J I P A

advertisement
Journal of Inequalities in Pure and
Applied Mathematics
CORRIGENDUM ON THE PAPER: ’LOWER BOUNDS FOR THE
INFIMUM OF THE SPECTRUM OF THE SCHRŐDINGER
OPERATOR IN RN AND THE SOBOLEV INEQUALITIES’
PUBLISHED IN JIPAM, VOL. 3, NO. 4. (2002), ARTICLE 63
volume 4, issue 5, article 109,
2003.
Received 28 October, 2003;
accepted 12 November, 2003.
E.J.M. VELING
Delft University of Technology
Faculty of Civil Engineering and Geosciences
Section for Hydrology and Ecology
P.O. Box 5048, NL-2600 GA Delft
The Netherlands.
EMail: [email protected]
Communicated by: S.S. Dragomir
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
cor037-02
Quit
Abstract
This paper is a corrigendum on a paper published in an earlier volume of JIPAM,
’Lower Bounds for the Infimum of the Spectrum of the Schrődinger Operator in
RN and the Sobolev Inequalities’ published in JIPAM, vol. 3, no. 4. (2002),
Article 63. It concerns of a number of misprints.
Corrigendum on the paper:
’Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities’
published in JIPAM, vol. 3, no.
4. (2002), Article 63
2000 Mathematics Subject Classification: 26D10, 26D15, 47A30.
Key words: Optimal lower bound, infimum spectrum Schrődinger operator, Sobolev
inequality.
Contents
1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 5
1.
Results
The following list of misprints have been brought to the attention of the author
by the review in Mathematical Reviews #1923362 by Jan Kříž. It appeared
that these misprints had crept in during the process of text-editing of an earlier
concept.
1. Page 2, formula (1.4) (definition of the form domain Q(h)):
Q(h) = H 1 (RN ) ∩ {u | u ∈ L2 (RN ),
q+1/2 ∈ L2 (RN )}.
to be replaced by [see second condition]
Q(h) = H 1 (RN ) ∩ {u | u ∈ L2 (RN ),
q+1/2 u ∈ L2 (RN )}.
2. Page 3, brackets between formulas (1.12) and (1.13): the line with
2θ
(P = 1/θ, Q = 1/(1 − θ), a = ηk∇wk2θ
2 , b = kwk2 /η).
Corrigendum on the paper:
’Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities’
published in JIPAM, vol. 3, no.
4. (2002), Article 63
E.J.M. Veling
to be replaced by [see exponent in expression for b]
2(1−θ)
(P = 1/θ, Q = 1/(1 − θ), a = ηk∇wk2θ
2 , b = kwk2
Title Page
/η).
3. Page 4, formula (1.22), integral in the numerator:
R
k∇uk22 + RN qkuk22 dx
l(N, θ) =
inf
inf
kq− k−1/(1−θ)
.
p
2
p
N
1
N
kuk2
q− ∈L (R ) u∈H (R )
to be replaced by [see |u|2 in integrand in integral in numerator]
R
k∇uk22 + RN q|u|2 dx
l(N, θ) =
inf
inf
kq− k−1/(1−θ)
.
p
kuk22
q− ∈Lp (RN ) u∈H 1 (RN )
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 5
4. Page 5, formulas (1.28) and (1.29), integrals (0,infinity) in the numerators:
R∞
ku0 k22 + 0 q|u|22 dx
inf
inf
= l(1, 1/(2p)),
kq− k−2p/(2p−1)
p
2
p
+
q− ∈L (R ) u∈D(T0 )
kuk2
R∞
ku0 k22 + 0 q|u|22 dx
inf
inf
kq− k−2p/(2p−1)
= 22/(2p−1) l(1, 1/(2p)).
p
q− ∈Lp (R+ ) u∈D(Tπ/2 )
kuk22
to be replaced by [see |u|2 in integrand integral numerator], respectively
R∞
ku0 k22 + 0 q|u|2 dx
inf
inf
kq− k−2p/(2p−1)
= l(1, 1/(2p)),
p
q− ∈Lp (R+ ) u∈D(T0 )
kuk22
R∞
ku0 k22 + 0 q|u|2 dx
inf
inf
kq− k−2p/(2p−1)
= 22/(2p−1) l(1, 1/(2p)).
p
q− ∈Lp (R+ ) u∈D(Tπ/2 )
kuk22
5. Page 7, Lemma 2.1: “defined in (6)” should be replaced by “defined in
(1.6)”.
Corrigendum on the paper:
’Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities’
published in JIPAM, vol. 3, no.
4. (2002), Article 63
E.J.M. Veling
Title Page
Contents
6. Page 9, formula (2.15):
−2/(1−θ)
h(u, u) = −bQ /Q = −(1 − θ)θθ/(1−θ) λN,θ
kq− k1/(1−θ)
kuk22 ,
p
to be replaced by [first equality sign to be replaced by inequality sign]
−2/(1−θ)
h(u, u) ≥ −bQ /Q = −(1 − θ)θθ/(1−θ) λN,θ
kq− k1/(1−θ)
kuk22 ,
p
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 5
7. Page 10, formula (2.25), integral in the numerator:
R∞
ku0j k22 + −∞ q|uj |22 dx
2
kqj k−2
1 = −(1 + 1/j) /4 > −1/4 = l(1, 1/2).
kuj k22
to be replaced by [see qj and |uj |2 in integrand integral numerator]
R∞
ku0j k22 + −∞ qj |uj |2 dx
2
kqj k−2
1 = −(1 + 1/j) /4 > −1/4 = l(1, 1/2).
kuj k22
Moreover, corrections have to be made in the following lines.
1. Page 1, Abstract, line 4: change
−1
“ΛN,θ (ν) = k∇vkθ2 kvk1−θ
2 kvkr , with ν element of the Sobolev space
−1
H 1 (RN )”, into “ΛN,θ (v) = k∇vkθ2 kvk1−θ
2 kvkr , with v element of the
Sobolev space H 1 (RN )”.
2. Page 1, line −2 : change “q = q+ + q− ” into “q = q+ − q− ” .
3. Page 6, line 12: change “l(3, 3/4) = −1.75018010−4 ” into “l(3, 3/4) '
−1.75018010−4 ”.
4. Page 8: label (2.4) refers the expression, one line higher; label (2.5) refers
to the expression two lines higher.
Corrigendum on the paper:
’Lower Bounds for the Infimum
of the Spectrum of the
Schrődinger Operator in RN
and the Sobolev Inequalities’
published in JIPAM, vol. 3, no.
4. (2002), Article 63
E.J.M. Veling
Title Page
Contents
JJ
J
II
I
Go Back
5. Page 9: change formula (2.16) “q = q− ” into “ q = −q− ”.
Close
6. Page 11, line 9: change “side of (31)” into “side of (1.31)”.
Quit
Page 5 of 5
Download