monotonicity methods for asymptotics of solutions …monotonicity methods for asymptotics of...
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Monotonicity methods for asymptotics ofsolutions to elliptic and parabolic equations
near singularities of the potential
Veronica Felli
Dipartimento di Matematica ed ApplicazioniUniversity of Milano–Bicocca
WIMCS – LMS Workshop on “Calculus of Variations and Nonlinear PDEs”
Swansea University, May 20, 2011
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Problem:describe the behavior at singularity of solutions to equationsassociated to Schrodinger operators with singularhomogeneous potentials
La := −∆− a(x/|x|)|x|2 , x ∈ R
N , a : SN−1 → R, N > 3
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Problem:describe the behavior at singularity of solutions to equationsassociated to Schrodinger operators with singularhomogeneous potentials
La := −∆− a(x/|x|)|x|2 , x ∈ R
N , a : SN−1 → R, N > 3
Examples: Dipole-potential
− ~2
2m∆+ e
x · D|x|3 a(θ) = λθ · D
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Problem:describe the behavior at singularity of solutions to equationsassociated to Schrodinger operators with singularhomogeneous potentials
La := −∆− a(x/|x|)|x|2 , x ∈ R
N , a : SN−1 → R, N > 3
Examples: Quantum many-body
M∑
j=1
−∆j
2mj+
M∑
j,m=1j<m
λjλm
|xj − xm|2 a(θ) =∑ λjλm
|θj − θm|2
xj ∈ Rd, N = Md, θj = xj/|x|, x = (x1, . . . , xM) ∈ R
N
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Problem:describe the asymptotic behavior at the singularity of solutionsto equations associated to Schrodinger operators with singularhomogeneous electromagnetic potentials
LA,a :=
(
− i∇+A(
x|x|
)
|x|
)2
−a(
x|x|
)
|x|2 , A ∈ C1(SN−1,RN)
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Problem:describe the asymptotic behavior at the singularity of solutionsto equations associated to Schrodinger operators with singularhomogeneous electromagnetic potentials
LA,a :=
(
− i∇+A(
x|x|
)
|x|
)2
−a(
x|x|
)
|x|2 , A ∈ C1(SN−1,RN)
Example: Aharonov-Bohm magnetic potentials associated tothin solenoids; if the radius of the solenoid tends to zero whilethe flux through it remains constant, then the particle is subjectto a δ-type magnetic field, which is called Aharonov-Bohm field.
A( x|x| )
|x| = α(
− x2
|x|2 ,x1
|x|2)
, x = (x1, x2) ∈ R2
A(θ1, θ2) = α(−θ2, θ1), (θ1, θ2) ∈ S1,
with α = circulation around the solenoid.
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Describe the behavior at singularities of solutions: Motivation
• regularity theory for elliptic operators with singularities ofFuchsian type[Mazzeo, Comm. PDE’s(1991)], [Pinchover , Ann. IHP(1994)]
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Describe the behavior at singularities of solutions: Motivation
• regularity theory for elliptic operators with singularities ofFuchsian type[Mazzeo, Comm. PDE’s(1991)], [Pinchover , Ann. IHP(1994)]
• informations about the “critical dimension” for existence ofsolutions to problems with critical growth (Brezis-Nirenberg type)[Jannelli , J. Diff. Eq.(1999)][Ferrero-Gazzola , J. Diff. Eq.(2001)]
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Describe the behavior at singularities of solutions: Motivation
• regularity theory for elliptic operators with singularities ofFuchsian type[Mazzeo, Comm. PDE’s(1991)], [Pinchover , Ann. IHP(1994)]
• informations about the “critical dimension” for existence ofsolutions to problems with critical growth (Brezis-Nirenberg type)[Jannelli , J. Diff. Eq.(1999)][Ferrero-Gazzola , J. Diff. Eq.(2001)]
• construction of solutions to equations with singular potentials[Felli-Terracini , Comm. PDE’s(2006)][Felli , J. Anal. Math.(2009)][Abdellaoui-Felli-Peral , Calc. Var. PDE’s(2009)]
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Describe the behavior at singularities of solutions: Motivation
• regularity theory for elliptic operators with singularities ofFuchsian type[Mazzeo, Comm. PDE’s(1991)], [Pinchover , Ann. IHP(1994)]
• informations about the “critical dimension” for existence ofsolutions to problems with critical growth (Brezis-Nirenberg type)[Jannelli , J. Diff. Eq.(1999)][Ferrero-Gazzola , J. Diff. Eq.(2001)]
• construction of solutions to equations with singular potentials[Felli-Terracini , Comm. PDE’s(2006)][Felli , J. Anal. Math.(2009)][Abdellaoui-Felli-Peral , Calc. Var. PDE’s(2009)]
• study of spectral properties (essential self-adjointness)[Felli-Marchini-Terracini , J. Funct. Anal.(2007)][Felli-Marchini-Terracini , Indiana Univ. Math. J.(2009)]
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Describe the behavior at singularities of solutions: References
[Felli-Schneider, Adv. Nonl. Studies (2003)] : Holder continuityresults for degenerate elliptic equations with singular weights;asymptotics of solutions for potentials λ|x|−2 (a(θ) constant).
[Felli-Marchini-Terracini, DCDS-A (2008)] : asymptotics of solutionsnear the pole for a(x/|x|)|x|−2 (a(θ) bounded), through separation ofvariables and comparison methods.
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Describe the behavior at singularities of solutions: References
[Felli-Schneider, Adv. Nonl. Studies (2003)] : Holder continuityresults for degenerate elliptic equations with singular weights;asymptotics of solutions for potentials λ|x|−2 (a(θ) constant).
[Felli-Marchini-Terracini, DCDS-A (2008)] : asymptotics of solutionsnear the pole for a(x/|x|)|x|−2 (a(θ) bounded), through separation ofvariables and comparison methods.
Difficulties:
• in the presence of a singular magnetic potential, comparisonmethods are no more available!
• in the many-particle/cylindrical case separation of variables (radialand angular) does not “eliminate” the singularity; the angular part ofthe operator is also singular!
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To overcome these difficulties
Almgren type monotonicity formulablow-up methods
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To overcome these difficulties
Almgren type monotonicity formulablow-up methods
[Felli-Ferrero-Terracini, J. Europ. Math. Soc. (2011)] : singularhomogeneous electromagnetic potentials of Aharonov-Bohm type, byan Almgren type monotonicity formula and blow-up methods.
[Felli-Ferrero-Terracini, Preprint 2010] : behavior at collisions ofsolutions to Schrodinger equations with many-particle and cylindricalpotentials by a nonlinear Almgren type formula and blow-up.
[Felli-Primo, DCDS-A, to appear] : local asymptotics for solutions toheat equations with inverse-square potentials.
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Outline of the talk
1. Elliptic monotonicity formula
2. Asymptotics at singularities (elliptic case)
3. Parabolic monotonicity formula
4. Asymptotics at singularities (parabolic case)
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Outline of the talk
1. Elliptic monotonicity formula
2. Asymptotics at singularities (elliptic case)
3. Parabolic monotonicity formula
4. Asymptotics at singularities (parabolic case)
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Outline of the talk
1. Elliptic monotonicity formula
2. Asymptotics at singularities (elliptic case)
3. Parabolic monotonicity formula
4. Asymptotics at singularities (parabolic case)
![Page 18: Monotonicity methods for asymptotics of solutions …Monotonicity methods for asymptotics of solutions to elliptic and parabolic equations near singularities of the potential Veronica](https://reader036.vdocuments.net/reader036/viewer/2022062920/5f0222847e708231d402be6f/html5/thumbnails/18.jpg)
Outline of the talk
1. Elliptic monotonicity formula
2. Asymptotics at singularities (elliptic case)
3. Parabolic monotonicity formula
4. Asymptotics at singularities (parabolic case)
![Page 19: Monotonicity methods for asymptotics of solutions …Monotonicity methods for asymptotics of solutions to elliptic and parabolic equations near singularities of the potential Veronica](https://reader036.vdocuments.net/reader036/viewer/2022062920/5f0222847e708231d402be6f/html5/thumbnails/19.jpg)
1. Elliptic monotonicity formula
Studying regularity of area-minimizing surfaces of codimension> 1, in 1979 F. Almgren introduced the frequency function
N (r) =r2−N
∫
Br|∇u|2 dx
r1−N∫
∂Bru2
and observed that, if u is harmonic, then N ր in r .
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1. Elliptic monotonicity formula
Studying regularity of area-minimizing surfaces of codimension> 1, in 1979 F. Almgren introduced the frequency function
N (r) =r2−N
∫
Br|∇u|2 dx
r1−N∫
∂Bru2
and observed that, if u is harmonic, then N ր in r .
Proof:
N ′(r) =2r
[(∫
∂Br
∣∣ ∂u∂ν
∣∣2
dS)(∫
∂Br|u|2dS
)
−(∫
∂Bru ∂u∂ν dS
)2]
(∫
∂Br|u|2dS
)2
Schwarz’s>
inequality0
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Why frequency?
If N ≡ γ is constant, then N ′(r) = 0, i.e.(∫
∂Br
∣∣∣∣
∂u∂ν
∣∣∣∣
2
dS
)
·(∫
∂Br
u2dS
)
−(∫
∂Br
u∂u∂ν
dS
)2
= 0
=⇒ u and ∂u∂ν are parallel as vectors in L2(∂Br), i.e. ∃ η(r) s.t.
∂u∂ν
(r, θ) = η(r)u(r, θ), i.e.ddr
log |u(r, θ)| = η(r).
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Why frequency?
If N ≡ γ is constant, then N ′(r) = 0, i.e.(∫
∂Br
∣∣∣∣
∂u∂ν
∣∣∣∣
2
dS
)
·(∫
∂Br
u2dS
)
−(∫
∂Br
u∂u∂ν
dS
)2
= 0
=⇒ u and ∂u∂ν are parallel as vectors in L2(∂Br), i.e. ∃ η(r) s.t.
∂u∂ν
(r, θ) = η(r)u(r, θ), i.e.ddr
log |u(r, θ)| = η(r).
After integration we obtain
u(r, θ)= e∫ r
1 η(s)dsu(1, θ) = ϕ(r)ψ(θ).
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Why frequency?u(r, θ) = ϕ(r)ψ(θ) and ∆u = 0
⇓(ϕ′′(r) + N−1
r ϕ′(r))ψ(θ) + r−2ϕ(r)∆SN−1ψ(θ) = 0.
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Why frequency?u(r, θ) = ϕ(r)ψ(θ) and ∆u = 0
⇓(ϕ′′(r) + N−1
r ϕ′(r))ψ(θ) + r−2ϕ(r)∆SN−1ψ(θ) = 0.
ψ is a spherical harmonic ⇒ ∃ k ∈ N : −∆SN−1ψ = k(N − 2 + k)ψ
−ϕ′′(r)− N−1r ϕ(r) + r−2k(N − 2 + k)ϕ(r) = 0.
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Why frequency?u(r, θ) = ϕ(r)ψ(θ) and ∆u = 0
⇓(ϕ′′(r) + N−1
r ϕ′(r))ψ(θ) + r−2ϕ(r)∆SN−1ψ(θ) = 0.
ψ is a spherical harmonic ⇒ ∃ k ∈ N : −∆SN−1ψ = k(N − 2 + k)ψ
−ϕ′′(r)− N−1r ϕ(r) + r−2k(N − 2 + k)ϕ(r) = 0.
ϕ(r) = c1rσ++ c2rσ
−with σ± = −N−2
2 ± 12(2k + N − 2)
σ+ = k, σ− = −(N − 2)− k
|x|σ−ψ( x
|x|) /∈ H1(B1) ; c2 = 0, ϕ(1) = 1 ; c1 = 1⇓
u(r, θ) = rkψ(θ)
From N (r) ≡ γ, we deduce that γ = k.
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Applications to elliptic PDE’s
The Almgren monotonicity formula was used in
• [Garofalo-Lin, Indiana Univ. Math. J. (1986)] :generalization to variable coefficient elliptic operators indivergence form (unique continuation)
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Applications to elliptic PDE’s
The Almgren monotonicity formula was used in
• [Garofalo-Lin, Indiana Univ. Math. J. (1986)] :generalization to variable coefficient elliptic operators indivergence form (unique continuation)
• [Athanasopoulos-Caffarelli-Salsa, Amer. J. Math. (2008) ]:regularity of the free boundary in obstacle problems.
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Applications to elliptic PDE’s
The Almgren monotonicity formula was used in
• [Garofalo-Lin, Indiana Univ. Math. J. (1986)] :generalization to variable coefficient elliptic operators indivergence form (unique continuation)
• [Athanasopoulos-Caffarelli-Salsa, Amer. J. Math. (2008) ]:regularity of the free boundary in obstacle problems.
• [Caffarelli-Lin, J. AMS (2008)] regularity of free boundary ofthe limit components of singularly perturbed ellipticsystems.
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Perturbed monotonicity
Example (Garofalo-Lin). Let u ∈ H1loc(B1) be a weak solution to
−∆u(x) + V(x)u = 0 in B1,
with V ∈ L∞loc(B1). Define
D(r) = 1rN−2
∫
Br
[∣∣∇u
∣∣2 + V|u|2
]
dx
H(r) = 1rN−1
∫
∂Br
|u|2 dS
Almgren type function
N (r) =D(r)H(r)
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Perturbed monotonicity
Example (Garofalo-Lin). Let u ∈ H1loc(B1) be a weak solution to
−∆u(x) + V(x)u = 0 in B1,
with V ∈ L∞loc(B1). Define
D(r) = 1rN−2
∫
Br
[∣∣∇u
∣∣2 + V|u|2
]
dx
H(r) = 1rN−1
∫
∂Br
|u|2 dS
Almgren type function
N (r) =D(r)H(r)
N ′(r) =2r
[(∫
∂Br
∣∣ ∂u∂ν
∣∣2
dS)(∫
∂Br|u|2dS
)
−(∫
∂Bru ∂u∂ν dS
)2]
(∫
∂Br|u|2dS
)2 +R(r)
6
0
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Perturbed monotonicity
R(r) = −2[ ∫
BrVu (x · ∇u) dx + N−2
2
∫
BrVu2 dx − r
2
∫
∂BrVu2 dS
]
∫
∂Br|u|2dS
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Perturbed monotonicity
R(r) = −2[ ∫
BrVu (x · ∇u) dx + N−2
2
∫
BrVu2 dx − r
2
∫
∂BrVu2 dS
]
∫
∂Br|u|2dS
|R(r)| 6 C1 r(N (r) + C2)
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Perturbed monotonicity
R(r) = −2[ ∫
BrVu (x · ∇u) dx + N−2
2
∫
BrVu2 dx − r
2
∫
∂BrVu2 dS
]
∫
∂Br|u|2dS
|R(r)| 6 C1 r(N (r) + C2)
N ′(r) > −C1r(N (r) + C2) i.e.ddr
log(N (r) + C2) > −C1r
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Perturbed monotonicity
R(r) = −2[ ∫
BrVu (x · ∇u) dx + N−2
2
∫
BrVu2 dx − r
2
∫
∂BrVu2 dS
]
∫
∂Br|u|2dS
|R(r)| 6 C1 r(N (r) + C2)
N ′(r) > −C1r(N (r) + C2) i.e.ddr
log(N (r) + C2) > −C1r
integrate between r and r N (r) + C2 6 (N (r) + C2)eC12 r2
6 const
In particular, N is bounded.
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Perturbed monotonicity
H(r) =1
rN−1
∫
∂Br
|u|2 dS =⇒ H′(r) =2
rN−1
∫
∂Br
u∂u∂ν
dS
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Perturbed monotonicity
H(r) =1
rN−1
∫
∂Br
|u|2 dS =⇒ H′(r) =2
rN−1
∫
∂Br
u∂u∂ν
dS
Test −∆u(x) + V(x)u = 0 with u =⇒∫
Br
[∣∣∇u
∣∣2 + V|u|2
]
dx =
∫
∂Br
u∂u∂ν
dS
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Perturbed monotonicity
H(r) =1
rN−1
∫
∂Br
|u|2 dS =⇒ H′(r) =2
rN−1
∫
∂Br
u∂u∂ν
dS
Test −∆u(x) + V(x)u = 0 with u =⇒
rN−2D(r) =∫
Br
[∣∣∇u
∣∣2 + V|u|2
]
dx =
∫
∂Br
u∂u∂ν
dS =rN−1
2H′(r)
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Perturbed monotonicity
H(r) =1
rN−1
∫
∂Br
|u|2 dS =⇒ H′(r) =2
rN−1
∫
∂Br
u∂u∂ν
dS
Test −∆u(x) + V(x)u = 0 with u =⇒
rN−2D(r) =∫
Br
[∣∣∇u
∣∣2 + V|u|2
]
dx =
∫
∂Br
u∂u∂ν
dS =rN−1
2H′(r)
Hence H′(r)H(r) = 2
rN (r) 6 constr , i.e. d
dr log H(r) 6 Cr .
Integrating between R and 2R
logH(2R)H(R)
6 C log 2 i.e.∫
∂B2R
u2 dS 6 const∫
∂BR
u2 dS
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Doubling and unique continuation
Doubling condition∫
B2R
u2 dx 6 Cdoub
∫
BR
u2 dx
with Cdoub independent of R
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Doubling and unique continuation
Doubling condition∫
B2R
u2 dx 6 Cdoub
∫
BR
u2 dx
with Cdoub independent of R
⇓
Strong unique continuation property
If u vanishes of infinite order at 0, i.e.∫
BR
u2 dx = O(Rm) as R → 0 ∀m ∈ N,
then u ≡ 0 in B1.
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2. Asymptotics at singularities (elliptic case)
describe the behavior at the singularity of solutions toequations associated to Schrodinger operators with singularhomogeneous potentials (with the same order of homogeneityof the operator)
La := −∆− a(x/|x|)|x|2 , x ∈ R
N , a : SN−1 → R, N > 3
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Hardy type inequalities
Λ(a) := supu∈D1,2(RN)\0
∫
RN|x|−2a(x/|x|) u2(x) dx∫
RN|∇u(x)|2 dx
• Classical Hardy’s inequality: Λ(1) =(
2N−2
)2.
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Hardy type inequalities
Λ(a) := supu∈D1,2(RN)\0
∫
RN|x|−2a(x/|x|) u2(x) dx∫
RN|∇u(x)|2 dx
• Classical Hardy’s inequality: Λ(1) =(
2N−2
)2.
• If a ∈ L∞(SN−1), classical Hardy’s inequality ⇒ Λ(a) < +∞.
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Hardy type inequalities
Λ(a) := supu∈D1,2(RN)\0
∫
RN|x|−2a(x/|x|) u2(x) dx∫
RN|∇u(x)|2 dx
• Classical Hardy’s inequality: Λ(1) =(
2N−2
)2.
• If a ∈ L∞(SN−1), classical Hardy’s inequality ⇒ Λ(a) < +∞.
• For many body potentialsa(x/|x|)|x|2 =
∑Mj<m
λjλm
|xj−xm|2
Maz’ja and Badiale-Tarantello
inequalities+3 Λ(a) < +∞.
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Hardy type inequalities
Λ(a) := supu∈D1,2(RN)\0
∫
RN|x|−2a(x/|x|) u2(x) dx∫
RN|∇u(x)|2 dx
• Classical Hardy’s inequality: Λ(1) =(
2N−2
)2.
• If a ∈ L∞(SN−1), classical Hardy’s inequality ⇒ Λ(a) < +∞.
• For many body potentialsa(x/|x|)|x|2 =
∑Mj<m
λjλm
|xj−xm|2
Maz’ja and Badiale-Tarantello
inequalities+3 Λ(a) < +∞.
The quadratic form associated to
La = −∆− a(x/|x|)|x|2
is positive definite in D1,2(RN)
⇐⇒ Λ(a) < 1
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Perturbations of La
linear /semilinear perturbations of La: Lau = h(x) u(x) + f(x,u(x))
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Perturbations of La
linear /semilinear perturbations of La: Lau = h(x) u(x) + f(x,u(x))
(H) h “negligible” with respect to a(x/|x|)|x|2 at singularity:
limr→0+
η0(r) = 0,η0(r)
r∈ L1,
1
r
∫ r
0
η0(s)
sds ∈ L1,
η1(r)
r∈ L1,
1
r
∫ r
0
η1(s)
sds ∈ L1,
where η0(r) = supu∈H1(Br)
u6≡0
∫
Br|h(x)|u2dx
∫
Br(|∇u|2 − a(x/|x|)
|x|2u2)dx + N−2
2r
∫
∂Bru2dS
,
η1(r) = supu∈H1(Br)
u6≡0
∫
Br|∇h · x|u2dx
∫
Br(|∇u|2 − a(x/|x|)
|x|2u2)dx + N−2
2r
∫
∂Bru2dS
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Perturbations of La
linear /semilinear perturbations of La: Lau = h(x) u(x) + f(x,u(x))
(H) h “negligible” with respect to a(x/|x|)|x|2 at singularity:
limr→0+
η0(r) = 0,η0(r)
r∈ L1,
1
r
∫ r
0
η0(s)
sds ∈ L1,
η1(r)
r∈ L1,
1
r
∫ r
0
η1(s)
sds ∈ L1,
Examples: • h, (x · ∇h) ∈ Ls for some s > N/2
• a ∈ L∞(SN−1) ; h(x) = O(|x|−2+ε)
•a(x/|x|)
|x|2=
∑
j,m
λjλm
|xj − xm|2; |h(x)|+ |∇h · x| = O
(∑
|xj − xm|−2+ε)
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Perturbations of La
linear /semilinear perturbations of La: Lau = h(x) u(x) + f(x,u(x))
(H) h “negligible” with respect to a(x/|x|)|x|2 at singularity:
limr→0+
η0(r) = 0,η0(r)
r∈ L1,
1
r
∫ r
0
η0(s)
sds ∈ L1,
η1(r)
r∈ L1,
1
r
∫ r
0
η1(s)
sds ∈ L1,
Examples: • h, (x · ∇h) ∈ Ls for some s > N/2
• a ∈ L∞(SN−1) ; h(x) = O(|x|−2+ε)
•a(x/|x|)
|x|2=
∑
j,m
λjλm
|xj − xm|2; |h(x)|+ |∇h · x| = O
(∑
|xj − xm|−2+ε)
(F) f at most critical: |f (x, s)s|+ |f ′s (x, s)s2|+ |∇xF(x, s) · x|6Cf (|s|
2 + |s|2∗
)
where F(x, s) =∫ s
0 f (x, t) dt, 2∗ = 2NN−2 .
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The angular operatorWe aim to describe the rate and the shape of the singularity ofsolutions, by relating them to the eigenvalues and theeigenfunctions of a Schrodinger operator on S
N−1
corresponding to the angular part of La:
La := −∆SN−1 − a
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The angular operatorWe aim to describe the rate and the shape of the singularity ofsolutions, by relating them to the eigenvalues and theeigenfunctions of a Schrodinger operator on S
N−1
corresponding to the angular part of La:
La := −∆SN−1 − a
If Λ(a) < 1, La admits a diverging sequence of real eigenvalues
µ1(a) < µ2(a) 6 · · · 6 µk(a) 6 · · ·
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The angular operatorWe aim to describe the rate and the shape of the singularity ofsolutions, by relating them to the eigenvalues and theeigenfunctions of a Schrodinger operator on S
N−1
corresponding to the angular part of La:
La := −∆SN−1 − a
If Λ(a) < 1, La admits a diverging sequence of real eigenvalues
µ1(a) < µ2(a) 6 · · · 6 µk(a) 6 · · ·
Positivity of the quadratic formassociated to La
Λ(a) < 1⇐⇒ µ1(a) > −
(N − 2
2
)2
(PD)
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The Almgren type frequency function
In an open bounded Ω ∋ 0, let u be a H1(Ω)-weak solution to
Lau = h(x) u(x) + f (x, u(x))
For small r > 0 define
D(r) =1
rN−2
∫
Br
(
|∇u(x)|2 −a( x
|x| )
|x|2 u2(x)− h(x) u2(x)− f (x, u(x))u(x)
)
dx,
H(r) =1
rN−1
∫
∂Br
|u|2 dS.
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The Almgren type frequency function
In an open bounded Ω ∋ 0, let u be a H1(Ω)-weak solution to
Lau = h(x) u(x) + f (x, u(x))
For small r > 0 define
D(r) =1
rN−2
∫
Br
(
|∇u(x)|2 −a( x
|x| )
|x|2 u2(x)− h(x) u2(x)− f (x, u(x))u(x)
)
dx,
H(r) =1
rN−1
∫
∂Br
|u|2 dS.
If (PD) holds and u 6≡ 0,⇒ H(r) > 0 for small r > 0
;
Almgren type frequency function
N (r) = Nu,h,f (r) =D(r)H(r)
is well defined in a suitably small interval (0, r0).
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“Perturbed monotonicity”N ′ is an integrable perturbation of a nonnegative function:enough to prove the existence of limr→0+ N (r)
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“Perturbed monotonicity”N ′ is an integrable perturbation of a nonnegative function:enough to prove the existence of limr→0+ N (r)
N ∈ W1,1loc (0, r0) and, in a distributional sense and for a.e. r ∈ (0, r0),
N ′(r) =2r
[(∫
∂Br
∣∣∣∂u∂ν
∣∣∣2
dS
)(∫
∂Br|u|2dS
)
−(∫
∂Bru ∂u∂ν
dS)2
]
(∫
∂Br|u|2dS
)2+ (Rh +R1
f +R2f )(r)
Rh(r) =−
∫
Br(2h(x) +∇h(x) · x)|u|2 dx
∫
∂Br|u|2 dS
, R1f (r) =
r∫
∂Br
(2F(x, u)− f (x, u)u
)dS
∫
∂Br|u|2 dS
R2f (r) =
∫
Br
((N − 2)f (x, u)u − 2NF(x, u)− 2∇xF(x, u) · x
)dx
∫
∂Br|u|2 dS
.
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“Perturbed monotonicity”N ′ is an integrable perturbation of a nonnegative function:enough to prove the existence of limr→0+ N (r)
N ∈ W1,1loc (0, r0) and, in a distributional sense and for a.e. r ∈ (0, r0),
N ′(r) =2r
[(∫
∂Br
∣∣∣∂u∂ν
∣∣∣2
dS
)(∫
∂Br|u|2dS
)
−(∫
∂Bru ∂u∂ν
dS)2
]
(∫
∂Br|u|2dS
)2+ (Rh +R1
f +R2f )(r)
︸ ︷︷ ︸
Schwarz’s inequality6
0
Rh(r) =−
∫
Br(2h(x) +∇h(x) · x)|u|2 dx
∫
∂Br|u|2 dS
, R1f (r) =
r∫
∂Br
(2F(x, u)− f (x, u)u
)dS
∫
∂Br|u|2 dS
R2f (r) =
∫
Br
((N − 2)f (x, u)u − 2NF(x, u)− 2∇xF(x, u) · x
)dx
∫
∂Br|u|2 dS
.
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“Perturbed monotonicity”N ′ is an integrable perturbation of a nonnegative function:enough to prove the existence of limr→0+ N (r)
N ∈ W1,1loc (0, r0) and, in a distributional sense and for a.e. r ∈ (0, r0),
N ′(r) =2r
[(∫
∂Br
∣∣∣∂u∂ν
∣∣∣2
dS
)(∫
∂Br|u|2dS
)
−(∫
∂Bru ∂u∂ν
dS)2
]
(∫
∂Br|u|2dS
)2+ (Rh +R1
f +R2f )(r)
︸ ︷︷ ︸
Schwarz’s inequality6
0
Rh(r) =−
∫
Br(2h(x) +∇h(x) · x)|u|2 dx
∫
∂Br|u|2 dS
, R1f (r) =
r∫
∂Br
(2F(x, u)− f (x, u)u
)dS
∫
∂Br|u|2 dS
R2f (r) =
∫
Br
((N − 2)f (x, u)u − 2NF(x, u)− 2∇xF(x, u) · x
)dx
∫
∂Br|u|2 dS
.
︸ ︷︷ ︸
integrable
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“Perturbed monotonicity”N ′ is an integrable perturbation of a nonnegative function:enough to prove the existence of limr→0+ N (r)
N ∈ W1,1loc (0, r0) and, in a distributional sense and for a.e. r ∈ (0, r0),
N ′(r) =2r
[(∫
∂Br
∣∣∣∂u∂ν
∣∣∣2
dS
)(∫
∂Br|u|2dS
)
−(∫
∂Bru ∂u∂ν
dS)2
]
(∫
∂Br|u|2dS
)2+ (Rh +R1
f +R2f )(r)
︸ ︷︷ ︸
Schwarz’s inequality6
0
N ′ = nonnegative function + L1-function on (0, r) =⇒
N (r) = N (r)−∫ r
r N ′(s) dsadmits a limit γ as r → 0+
which is necessarily finite
︸ ︷︷ ︸integrable
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Blow-up:Set
wλ(x) =u(λx)√
H(λ), so that
∫
∂B1
|wλ|2dS = 1.
wλλ∈(0,λ) is bounded in H1(B1) =⇒ for any λn → 0+, wλnk w inH1(B1) along a subsequence λnk → 0+, and
∫
∂B1|w|2dS = 1.
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Blow-up:Set
wλ(x) =u(λx)√
H(λ), so that
∫
∂B1
|wλ|2dS = 1.
wλλ∈(0,λ) is bounded in H1(B1) =⇒ for any λn → 0+, wλnk w inH1(B1) along a subsequence λnk → 0+, and
∫
∂B1|w|2dS = 1.
(Ek) − Lawλnk = λ2nk
h(λnk x)wλnk +λ2
nk√
H(λnk)f(
λnk x,√
H(λnk)wλnk
)
limit
;
weak
(E) Law(x) = 0
(Ek)− (E) tested with wλnk − w ⇒ wλnk → w in H1(B1)
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Blow-up:Set
wλ(x) =u(λx)√
H(λ), so that
∫
∂B1
|wλ|2dS = 1.
wλλ∈(0,λ) is bounded in H1(B1) =⇒ for any λn → 0+, wλnk w inH1(B1) along a subsequence λnk → 0+, and
∫
∂B1|w|2dS = 1.
(Ek) − Lawλnk = λ2nk
h(λnk x)wλnk +λ2
nk√
H(λnk)f(
λnk x,√
H(λnk)wλnk
)
limit
;
weak
(E) Law(x) = 0
(Ek)− (E) tested with wλnk − w ⇒ wλnk → w in H1(B1)
If Nk(r) the Almgren frequency function associated to (Ek) andNw(r) is the Almgren frequency function associated to (E), then
limk→∞
Nk(r) = Nw(r) for all r ∈ (0, 1).
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Blow-up:
By scalingNk(r) = N (λnk r)
⇓Nw(r) = lim
k→∞N (λnk r)= γ ∀r ∈ (0, 1)
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Blow-up:
By scalingNk(r) = N (λnk r)
⇓Nw(r) = lim
k→∞N (λnk r)= γ ∀r ∈ (0, 1)
Then Nw is constant in (0, 1) and hence N ′w(r) = 0 for any r ∈ (0, 1)
⇓(∫
∂Br
∣∣∣∣
∂w∂ν
∣∣∣∣
2
dS
)
·(∫
∂Br
|w|2dS
)
−(∫
∂Br
w∂w∂ν
dS
)2
= 0
Therefore w and ∂w∂ν are parallel as vectors in L2(∂Br), i.e. ∃ a real
valued function η = η(r) such that ∂w∂ν (r, θ) = η(r)w(r, θ) for r ∈ (0, 1).
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Blow-up:
By scalingNk(r) = N (λnk r)
⇓Nw(r) = lim
k→∞N (λnk r)= γ ∀r ∈ (0, 1)
Then Nw is constant in (0, 1) and hence N ′w(r) = 0 for any r ∈ (0, 1)
⇓(∫
∂Br
∣∣∣∣
∂w∂ν
∣∣∣∣
2
dS
)
·(∫
∂Br
|w|2dS
)
−(∫
∂Br
w∂w∂ν
dS
)2
= 0
Therefore w and ∂w∂ν are parallel as vectors in L2(∂Br), i.e. ∃ a real
valued function η = η(r) such that ∂w∂ν (r, θ) = η(r)w(r, θ) for r ∈ (0, 1).
After integration we obtain
w(r, θ)= e∫ r
1 η(s)dsw(1, θ) = ϕ(r)ψ(θ).
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Blow-up:
w(r, θ) = ϕ(r)ψ(θ)
Rewriting equation (E) Law(x) = 0 in polar coordinates we obtain
(−ϕ′′(r)− N−1
r ϕ′(r))ψ(θ) + r−2ϕ(r)Laψ(θ) = 0.
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Blow-up:
w(r, θ) = ϕ(r)ψ(θ)
Rewriting equation (E) Law(x) = 0 in polar coordinates we obtain
(−ϕ′′(r)− N−1
r ϕ′(r))ψ(θ) + r−2ϕ(r)Laψ(θ) = 0.
Then ψ is an eigenfunction of the operator La.Let µk0(a) be the corresponding eigenvalue =⇒ ϕ(r) solves
−ϕ′′(r)− N−1r ϕ(r) + r−2µk0(a)ϕ(r) = 0.
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Blow-up:
w(r, θ) = ϕ(r)ψ(θ)
Rewriting equation (E) Law(x) = 0 in polar coordinates we obtain
(−ϕ′′(r)− N−1
r ϕ′(r))ψ(θ) + r−2ϕ(r)Laψ(θ) = 0.
Then ψ is an eigenfunction of the operator La.Let µk0(a) be the corresponding eigenvalue =⇒ ϕ(r) solves
−ϕ′′(r)− N−1r ϕ(r) + r−2µk0(a)ϕ(r) = 0.
Then ϕ(r) = rσ+
with σ+ = −N−22 +
√(
N−22
)2+ µk0(a).
⇓w(r, θ) = rσ
+
ψ(θ)
From Nw(r) ≡ γ, we deduce that γ = σ+.
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Asymptotics at the singularity
Step 1: any λn → 0+ admits a subsequence λnkk∈N s.t.
u(λnk x)√
H(λnk)→ |x|γψ
( x|x|)
strongly in H1(B1)
γ = −N−22 +
√(
N−22
)2+ µk0(a), ψ eigenfunction associated to µk0 .
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Asymptotics at the singularity
Step 1: any λn → 0+ admits a subsequence λnkk∈N s.t.
u(λnk x)√
H(λnk)→ |x|γψ
( x|x|)
strongly in H1(B1)
γ = −N−22 +
√(
N−22
)2+ µk0(a), ψ eigenfunction associated to µk0 .
Step 2: limr→0+H(r)r2γ is finite and > 0 (Step 1 + separation of variables)
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Asymptotics at the singularity
Step 1: any λn → 0+ admits a subsequence λnkk∈N s.t.
u(λnk x)√
H(λnk)→ |x|γψ
( x|x|)
strongly in H1(B1)
γ = −N−22 +
√(
N−22
)2+ µk0(a), ψ eigenfunction associated to µk0 .
Step 2: limr→0+H(r)r2γ is finite and > 0 (Step 1 + separation of variables)
Step 3: So λ−γnk
u(λnkθ) →j0+m−1∑
i=j0
βiψi(θ) in L2(SN−1)
where ψij0+m−1i=j0 is an L2(SN−1)-orthonormal basis for the
eigenspace associated to µk0 .
Expanding u(λ θ) =∞∑
k=1ϕk(λ)ψk(θ), we compute the βi’s.
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Asymptotics at the singularity
βi = limk→∞
λ−γnkϕi(λnk )
=
∫
SN−1
[
R−γu(Rθ) +∫ R
0
h(sθ)u(sθ) + f(sθ, u(sθ)
)
2γ + N − 2
(
s1−γ −sγ+N−1
R2γ+N−2
)
ds
]
ψi(θ) dS(θ)
depends neither on the sequence λnn∈N nor on its subsequence λnkk∈N
=⇒ the convergences actually hold as λ → 0+.
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Asymptotics at the singularity
βi = limk→∞
λ−γnkϕi(λnk )
=
∫
SN−1
[
R−γu(Rθ) +∫ R
0
h(sθ)u(sθ) + f(sθ, u(sθ)
)
2γ + N − 2
(
s1−γ −sγ+N−1
R2γ+N−2
)
ds
]
ψi(θ) dS(θ)
Theorem [F.-Ferrero-Terracini (2010)] Let Ω ∋ 0 be a bounded open setin R
N , N > 3, (PD), (H), (F) hold. If u 6≡ 0 weakly solves Lau = hu + f (x, u)in Ω, then ∃ k0 ∈ N, k0 > 1, s. t.
γ = limr→0+ Nu,h,f (r) = −N−22 +
√(
N−22
)2+ µk0(a).
Furthermore, as λ→ 0+,
λ−γu(λx) → |x|γj0+m−1∑
i=j0
βiψi
(x
|x|
)
in H1(B1).
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3. Parabolic monotonicity formula
[Poon, Comm. PDE’s(1996)]
For u solving ut +∆u = 0 in RN × (0, T), define
D(t) = t∫
RN
|∇u(x, t)|2G(x, t) dx
H(t) =∫
RN
u2(x, t)G(x, t) dx
Parabolic Almgren frequency
N (t) =D(t)H(t)
where
G(x, t) = t−N/2 exp(
− |x|24t
)
is the heat kernel satisfying
Gt −∆G = 0 and ∇G(x, t) = − x2t
G(x, t).
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3. Parabolic monotonicity formula
N ր in t
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3. Parabolic monotonicity formula
N ր in t
Proof.
N ′(t) =2t
[( ∫
RN
∣∣ut +
∇u·x2t
∣∣2G dx
)( ∫
RN u2 G dx)
−( ∫
RN
(ut +
∇u·x2t
)u G dx
)2]
H2(t)︸ ︷︷ ︸
Schwarz’s inequality
6
0
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3. Parabolic monotonicity formula
N ր in t
Proof.
N ′(t) =2t
[( ∫
RN
∣∣ut +
∇u·x2t
∣∣2G dx
)( ∫
RN u2 G dx)
−( ∫
RN
(ut +
∇u·x2t
)u G dx
)2]
H2(t)︸ ︷︷ ︸
Schwarz’s inequality
6
0
Doubling properties and unique continuation:[Escauriaza-Fern andez, Ark. Mat. (2003)][Fern andez, Comm. PDEs (2003)][Escauriaza-Fern andez-Vessella, Appl. Analysis (2006)][Escauriaza-Kenig-Ponce-Vega, Math. Res. Lett. (2006)]
See also [Caffarelli-Karakhanyan-Lin, J. Fixed Point Theory Appl. (2009)]
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4. Asymptotics at singularities (parabolic case)
ut +∆u +a(x/|x|)|x|2 u + h(x, t)u = 0, in R
N × (0, T)
where T > 0, N > 3, a ∈ L∞(SN−1), and
(H)
|h(x, t)| 6 Ch(1 + |x|−2+ε) for all t ∈ (0, T), a.e. x ∈ RN
h, ht ∈ Lr((0, T), LN/2
), r > 1, ht ∈ L∞
loc
((0, T), LN/2
).
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4. Asymptotics at singularities (parabolic case)
ut +∆u +a(x/|x|)|x|2 u + h(x, t)u = 0, in R
N × (0, T)
where T > 0, N > 3, a ∈ L∞(SN−1), and
(H)
|h(x, t)| 6 Ch(1 + |x|−2+ε) for all t ∈ (0, T), a.e. x ∈ RN
h, ht ∈ Lr((0, T), LN/2
), r > 1, ht ∈ L∞
loc
((0, T), LN/2
).
Remark: also subcritical semilinear perturbationsf (x, t, s) can be treated
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4. Asymptotics at singularities (parabolic case)
ut +∆u +a(x/|x|)|x|2 u + h(x, t)u = 0, in R
N × (0, T)
where T > 0, N > 3, a ∈ L∞(SN−1), and
(H)
|h(x, t)| 6 Ch(1 + |x|−2+ε) for all t ∈ (0, T), a.e. x ∈ RN
h, ht ∈ Lr((0, T), LN/2
), r > 1, ht ∈ L∞
loc
((0, T), LN/2
).
Parabolic Almgren type frequency function
N (t) =t∫
RN
(|∇u(x, t)|2 − a(x/|x|)
|x|2 u2(x, t)− h(x, t)u2(x, t))G(x, t) dx
∫
RN u2(x, t)G(x, t) dx
where G(x, t) = t−N/2 exp(− |x|2
4t
)
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“Perturbed monotonicity”
N ′ is an L1-perturbation of a nonnegative function as t → 0+
⇓∃ lim
t→0+N (t) = γ.
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“Perturbed monotonicity”
N ′ is an L1-perturbation of a nonnegative function as t → 0+
⇓∃ lim
t→0+N (t) = γ.
Blow-up for scaling uλ(x, t) = u(λx, λ2t)
;γ is an eigenvalue of
Ornstein-Uhlenbeck type operator
La = −∆+ x2 · ∇ − a(x/|x|)
|x|2
i.e.
γ = γm,k = m − αk2 , αk =
N−22 −
√(
N−22
)2+ µk(a),
for some k,m ∈ N, k > 1.
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Asymptotics at the singularity
Theorem [F.-Primo, to appear in DCDS-A]
Let u 6≡ 0 be a weak solution to ut +∆u + a(x/|x|)|x|2 u + h(x, t)u = 0, with
h satisfying (H) and a ∈ L∞(S
N−1)
satisfying (PD). Then ∃ m0, k0 ∈ N,k0 > 1, such that limt→0+ N (t) = γm0,k0 . Furthermore ∀ τ ∈ (0, 1)
limλ→0+
∫ 1
τ
∥∥∥∥λ−2γm0,k0 u(λx, λ2t)− tγm0,k0 V(x/
√t)
∥∥∥∥
2
Ht
dt = 0,
limλ→0+
supt∈[τ,1]
∥∥∥∥λ−2γm0,k0 u(λx, λ2t)− tγm0,k0 V(x/
√t)
∥∥∥∥Lt
= 0,
where V is an eigenfunction of La associated to the eigenvalue γm0,k0 .
‖u‖Ht =
(∫
RN
(t|∇u(x)|2 + |u(x)|2
)G(x, t) dx
)1/2
, ‖u‖Lt =
(∫
RN|u(x)|2G(x, t) dx
)1/2
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Eigenfunctions of La
A basis of the eigenspace corresponding to γm,k = m − αk2 is
Vn,j : j, n ∈ N, j > 1, γm,k = n − αj
2
,
where
Vn,j(x) = |x|−αjPj,n
( |x|24
)
ψj
( x|x|)
,
ψj is an eigenfunction of the operator La = −∆SN−1 − a(θ)associated to the j-th eigenvalue µj(a), and
Pj,n(t) =n∑
i=0
(−n)i(
N2 − αj
)
i
ti
i!,
(s)i =∏i−1
j=0(s + j), (s)0 = 1.