi )
. Let
R
\mathcal {R}
be the ellipsoid
{
f
:
(
R
f
,
f
)
≦
1
}
\{ f:(Rf,f) \leqq 1\}
. The
L
2
{L^2}
n-widths
d
n
(
R
)
{d_n}(\mathcal {R})
satisfy
d
n
(
R
)
∼
(
c
/
n
)
v
/
2
m
{d_n}(\mathcal {R}) \sim {(c/n)^{v/2m}}
where
c
=
∫
Ω
(
∫
0
>
a
(
x
,
ξ
)
>
1
d
ξ
)
d
x
c = \smallint _\Omega {(\smallint _{0 > a(x,\xi ) > 1} {d\xi )\;dx} }
. If
B
(
u
,
v
)
B(u,v)
is a nonnegative Hermitian coercive form over a subspace
V
\mathcal {V}
of the Sobolev space
W
k
,
2
(
Ω
)
{W^{k,2}}(\Omega )
, then the n-widths of
B
=
{
f
∈
V
:
B
(
f
,
f
)
≦
1
}
\mathcal {B} = \{ f \in \mathcal {V}:B(f,f) \leqq 1\}
satisfy,
0
>
(
c
′
)
k
/
m
≦
lim
inf
d
n
(
B
)
n
k
/
m
≦
lim
sup
d
n
(
B
)
n
k
/
m
≦
(
c
)
k
/
m
0 > {(c’)^{k/m}} \leqq \lim \inf {d_n}(\mathcal {B}){n^{k/m}} \leqq \lim \sup {d_n}(\mathcal {B}){n^{k/m}} \leqq {(c)^{k/m}}
. In some cases
c
′
=
c
=
c
c’ = c = c
where c is defined in terms of an elliptic operator of order 2k. The n-widths of
B
\mathcal {B}
in
W
j
,
2
(
Ω
)
,
0
≦
j
≦
k
{W^{j,2}}(\Omega ),0 \leqq j \leqq k
, are of order
O
(
n
−
(
k
−
j
)
/
m
)
,
n
→
∞
O({n^{ - (k - j)/m}}),n \to \infty
.