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Dr. Detalla Math Theory
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INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MISSING TERMS IN CLASSICAL
INEQUALITIES
ALNAR L. DETALLA
Department of MathematicsCollege of Arts and SciencesCentral Mindanao University
8710 Musuan, Bukidnon
May 21, 2010
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
f : is a non-negative measurable function on (0,∞). Then
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
f : is a non-negative measurable function on (0,∞). Then
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
INRODUCTION
In 1920, G. H. Hardy proved the following:Let 1 < p < ∞ and denote
F (t) :=
∫ t
0f(x)dx, for ǫ < p− 1
∫∞t
f(x)dx, for ǫ > p− 1
f : is a non-negative measurable function on (0,∞). Then
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
where C > 0 independent of f .
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
∫ ∞
0
|u(t)|ptǫ−pdt ≤(
p
|ǫ− p+ 1|
)p ∫ ∞
0
|u′(t)|ptǫdt (2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
∫ ∞
0
|u(t)|ptǫ−pdt ≤(
p
|ǫ− p+ 1|
)p ∫ ∞
0
|u′(t)|ptǫdt (2)
where u′(t) := dudt, u ∈ Cc((0,∞)), p > 1, ǫ 6= p− 1.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫ ∞
0
F p(t)tǫ−pdt ≤ C
∫ ∞
0
f p(t)tǫdt (1)
C =(
p
|ǫ−p+1|
)p
by: E. Landau (1930)
NOTE: (1) can be written as
∫ ∞
0
|u(t)|ptǫ−pdt ≤(
p
|ǫ− p+ 1|
)p ∫ ∞
0
|u′(t)|ptǫdt (2)
where u′(t) := dudt, u ∈ Cc((0,∞)), p > 1, ǫ 6= p− 1.
We call (2) the classical Hardy inequality
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)
Lp-version of (4) where 1 ≤ p < n is
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
for ǫ = 0, (2) becomes
∫ ∞
0
|u′(t)|pdt ≥(
p− 1
p
)p ∫ ∞
0
|u(t)|p|t|p dt (3)
from (3) a Hardy inequality of higher dimension can bederived which is a classical Sobolev embedding inequality.For n > 2,
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω). (4)
Lp-version of (4) where 1 ≤ p < n is
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω).
(5)A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by Rellich
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by Rellich
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx,
∀u ∈ W 2,20 (Ω).
(6)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx,
∀u ∈ W 2,20 (Ω).
(6)
Lp-version of (6) is
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A higher order generalization of (4) was proven by RellichFor n > 4,
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx,
∀u ∈ W 2,20 (Ω).
(6)
Lp-version of (6) is
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx,
∀u ∈ W 2,p0 (Ω).
(7)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
||u||W l,p(Ω) =∑
|γ|≤l
(∫
Ω
|∂γu(x)|pdx)
1p
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
W l,p(Ω):function space on Ω whose generalized derivatives∂γu of order ≤ l satisfies
||u||W l,p(Ω) =∑
|γ|≤l
(∫
Ω
|∂γu(x)|pdx)
1p
< ∞
W l,p0 (Ω) denotes the completion of C∞
0 (Ω) in W l,p(Ω).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Hardy-Sobolev Inequalities
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx, ∀u ∈ W 1,20 (Ω) (4)
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx, ∀u ∈ W 1,p
0 (Ω) (5)
Hardy-Sobolev-Rellich Inequalities
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)
and
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (4)
NOTATIONS
For t > 0 and ρ ≥ 2,
A1(t) = log Rt, A2(t) = logA1(t), . . . Aρ(t) = logAρ−1(t)
and
e1 = e, e2 = ee1 , . . . eρ = eeρ−1
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM 1 ( A. Detalla, T. Horiuchi and H. Ando(2005))
Assume γ ≥ 2 and n ≥ 2. If R ≥ supΩ |x|ek then thereexists sharp remainder terms such that
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx.
(8)
for any u ∈ W 1,20 (Ω).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx (8)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx (8)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∇u(x)|2dx ≥(
n− 2
2
)2 ∫
Ω
u(x)2
|x|2 dx+1
4
∫
Ω
u(x)2
|x|2[
A1(|x|)−γ +
(
A1(|x|)A2(|x|))−γ
+ · · ·+(
A1(|x|)A2(|x|) . . . Ak(|x|))−γ]
dx (8)
REMARK
In inequality (8), 14is best constant for all k-missing terms
and γ ≥ 2 is sharp
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
Let R ≥ supΩ
(
|x|e 2p
)
. Then there exist K > 0 depending on
n, p, and R such that for any u ∈ W 1,p0 (Ω) and γ ≥ 2
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Noncritical Case(1 < p < n)
Let R ≥ supΩ
(
|x|e 2p
)
. Then there exist K > 0 depending on
n, p, and R such that for any u ∈ W 1,p0 (Ω) and γ ≥ 2
∫
Ω
|∇u(x)|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|p dx+
K
∫
Ω
|u(x)|p|x|p
(
logR
|x|
)−γ
dx(9)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
Let R ≥ supΩ
(
|x|e 2n
)
. Then for any u ∈ W 1,n0 (Ω)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (5)
THEOREM 2 ( Adimurthi, N. Chaudhuri and M.Ramaswamy (2001))
Let Ω be a bounded domain in Rn with 0 ∈ Ω and n ≥ 2.
Critical Case (p = n)
Let R ≥ supΩ
(
|x|e 2n
)
. Then for any u ∈ W 1,n0 (Ω)
∫
Ω
|∇u(x)|ndx ≥(
n− 1
n
)n ∫
Ω
|u(x)|n|x|n
(
logR
|x|
)−n
dx
(10)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 1,p0 (Ω)
∫
Ω
|∇u|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|2p dx+λ(f)
∫
Ω
|u(x)|pf(x)dx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 1,p0 (Ω)
∫
Ω
|∇u|pdx ≥(
n− p
p
)p ∫
Ω
|u(x)|p|x|2p dx+λ(f)
∫
Ω
|u(x)|pf(x)dx
If f /∈ Fp and if |x|pf(x)(
log 1|x|
)2
tends to ∞ as |x| → 0,
then no inequality of the above type can hold.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
The results will be used to analyze the behaviour of thefirst eigenvalue of the weighted eigenvalue problem for theoperator
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
as µ →(
n−p
p
)p
.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
infu∈K
∫
Ω
(
|∇u|p − µ|u|p|x|p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
infu∈K
∫
Ω
(
|∇u|p − µ|u|p|x|p
)
dx
where K is given by
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
L∇µis related to the variational problem
infu∈K
∫
Ω
(
|∇u|p − µ|u|p|x|p
)
dx
where K is given by
K =
u ∈ W 1,p0 (Ω) :
∫
Ω
|u(x)|pf(x)dx = 1
,
f : is a weight function.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
µ →(
n−p
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∇µu = −
(
div(
|∇u|p−2∇u)
+µ
|x|p |u|p−2u
)
Consider the weighted eigenvalue problem
L∇µu = λ|u|p−2uf in Ω
u = 0 on ∂Ω
µ →(
n−p
p
)p
The expression L∇µu = λ|u|p−2uf is related to the
improved Hardy-Sobolev inequality.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
IMPROVEMENT OF INEQUALITY (6)
∫
Ω
|∆u(x)|2dx ≥ n2(n− 4)2
16
∫
Ω
|u(x)|2|x|4 dx, ∀u ∈ W 2,2
0 (Ω) (6)
ON GOING RESEARCH
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
The candidates of the extremals are singular at the originwhich are not in W 2,p
0 (Ω).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
The candidates of the extremals are singular at the originwhich are not in W 2,p
0 (Ω).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u(x)|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx (7)
Here the best constant(
n−2pp
)p (np−n
p
)p
is given by the
infimum of I(u) =∫Ω |∆u|pdx
∫Ω
|u(x)|p
|x|2pdx.
No extremal function in W 2,p0 (Ω) which attains the infimum
of I(u).
The candidates of the extremals are singular at the originwhich are not in W 2,p
0 (Ω).
Therefore it is natural to consider that there exists amissing terms in the right hand side of (7).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a
singular weight of type(
log 1|x|
)−2
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a
singular weight of type(
log 1|x|
)−2
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OBJECTIVE
To achieve an optimal improvement of inequality (7) byadding sharp terms in the right hand side involving a
singular weight of type(
log 1|x|
)−2
Optimal in the sense that the improved inequality holds for
this weight function(
log 1|x|
)−2
but fails for any weight
more singular than this.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx(11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
There exists K∗ = K∗(n) > 0 and C∗ = C∗(n) > 0 such
that if R > K∗ supΩ |x| and for any u ∈ W2,n
20 (Ω),
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
MAIN RESULTS
THEOREM 3 ( A. Detalla, T. Horiuchi and H. Ando(2004))
Let n ≥ 3, 0 ∈ Ω and Ω is a bounded domain in Rn.
Critical Case(p = n2)
There exists K∗ = K∗(n) > 0 and C∗ = C∗(n) > 0 such
that if R > K∗ supΩ |x| and for any u ∈ W2,n
20 (Ω),
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx(12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
REMARK 1
In inequality (11) the exponent 2 of the weight function isoptimal.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 2
In inequality (12) the exponent n2of the weight function is
optimal and the constant(
n−2√n
)n
is sharp.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 3
C and C∗ depends on R in a weak sense since(
log R|x|
)−2
and(
log R|x|
)−n2−1
tends to zero as R → ∞.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Noncritical Case(1 < p < n2)
There exists C = C(n,R) > 0 such that if R > supΩ |x| andfor any u ∈ W 2,p
0 (Ω),
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
REMARK 4
At first we prove inequality (11) assuming that
R > e1p supΩ |x| because of technical reason. Namely, the
weight function g(r) = r−2p(
log Rr
)−2should be monotone
decreasing. Then we can extend (11) for any R > supΩ |x|
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 4
In the proof of the critical case (12), we used decreasing
rearrangement argument, hence g∗(r) = r−n(
log Rr
)−n2−1
should be monotone decreasing and R ≥ re12+ 1
n .
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 4
In the proof of the critical case (12), we used decreasing
rearrangement argument, hence g∗(r) = r−n(
log Rr
)−n2−1
should be monotone decreasing and R ≥ re12+ 1
n .
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Critical Case(p = n2): There exists K∗ = K∗(n) > 0 and
C∗ = C∗(n) > 0 such that if R > K∗ supΩ |x|,∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
REMARK 4
In the proof of the critical case (12), we used decreasing
rearrangement argument, hence g∗(r) = r−n(
log Rr
)−n2−1
should be monotone decreasing and R ≥ re12+ 1
n .Moreoverwe need the condition to absorb the error terms in the righthand side of (12) with C∗ > 0, hence K∗ ≥ e
12+ 1
n
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 2,p0 (Ω)
∫
Ω
|∆u|pdx ≥ Λn,p
∫
Ω
|u(x)|p|x|2p dx+ λ(f)
∫
Ω
|u(x)|pf(x)dx(13)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 2,p0 (Ω)
∫
Ω
|∆u|pdx ≥ Λn,p
∫
Ω
|u(x)|p|x|2p dx+ λ(f)
∫
Ω
|u(x)|pf(x)dx(13)
If f /∈ Fp and if |x|2pf(x)(
log 1|x|
)2
tends to ∞ as |x| → 0,
then no inequality of type (13) can hold.A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
APPLICATION
We use the results to determine exactly when the firsteigenvalue of the weighted eigenvalue problem for theoperator
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
will tend to 0 as µ →(
n−2pp
)p (np−n
p
)p
.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
infu∈K
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
infu∈K
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
where K is given by
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
L∆µis related to the variational problem
infu∈K
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
where K is given by
K =
u ∈ W 2,p(Ω) ∩W 1,p0 (Ω) :
∫
Ω
|u(x)|pf(x)dx = 1
,
f : is a weight function which will be specified later.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
µ →(
n−2pp
)p (np−n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
µ →(
n−2pp
)p (np−n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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L∆µu = ∆
(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u
Consider the weighted eigenvalue problem
L∆µu = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω
µ →(
n−2pp
)p (np−n
p
)p
The expression L∆µu = λ|u|p−2uf is related to the
improved Rellich inequality.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
u∗(x) : spherically symmetric decreasing rearrangement offunction u(x)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
REARRANGEMENT
For a domain Ω we define a ball Ω∗ such that |Ω∗| = |Ω|with center at the origin.
u∗(x) : spherically symmetric decreasing rearrangement offunction u(x)
u∗(x) = inf
t ≥ 0 : µ(t) < B|x|n
in Ω∗
µ(t) = |x ∈ Ω : |u(x)| > t|
B: Volume of unit ball
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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LEMMA (Talente)
Let f ∈ C∞0 (Ω). If u is a solution of
−∆u = f in Ω
u = 0 on ∂Ω
and v is a solution of
−∆v = f ∗ in Ω∗
v = 0 on ∂Ω∗
⇒v≥ u∗.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
Hence we have∫
Ω|∆u|pdx
∫
Ω|u|p|x|2pdx
≥∫
Ω∗ |∆v|pdx∫
Ω∗
|v|p|x|2pdx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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From this lemma we can show that∫
Ω
|∆u|pdx =
∫
Ω∗
|∆v|pdx
∫
Ω
|u|p|x|2pdx ≤
∫
Ω∗
|v|p|x|2pdx
Hence we have∫
Ω|∆u|pdx
∫
Ω|u|p|x|2pdx
≥∫
Ω∗ |∆v|pdx∫
Ω∗
|v|p|x|2pdx
From this we can assume that u is radial and Ω is a ball.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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ASSUMPTIONS
Ω : unit ball B1
u : radially nonincreasing,
−∆u > 0
u > 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
1a) We prove inequality (11)
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
1a) We prove inequality (11)
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
1b) We show the sharpness of(
n−2pp
)p (np−n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
1. Noncritical Case(1 < p < n2)
1a) We prove inequality (11)
∫
Ω
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
Ω
|u(x)|p|x|2p dx
+ C
∫
Ω
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
1b) We show the sharpness of(
n−2pp
)p (np−n
p
)p
1c) We show the optimality of the exponent 2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
2a) We prove inequality (12)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
2a) We prove inequality (12)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
2b) We show the sharpness of(
n−2√n
)n
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
OUTLINE OF THE PROOF
2. Critical Case(p = n2)
2a) We prove inequality (12)
∫
Ω
|∆u|n2 dx ≥(
n− 2√n
)n ∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
Ω
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
2b) We show the sharpness of(
n−2√n
)n
2c) We show the optimality of the exponent n2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA 2
For any R > 1, q ≤ 0, ν ∈ (0, 1) satisfying 2ν − 1 + q = 0
∫ 1
0
|h′(r)|2(
logR
r
)q
rdr ≥ ν2
∫ 1
0
|h(r)|2(
logR
r
)q−2dr
r(14)
holds for any h ∈ C([0, 1]) ∩ C1(0, 1), with h(0) = h(1) = 0.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
For u ∈ C20(B1), u > 0, radially nonincreasing , we define
v(r) = u(r)rnp−2 r = |x|.
then apply this to
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
For u ∈ C20(B1), u > 0, radially nonincreasing , we define
v(r) = u(r)rnp−2 r = |x|.
then apply this to
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
For u ∈ C20(B1), u > 0, radially nonincreasing , we define
v(r) = u(r)rnp−2 r = |x|.
then apply this to
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
we get
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
≥ 4ωn
α
(
n− 2p
p
)p(np− n
p
)p(p− 1
p
)∫ 1
0
|(
vn2 (r)
)′ |2rdr
By Lemma 2(
v = 12, q = 0
)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
≥ 4ωn
α
(
n− 2p
p
)p(np− n
p
)p(p− 1
p
)∫ 1
0
|(
vn2 (r)
)′ |2rdr
By Lemma 2(
v = 12, q = 0
)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
∫
B1
|∆u|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
≥ 4ωn
α
(
n− 2p
p
)p(np− n
p
)p(p− 1
p
)∫ 1
0
|(
vn2 (r)
)′ |2rdr
By Lemma 2(
v = 12, q = 0
)
LEMMA 2
∫ 1
0
|h′(r)|2(
logR
r
)q
rdr ≥ ν2
∫ 1
0
|h(r)|2(
logR
r
)q−2dr
r
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
we get
∫
B1
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
+ C
∫
B1
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
we get
∫
B1
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
+ C
∫
B1
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1a) PROOF OF INEQUALITY (11)
we get
∫
B1
|∆u|pdx ≥(
n− 2p
p
)p(np− n
p
)p ∫
B1
|u(x)|p|x|2p dx
+ C
∫
B1
|u(x)|p|x|2p
(
logR
|x|
)−2
dx (11)
where C =(
n−2pp
)p (np−n
p
)p (p−1p
)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
For ǫ > 0 sufficiently small, let us define
uǫ =
0, 0 < r < ǫ2
log r
ǫ2
rn−pp log 1
ǫ
, ǫ2 < r < ǫ
log 1r
rn−pp log 1
ǫ
, ǫ < r < 1
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Let wǫ =∫ 1
ruǫ(ρ)dρ. Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Let wǫ =∫ 1
ruǫ(ρ)dρ. Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Let wǫ =∫ 1
ruǫ(ρ)dρ. Direct calculation gives
wǫ =
(
p
n−2p
)21−2ǫ
2−np +ǫ
2(2−np )
log 1ǫ
, 0 < r < ǫ2
p
n−2p
r2−n
p log r
ǫ2
log 1ǫ
+(
p
n−2p
)21−2ǫ
2−np +r
2−np
log 1ǫ
, ǫ2 < r < ǫ
p
n−2p
r2−n
p log 1r
log 1ǫ
+(
p
n−2p
)21−r
2−np
log 1ǫ
, ǫ < r < 1
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∆wǫ =
0, 0 < r < ǫ2
1p
r−np
(
−p+ n(1− p) log rǫ2
)
(
log 1ǫ
)−1, ǫ2 < r < ǫ
1p
r−np
(
p+ n(1− p) log 1r
)
(
log 1ǫ
)−1, ǫ < r < 1
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Direct calculation gives
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
Direct calculation gives
∫
B1
|∆wǫ|pdx =2
p+ 1
(
n(p− 1)
p
)p
ωn log1
ǫ+O
(
(
log1
ǫ
)−1)
∫
B1
|wǫ|p|x|2p dx ≥ 2
p+ 1
(
p
n− 2p
)p
ωn log1
ǫ+O
(
(
log1
ǫ
)−1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
and
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
∫
B1
|∆wǫ|pdx−(
n− 2p
p
)p(np− n
p
)p ∫
B
|wǫ|p|x|2p dx
≤ O
(
(
log1
ǫ
)−1)
and∫
B1
|wǫ|p|x|2p
(
logR
|x|
)−γ
≥ O
(
(
log1
ǫ
)1−γ)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
But by Rellich inequality
limǫ→0
I(wǫ) ≥(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1b. SHARPNESS OF(
n−2pp
)p (np−n
p
)p
limǫ→0
I(wǫ) = limǫ→0
∫
B1|∆wǫ|pdx
∫
B1
|wǫ|p|x|2p dx
≤(
n− 2p
p
)p(np− n
p
)p
But by Rellich inequality
limǫ→0
I(wǫ) ≥(
n− 2p
p
)p(np− n
p
)p
hence
limǫ→0
I(wǫ) =
(
n− 2p
p
)p(np− n
p
)p
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
Hence using a family wǫ ∈ W 2,p0 (B1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
Hence using a family wǫ ∈ W 2,p0 (B1) we have
limǫ→0 Iγ(wǫ) = 0.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
1c. OPTIMALITY OF THE EXPONENT 2
Assume 0 < γ < 2. Optimality will follow if we can showfor a unit ball B1 that inf
u∈W 2,p0 (B1)\0
Iγ(u) = 0 where
Iγ(u) =
∫
B1|∆u|pdx−
(
n−2pp
)p (np−n
p
)p∫
B1
|u|p|x|2pdx
∫
B1
|u|p|x|2p
(
log R|x|
)−γ
dx
Hence using a family wǫ ∈ W 2,p0 (B1) we have
limǫ→0 Iγ(wǫ) = 0. Thus optimality follow. i.e. γ ≥ 2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA 3
Assume f ∈ C2(B1) and u ∈ C20(B1) are radial satisfying
f(r) > 0,∆f(r) ≤ 0, u(r) > 0, and −∆u > 0 where r = |x|.Set u(r) = f(r)v(r), then for any u ∈ C2
0(B1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
LEMMA 3
∫
B1
|∆u|n2 dx ≥
n(n− 2)
4ωn
∫ 1
0
(v′(r))2v
n−42 (r)rn−1|∆f(r)|n−2
2 f(r)dr
+ ωn
∫ 1
0
vn2 (r)
rn−1|∆f(r)|n2 +
∂r
[
rn−1
(
|∆f(r)|n−22 f ′(r)− ∂r|∆f(r)|n−2
2 f(r)
)]
dr
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
From Lemma 3, we denote
S1 =n(n− 2)
4ωn
∫ 1
0
(v′(r))2v
n−42 (r)rn−1|∆f(r)|n−2
2 f(r)dr
and
S2 =ωn
∫ 1
0
vn2 (r)
rn−1|∆f(r)|n2 +
∂r
[
rn−1
(
|∆f(r)|n−22 f ′(r)− ∂r|∆f(r)|n−2
2 f(r)
)]
dr
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
S1 ≥a
n−22 (1− a)2(n− 2)
n2
4ωn
∫ 1
0
un2
(
logR
r
)−n2−1
dr
r
S2 ≥a
n−22 (1− a)(n− 2)
n2+1
2ωn
∫ 1
0
un2
(
logR
r
)−n2
(
1 +O
(
logR
r
)−2)
dr
r
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
S1 ≥a
n−22 (1− a)2(n− 2)
n2
4ωn
∫ 1
0
un2
(
logR
r
)−n2−1
dr
r
S2 ≥a
n−22 (1− a)(n− 2)
n2+1
2ωn
∫ 1
0
un2
(
logR
r
)−n2
(
1 +O
(
logR
r
)−2)
dr
r
set Q(a) = an−22 (1− a),
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
Then∫
B1|∆u|n2 dx ≥ S1 + S2. For f(r) =
(
log Rr
)a,
0 < a < 1
S1 ≥a
n−22 (1− a)2(n− 2)
n2
4ωn
∫ 1
0
un2
(
logR
r
)−n2−1
dr
r
S2 ≥a
n−22 (1− a)(n− 2)
n2+1
2ωn
∫ 1
0
un2
(
logR
r
)−n2
(
1 +O
(
logR
r
)−2)
dr
r
set Q(a) = an−22 (1− a), Q takes its maximum at a = n−2
2
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
where C∗ =(
n−2√n
)n (
(n− 2)−1 − k(logR)−1)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
where C∗ =(
n−2√n
)n (
(n− 2)−1 − k(logR)−1)
> 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2a. PROOF OF INEQUALITY (12)
∫
B1
|∆u|n2 dx ≥(
n− 2√n
)n ∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2
dx
+ C∗∫
B1
|u(x)|n2|x|n
(
logR
|x|
)−n2−1
dx (12)
where C∗ =(
n−2√n
)n (
(n− 2)−1 − k(logR)−1)
> 0 if
R > e(n−2)k, k = k(n)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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2b. SHARPNESS OF(
n−2√n
)n
To show sharpness, we use the test function
zǫ =
(
log1
r + ǫ
)n−2n
−(
log1
1 + ǫ
)n−2n
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2b. SHARPNESS OF(
n−2√n
)n
To show sharpness, we use the test function
zǫ =
(
log1
r + ǫ
)n−2n
−(
log1
1 + ǫ
)n−2n
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
2b. SHARPNESS OF(
n−2√n
)n
To show sharpness, we use the test function
zǫ =
(
log1
r + ǫ
)n−2n
−(
log1
1 + ǫ
)n−2n
then we can show
limǫ→0
∫
B1|∆zǫ|
n2 dx
∫
B1
|zǫ|n2
|x|n
(
log R|x|
)−n2dx
=
(
n− 2√n
)n
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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2c. OPTIMALITY OF THE EXPONENT n2
We use the same test function uǫ with p = n2, and
wǫ =∫ 1
ruǫ(ρ)dρ. Then for 0 < γ < n
2
limǫ→0
∫
B1|∆wǫ|
n2 dx
∫
B1
|wǫ|n2
|x|n
(
log R|x|
)γ
dx= 0
Thus optimality follow. i.e. γ ≥ n2
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APPLICATION
Consider the weighted eigenvalue problem with a singularweight
∆(
|∆u|p−2∆u)
− µ
|x|2p |u|p−2u = λ|u|p−2uf in Ω
u = ∆u = 0 on ∂Ω (15)
Here f ∈ Fp
Fp =
f : Ω → R+| lim
|x|→0|x|2pf(x) = 0, f ∈ L∞
loc
(
Ω \ 0)
,
1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
and λ ∈ R.
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We look for a weak solution
u ∈ W = W 2,p(Ω) ∩W 1,p0 (Ω)
of problem ♯15 and study the asymptotic behaviour of thefirst eigenvalues for different singular weights as µ increases
to(
n−2pp
)p (np−n
p
)p
.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
We look for a weak solution
u ∈ W = W 2,p(Ω) ∩W 1,p0 (Ω)
of problem ♯15 and study the asymptotic behaviour of thefirst eigenvalues for different singular weights as µ increases
to(
n−2pp
)p (np−n
p
)p
.
Definition
u ∈ W is said to be a weak solution of (15) iff for anyφ ∈ C2(Ω) with φ = 0 on ∂Ω
∫
Ω
(
|∆u|p−2∆u∆φ− µ
|x|2p |u|p−2uφ
)
dx = λ
∫
Ω
|u|p−2ufφdx.
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INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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LEMMA
For u ∈ W ∃ v ∈ W such that v > 0 and satisfies
∫
Ω|∆u|pdx− λ
∫
Ω|u|p|x|2pdx
∫
Ω|u|pfdx ≥
∫
Ω|∆v|pdx− λ
∫
Ω|v|p|x|2pdx
∫
Ω|v|pfdx .
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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LEMMA
For u ∈ W ∃ v ∈ W such that v > 0 and satisfies
∫
Ω|∆u|pdx− λ
∫
Ω|u|p|x|2pdx
∫
Ω|u|pfdx ≥
∫
Ω|∆v|pdx− λ
∫
Ω|v|p|x|2pdx
∫
Ω|v|pfdx .
REMARK
Since λ is first eigenvalue and u is the correspondingeigenfunction, by using the above lemma, we can assumeu > 0 in Ω. Then by the elliptic regularity theory, u issmooth near the boundary. From the definition of weaksolution one can derive the boundary condition of (15) byusing integration by parts.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
If lim sup|x|→0
|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
If lim sup|x|→0
|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0
If lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞, ⇒ λ(f) > 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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THEOREM
For all 1 < p < n2, 0 ≤ µ <
(
n−2pp
)p (np−n
p
)p
, the above
problem ♯15 admits a positive weak solution u ∈ Wcorresponding to the first eigenvalue λ = λ1
µ(f) > 0.
Moreover, as µ →(
n−2pp
)p (np−n
p
)p
,
If lim sup|x|→0
|x|2pf(x) = 0, ⇒ λ1µ(f) → λ(f) ≥ 0
If lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞, ⇒ λ(f) > 0
If lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
= ∞, ⇒ λ(f) = 0
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INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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REMARK
In the proof of the theorem u will be characterize as asolution of variational problem defined by
Jµ(u) =
∫
Ω
(
|∆u|p − µ|u|p|x|2p
)
dx
and the problem (♯15) stated earlier becomes Euler-Lagrange equation of this variational problem.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
We minimize Jµ over M = u ∈ W |∫
Ω|u(x)|pf(x)dx = 1
and let λ1µ > 0 be the infimum.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
We minimize Jµ over M = u ∈ W |∫
Ω|u(x)|pf(x)dx = 1
and let λ1µ > 0 be the infimum.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
We define Jµ(u) =∫
Ω
(
|∆u|p − µ |u|p|x|2p
)
dx
We minimize Jµ over M = u ∈ W |∫
Ω|u(x)|pf(x)dx = 1
and let λ1µ > 0 be the infimum.
We choose minimizing sequence (um)m∈N ⊂ M such thatJµ(um) → λ1
µ.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
For a subsequence umkof um, umk
u weakly in W whereu ∈ W ∩M and
Jµ(umk) → λ1
µ = λ
J ′µ(umk
) → 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
For a subsequence umkof um, umk
u weakly in W whereu ∈ W ∩M and
Jµ(umk) → λ1
µ = λ
J ′µ(umk
) → 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
For a subsequence umkof um, umk
u weakly in W whereu ∈ W ∩M and
Jµ(umk) → λ1
µ = λ
J ′µ(umk
) → 0
By Fatou’s lemma, we get
umk→ u strongly in W
umk→ u strongly in Lp (Ω, |x|−2p)
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
SKETCH OF THE PROOF OF THE THEOREM
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
Since u ∈ W , it is a weak solution of problem (♯15).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
Since u ∈ W , it is a weak solution of problem (♯15).
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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SKETCH OF THE PROOF OF THE THEOREM
Hence
Jµ(umk) → Jµ(u) = λ1
µ = λ
J ′µ(umk
) = 0
u satisfies Euler- Lagrange equation in a distribution sense.
Since u ∈ W , it is a weak solution of problem (♯15).
The remaining part of the proof follows from the corollaryof the main theorem.
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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Corollary
Let 1 < p < n2, and let
Fp =
f : Ω → R+|f ∈ L∞
loc(Ω \ 0) with
lim sup|x|→0
|x|2pf(x)(
log1
|x|
)2
< ∞
If f ∈ Fp, ∃ λ(f) > 0 such that for u ∈ W 2,p0 (Ω)
∫
Ω
|∆u|pdx ≥ Λn,p
∫
Ω
|u(x)|p|x|2p dx+λ(f)
∫
Ω
|u(x)|pf(x)dx (13)
If f /∈ Fp and if |x|2pf(x)(
log 1|x|
)2
tends to ∞ as |x| → 0,
then no inequality of type (13) can hold.A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
λ(f) = infu∈W (Ω\0)
∫
Ω
(
|∆u|p −(
n−2pp
)p (np−n
p
)p |u|p|x|2p
)
dx∫
Ω|u|pfdx ≥ 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
λ(f) = infu∈W (Ω\0)
∫
Ω
(
|∆u|p −(
n−2pp
)p (np−n
p
)p |u|p|x|2p
)
dx∫
Ω|u|pfdx ≥ 0
if f ∈ Fp ⇒ λ(f) > 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
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As µ →(
n−2pp
)p (np−n
p
)p
, λ1µ → λ(f) where
λ(f) = infu∈W (Ω\0)
∫
Ω
(
|∆u|p −(
n−2pp
)p (np−n
p
)p |u|p|x|2p
)
dx∫
Ω|u|pfdx ≥ 0
if f ∈ Fp ⇒ λ(f) > 0if f 6= Fp ⇒ λ(f) = 0
A. L. Detalla Missing Terms in Classical Inequalities
INTRODUCTIONIMPROVED HARDY-SOBOLEV INEQUALITIES
IMPROVED RELLICH INEQUALITIES
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A. L. Detalla Missing Terms in Classical Inequalities