of the Non-critical Caffarelli-Kohn-Nirenberg type inequalities by a linearization method

On the Symmetry Breaking

of the Non-critical Caffarelli-Kohn-Nirenberg type inequalities by a linearization method

堀内 利郎

;

( [email protected])

本講演では非臨界な

CKN

[2]

CKN

1 CKN ^{型不等式の導入}

ここでは

CKN

p, q

γ

1 ≤ p ≤ q < ∞ , γ ∈ R \ { 0 } . (1.1)

γ > 0

γ < 0

γ = 0

臨界と言われる。

Definition 1.1. α ∈ R, R ≥ 1

I

(x) = I

( | x | ) = 1

| x |

for x ∈ R

\{ 0 } , (1.2) A

(x) = A

( | x | ) = log R

| x | for x ∈ B

\{ 0 } , (1.3)

< α < n

I

α

Riesz kernel

このとき、

CKN

S

C

,

γ 6 = 0

∫

Rⁿ

|∇ u(x) |

I

(x)dx ≥ S

(∫

Rⁿ

| u(x) |

I

(x)dx )

, (1.4)

γ = 0

∫

B₁

|∇ u(x) |

I

(x)dx ≥ C

(∫

B₁

| u(x) |

I

(x) A

(x)

dx

)

. (1.5)

ただし、

S

= inf

u∈C₀^∞(Rⁿ)\{0}

∫

Rⁿ

|∇ u(x) |

I

(x)dx (∫

Rⁿ

| u(x) |

I

(x)dx

)

(1.6)

C

= inf

u∈C₀^∞(Rⁿ)\{0}

∫

Rⁿ

|∇ u(x) |

I

(x)dx (∫

B₁

| u(x) |

I

(x) A

(x)

dx

)

.

(1.7)

S

C

C

(R

)

2 ^非臨界 CKN- ^{型不等式について}

Definition 2.1. 1 < p ≤ q < ∞ , p

= p/(p − 1), τ

= 1 p − 1

q , B( · , · )

γ

= n − 1

1 + q/p

, S

= (p

)

q

( ω

τ

B ( 1

p τ

, 1 p

τ

))

if p < q. (2.1)

1

この

S

S

= 1

Theorem 2.1. 1 < p ≤ q ≤ q < ∞ , τ

≤ 1

n

1.

= S

, S

= S

for γ 6 = 0.

2. S

= S

| γ |

for γ 6 = 0.

3.

S

= S

= S

| γ |

for 0 < | γ | ≤ γ

.

4. γ

γ

S

≤ S

≤ γ γ

S

for 0 < | γ | ≤ | γ | .

5. 1

(2 − γ

/γ)

S

≤ S

≤ S

= S

for | γ | ≥ γ

= n − p

p if p < n.

6. S

= S

= S

for | γ | ≥ γ

= n − 2

2 if p = 2 < n.

7. S

≥ ( | γ |

(S

)

)

for γ 6 = 0.

特に,

S

≥ | γ |

(S

)

for γ 6 = 0 if p < n.

Remark 2.1. 1. S

= S

= | γ |

for γ 6 = 0.

2. For 1 < p < n, S

= S

= n ( n − p

p − 1 )

(

ω

p

B

( n p , n

p

))

（ソボレフ最良定数）

Theorem 2.2. For 1 < p < ∞ ,

([ p, p

] \{∞} ) × (R \{ 0 } ) 3 (q ; γ) 7→ S

, S

∈ R (2.2)

S

→ S

= | γ |

as q → p.

Theorem 2.3. 1 < p ≤ q < ∞ , τ

≤ 1/n, p

= np/(n − p)

γ 6 = 0

1.

p < q

S

W

(R

)

\{ 0 } .

2.

p < q < p

S

W

(R

) \{ 0 } .

3.

p < n, q = p

| γ | ≤ (n − p)/p = γ

S

= S

W

(R

)

\{ 0 }

4.

p = 2 < n, q = 2

= 2n/(n − 2) and | γ | > (n − 2)/2 = γ

, S

= S

が成り立ち、

S

W

(R

) \{ 0 } .

Theorem 2.4. (球対称な極値関数 ) 1 < p < q < + ∞

γ > 0

S

u(x) = u(r), (r = | x | )

u(r) = λ

[1 + r

]

,

, h = qγτ

> 0, λ = ( p

p − 1 )

γ

q. (2.3)

さらに

u

Euler-Lagrange

:

L

(u) ≡ − div (I

(x) |∇ u |

∇ u) = I

(x) | u |

u. (2.4)

3 ^臨界 CKN- ^型不等式

Theorem 3.1. 1 < p ≤ q < ∞ , τ

≤ 1/n

もし

R > 1

C

≥ C

> 0.

もし

R = 1

1.

n = 1

C

≥ C

> 0

2.

n ≥ 2

, C

> 0

.

C

= 0

. Definition 3.1. 1 < p ≤ q < ∞

R

= exp 1 + q/p

(n − 1)p

if n ≥ 2, C

= S

(p

)

. (3.1) Theorem 3.2. 1 < p ≤ q < ∞

τ

≤ 1/n

1. C

= C

for R ≥ 1.

2. C

= C

= C

for R ≥

{ 1 if n = 1, R

if p ≥ n ≥ 2.

3. C

≤ C

≤ ( log R

log R )

C

for 1 < R ≤ R.

Theorem 3.3. 1. 1 < p < ∞

([ p, p

] \{∞} ) × (1, ∞ ) 3 (q ; R) 7→ C

, C

∈ R (3.2) 2. n = 1

1 < p < ∞

[ p, ∞ ) × [1, ∞ ) 3 (q ;R ) 7→ C

= C

∈ R (3.3) Theorem 3.4. 1 < p < q < ∞ , τ

≤ 1/n

R ≥ 1

1. R = 1

, C

W

(B

)

\{ 0 }

.

2. n = 1, R = 1

C

= C

W

(( − 1,1))

\{ 0 }

3. R > 1

C

W

(B

)

\{ 0 }

Proposition 3.1. 1 < p = q < ∞

τ

≤ 1/n

R > 1

C

, C

W

(B

) \{ 0 } , W

(B

)

\{ 0 }

4 線形化 CKN-type 作用素

Definition 4.1.

ϕ ∈ C

(R

)

L

(u)ϕ = − div

(

I

(x) |∇ u |

(

∇ ϕ + (p − 2) ( ∇ u, ∇ ϕ)

|∇ u |

∇ u ))

(4.1)

5 主要な結果

Theorem 5.1. ( The symmetry breaking ) n > 1

. q

p < q < p

| γ |

S

W

(R

)

証明は球対称関数

u

w

and w

u

γ

G(η, s) =

∫

Rⁿ

| u + ηw

+ sw

|

I

(x) dx (5.1)

References

[1] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Com- positio Math.,Vol. 53, 1984, No. 3, pp259-275.

[2] T. Horiuchi, P. Kumlin, “On the Caffarelli-Kohn-Nirenberg type inequalities in-

volving Critical and Supercritical Weights”, to appear in Journal of Math-

ematics of Kyoto University.

[www.math.chalmers.se/Math/Research/Preprints/2011/20.pdf]