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The Pohozaev identity for the fractional Laplacian
Xavier Ros-Oton
Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya
(joint work with Joaquim Serra)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 1 / 18
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Outline of the talk
The classical Pohozaev identity; applications
The Dirichlet semilinear problem for the fractional Laplacian
The Pohozaev identity for the fractional Laplacian
Applications
Sketch of the proof
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 2 / 18
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The classical Pohozaev identity
Ω bounded Lipschitz domain, −∆u = f (u) in Ωu = 0 on ∂Ω, (1)Theorem (Pohozaev)
(2− n)∫
Ω
u f (u)dx + 2n
∫Ω
F (u)dx =
∫∂Ω
|∇u|2(x · ν)dσ
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 3 / 18
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Applications of the classical Pohozaev identity
(2− n)∫
Ω
u f (u)dx + 2n
∫Ω
F (u)dx =
∫∂Ω
|∇u|2(x · ν)dσ
Nonexistence of solutions: critical exponent −∆u = u n+2n−2
Ground states in Rn: monotonicity formulas, estimates
Radial symmetry: proof of P.-L. Lions combining the Pohozaev identity with
the isoperimetric inequality
Stable solutions: uniqueness, H1 interior regularity
etc.
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 4 / 18
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Proof of the classical Pohozaev identity
First note that
∆(x · ∇u) = 2∆u + x · ∇(∆u).
Then, integrating by parts twice and using that u ≡ 0 on ∂Ω, we obtain∫Ω
(x · ∇u)∆u = 2∫
Ω
u∆u +
∫Ω
u x · ∇(∆u) +∫∂Ω
(x · ∇u)(∇u · ν)dσ
= (2− n)∫
Ω
u∆u −∫
Ω
(x · ∇u)∆u +∫∂Ω
|∇u|2(x · ν)dσ
We have used that ∇u · ν = |∇u| on ∂Ω. Finally, since −∆u = f (u), then
2
∫Ω
(x · ∇u)∆u = −2∫
Ω
x · ∇F (u) = 2n∫
Ω
F (u),
and the identity follows.
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 5 / 18
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The Dirichlet semilinear problem with (−∆)s
Ω bounded C 1,1 domain, δ(x) := dist (x , ∂Ω), f ∈ C 1 (−∆)su = f (u) in Ωu = 0 in Rn\Ω,Theorem (X.R., J. Serra)
(i) u ∈ C s(Rn)
(ii) u/δs ∈ Cα(Ω)
(iii) [u]Cβ(Bρ/2) ≤ Cρs−β
(iv)[u/δs
]Cβ(Bρ/2)
≤ Cρα−β
Ω
Bρ
Bρ/2
u ≡ 0
(−∆)su = g
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 6 / 18
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The Pohozaev identity for the fractional Laplacian
Ω bounded C 1,1 domain, (−∆)su = f (u) in Ωu = 0 in Rn \ Ω,Theorem (X. R., J. Serra)
Denote δ(x) := dist (x , ∂Ω). Then u/δs ∈ Cα(Ω) and
(2s − n)∫
Ω
uf (u)dx + 2n
∫Ω
F (u)dx = Γ(1 + s)2∫∂Ω
( uδs
)2(x · ν)dσ,
where Γ is the gamma function.
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 7 / 18
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Corollary: nonexistence results
Ω bounded C 1,1 domain, (−∆)su = f (u) in Ωu = 0 in Rn \ Ω,Corollary
Assume that Ω is star-shaped and F (t) < n−2s2n t f (t) for all t. Then the problem
admits no nontrivial solution.
For example, for f (u) = up we obtain nonexistence for p ≥ n+2sn−2s .For positive solutions, this was done by [Fall-Weth,’12] with moving planes.
Existence for subcritical p by [Servadei-Valdinoci,’12].
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 8 / 18
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Pohozaev identity with (−∆)s
Proposition (X. R., J. Serra)
Assume
1 Ω bounded C 1,1 domain
2 u ∈ C s(Rn), u ≡ 0 outside Ω, u/δs ∈ Cα(Ω)3 Interior Cβ estimates for u and u/δs , β < 1 + 2s
4 (−∆)su is bounded in Ω
Then∫Ω
(x · ∇u)(−∆)su = 2s − n2
∫Ω
u(−∆)su − Γ(1 + s)2
2
∫∂Ω
( uδs
)2(x · ν)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 9 / 18
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Main consequences
Changing the origin in our identity, we deduce the following
Theorem (X. R., J. Serra)
Under the same hypotheses of the Proposition,∫Ω
(−∆)su vxi = −∫
Ω
uxi (−∆)sv + Γ(1 + s)2∫∂Ω
u
δsv
δsνi
It has a local boundary term!
Note the contrast with the nonlocal flux in the formula for∫
Ωf (x , u)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 10 / 18
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Sketch of the Proof (Star-shaped domains)
1 uλ(x) = u(λx) ⇒∫Ω
(x · ∇u)(−∆)su = ddλ
∣∣∣∣λ=1+
∫Ω
uλ(−∆)su
2 Ω star-shaped ⇒ uλ vanishes outside Ω for λ > 1 ⇒∫Ω
uλ(−∆)su =∫Rn
(−∆) s2 uλ(−∆)s2 u
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 11 / 18
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∫Rn
(−∆) s2 uλ(−∆)s2 u = λs
∫Rn
((−∆) s2 u
)(λx)(−∆) s2 u(x) dx
= λs∫Rn
w(λx)w(x) dx
= λ2s−n
2
∫Rn
w(λ12 y)w(λ−
12 y) dy
where w = (−∆) s2 u . Therefore,
∫Ω
(x · ∇u)(−∆)su = 2s − n2
∫Rn
w2 +1
2
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ
where wλ(x) = w(λx).
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 12 / 18
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∫Rn
(−∆) s2 uλ(−∆)s2 u = λs
∫Rn
((−∆) s2 u
)(λx)(−∆) s2 u(x) dx
= λs∫Rn
w(λx)w(x) dx
= λ2s−n
2
∫Rn
w(λ12 y)w(λ−
12 y) dy
where w = (−∆) s2 u . Therefore,
∫Ω
(x · ∇u)(−∆)su = 2s − n2
∫Ω
u(−∆)su + 12
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ
where wλ(x) = w(λx).
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 12 / 18
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What about ddλ∣∣λ=1+
∫Rn wλw1/λ?
I(ϕ) = − ddλ
∣∣∣∣λ=1+
∫Rnϕ(λx)ϕ(x/λ) dx
Important properties:
1 I(ϕ) ≥ 0 since∫Rnϕ(λx)ϕ(x/λ)dx ≤
(∫Rnϕ2(λx)dx
) 12(∫
Rnϕ2(x/λ)dx
) 12
=
∫Rnϕ2
2 ψ smooth ⇒ I(ψ) = 03 If I(ψ) = 0 ⇒ I(ϕ+ ψ) = I(ϕ)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 13 / 18
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What about ddλ∣∣λ=1+
∫Rn wλw1/λ?
I(ϕ) = − ddλ
∣∣∣∣λ=1+
∫Rnϕ(λx)ϕ(x/λ) dx
Important properties:
1 I(ϕ) ≥ 0 since∫Rnϕ(λx)ϕ(x/λ)dx ≤
(∫Rnϕ2(λx)dx
) 12(∫
Rnϕ2(x/λ)dx
) 12
=
∫Rnϕ2
2 ψ smooth ⇒ I(ψ) = 03 If I(ψ) = 0 ⇒ I(ϕ+ ψ) = I(ϕ)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 13 / 18
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What about ddλ∣∣λ=1+
∫Rn wλw1/λ?
I(ϕ) = − ddλ
∣∣∣∣λ=1+
∫Rnϕ(λx)ϕ(x/λ) dx
Important properties:
1 I(ϕ) ≥ 0 since∫Rnϕ(λx)ϕ(x/λ)dx ≤
(∫Rnϕ2(λx)dx
) 12(∫
Rnϕ2(x/λ)dx
) 12
=
∫Rnϕ2
2 ψ smooth ⇒ I(ψ) = 03 If I(ψ) = 0 ⇒ I(ϕ+ ψ) = I(ϕ)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 13 / 18
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What about ddλ∣∣λ=1+
∫Rn wλw1/λ?
I(ϕ) = − ddλ
∣∣∣∣λ=1+
∫Rnϕ(λx)ϕ(x/λ) dx
Important properties:
1 I(ϕ) ≥ 0 since∫Rnϕ(λx)ϕ(x/λ)dx ≤
(∫Rnϕ2(λx)dx
) 12(∫
Rnϕ2(x/λ)dx
) 12
=
∫Rnϕ2
2 ψ smooth ⇒ I(ψ) = 03 If I(ψ) = 0 ⇒ I(ϕ+ ψ) = I(ϕ)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 13 / 18
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What about ddλ∣∣λ=1+
∫Rn wλw1/λ?
We want to compute:
I(w) = − ddλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ
Reduce to a 1− D calculationUse “star-shaped” (t, z)-coordinates
x = tz , z ∈ ∂Ω, t > 0
0Ω
zz̃
(2/3, z)
(1/3, z)
(1/2, z̃)
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ =d
dλ
∣∣∣∣λ=1+
∫∂Ω
(z · ν)dσ(z)∫ ∞
0
tn−1w(λtz)w( tzλ
)dt
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 14 / 18
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What about ddλ∣∣λ=1+
∫Rn wλw1/λ?
We want to compute:
I(w) = − ddλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ
Reduce to a 1− D calculationUse “star-shaped” (t, z)-coordinates
x = tz , z ∈ ∂Ω, t > 0
0Ω
zz̃
(2/3, z)
(1/3, z)
(1/2, z̃)
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ =
∫∂Ω
(z · ν)dσ(z) ddλ
∣∣∣∣λ=1+
∫ ∞0
tn−1w(λtz)w( tzλ
)dt
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 14 / 18
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What do we know about w = (−∆)s/2u?
Proposition (X. R., J. Serra)
Fix z ∈ ∂Ω. Then,
w(tz) = (−∆)s/2u(tz) = c1{
log− |t − 1|+ c2χ(0,1)(t)} uδs
(z) + h(t)
whered
dλ
∣∣∣∣λ=1+
∫ ∞0
tn−1h(λt)h( tλ
)dt = 0
c1 =Γ(1 + s) sin
(πs2
)π
, and c2 =π
tan(πs2
)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 15 / 18
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Summarising...
w(tz) = c1{
log− |t − 1|+ c2χ(0,1)(t)} uδs
(z) + h(t)
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ =
∫∂Ω
(z · ν)dσ(z) ddλ
∣∣∣∣λ=1+
∫ ∞0
tn−1w(λtz)w( tzλ
)dt
=
∫∂Ω
(z · ν)dσ(z) ddλ
∣∣∣∣λ=1+
( uδs
(z))2 ∫ ∞
0
tn−1φs(λt)φs( tλ
)dt
=
∫∂Ω
(z · ν)dσ(z)( uδs
(z))2
C (s)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 16 / 18
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Summarising...
w(tz) = φs(t)u
δs(z) + h(t)
where φs(t) = c1{
log− |t − 1|+ c2χ(0,1)(t)}
d
dλ
∣∣∣∣λ=1+
∫Rn
wλw1/λ =
∫∂Ω
(z · ν)dσ(z) ddλ
∣∣∣∣λ=1+
∫ ∞0
tn−1w(λtz)w( tzλ
)dt
=
∫∂Ω
(z · ν)dσ(z) ddλ
∣∣∣∣λ=1+
( uδs
(z))2 ∫ ∞
0
tn−1φs(λt)φs( tλ
)dt
=
∫∂Ω
(z · ν)dσ(z)( uδs
(z))2
C (s)
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 16 / 18
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And if the domain is not star-shaped...
Key observations:
1 Pohozaev identity is quadratic in u and it “comes from a bilinear identity”∫Ω
(x · ∇u)(−∆)su = 2s−n2∫
Ωu(−∆)su − Γ(1+s)
2
2
∫∂Ω
(uδs
)2(x · ν)
∫Ω
(x · ∇u)(−∆)sv +∫
Ω(x · ∇v)(−∆)su =
2s−n2
∫Ωu(−∆)sv + 2s−n2
∫Ωv(−∆)su − Γ(1 + s)2
∫∂Ω
uδs
vδs (x · ν)
2 every C 1,1 domain is locally star-shaped
3 the bilinear identity holds easily when u and v have disjoint support
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 17 / 18
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And if the domain is not star-shaped...
Key observations:
1 Pohozaev identity is quadratic in u and it “comes from a bilinear identity”∫Ω
(x · ∇u)(−∆)su = 2s−n2∫
Ωu(−∆)su − Γ(1+s)
2
2
∫∂Ω
(uδs
)2(x · ν)
∫Ω
(x · ∇u)(−∆)sv +∫
Ω(x · ∇v)(−∆)su =
2s−n2
∫Ωu(−∆)sv + 2s−n2
∫Ωv(−∆)su − Γ(1 + s)2
∫∂Ω
uδs
vδs (x · ν)
2 every C 1,1 domain is locally star-shaped
3 the bilinear identity holds easily when u and v have disjoint support
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 17 / 18
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And if the domain is not star-shaped...
Key observations:
1 Pohozaev identity is quadratic in u and it “comes from a bilinear identity”∫Ω
(x · ∇u)(−∆)su = 2s−n2∫
Ωu(−∆)su − Γ(1+s)
2
2
∫∂Ω
(uδs
)2(x · ν)
∫Ω
(x · ∇u)(−∆)sv +∫
Ω(x · ∇v)(−∆)su =
2s−n2
∫Ωu(−∆)sv + 2s−n2
∫Ωv(−∆)su − Γ(1 + s)2
∫∂Ω
uδs
vδs (x · ν)
2 every C 1,1 domain is locally star-shaped
3 the bilinear identity holds easily when u and v have disjoint support
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 17 / 18
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And if the domain is not star-shaped...
Key observations:
1 Pohozaev identity is quadratic in u and it “comes from a bilinear identity”∫Ω
(x · ∇u)(−∆)su = 2s−n2∫
Ωu(−∆)su − Γ(1+s)
2
2
∫∂Ω
(uδs
)2(x · ν)
∫Ω
(x · ∇u)(−∆)sv +∫
Ω(x · ∇v)(−∆)su =
2s−n2
∫Ωu(−∆)sv + 2s−n2
∫Ωv(−∆)su − Γ(1 + s)2
∫∂Ω
uδs
vδs (x · ν)
2 every C 1,1 domain is locally star-shaped
3 the bilinear identity holds easily when u and v have disjoint support
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 17 / 18
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The end
Thank you!
Xavier Ros-Oton (UPC) The Pohozaev identity for the fractional Laplacian BCAM, Bilbao, February 2013 18 / 18