The Pohozaev identity for the fractional Laplacian...Xavier Ros-Oton Departament Matem atica Aplicada I, Universitat Polit ecnica de Catalunya (joint work with Joaquim Serra) Xavier

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

Outline of the talk

The classical Pohozaev identity; applications

The Dirichlet semilinear problem for the fractional Laplacian


Applications

Sketch of the proof


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


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.


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.


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



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.


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].


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)


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)


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


∫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).


∫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).


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)



∫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



∫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


)dt


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

)


Summarising...

w(tz) = c1{

log− |t − 1|+ c2χ(0,1)(t)} uδs

(z) + h(t)

d

dλ

∣∣∣∣λ=1+

∫Rn

wλw1/λ =

∫∂Ω


∣∣∣∣λ=1+

∫ ∞0


)dt

=

∫∂Ω


∣∣∣∣λ=1+

( uδs

(z))2 ∫ ∞

0

tn−1φs(λt)φs( tλ

)dt

=

∫∂Ω

(z · ν)dσ(z)( uδs

(z))2

C (s)


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


∣∣∣∣λ=1+

∫ ∞0


)dt

=

∫∂Ω


∣∣∣∣λ=1+

( uδs

(z))2 ∫ ∞

0

tn−1φs(λt)φs( tλ

)dt

=

∫∂Ω

(z · ν)dσ(z)( uδs

(z))2

C (s)


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


The end

Thank you!


Documents

The Pohozaev identity for the fractional Laplacian...Xavier Ros-Oton Departament Matem atica Aplicada I, Universitat Polit ecnica de Catalunya (joint work with Joaquim Serra) Xavier