On the behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential

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On the behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential

Veronica Felli

Dipartimento di Matematica ed Applicazioni University of Milano–Bicocca

[email protected]

joint work with Alberto Ferrero and Susanna Terracini

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 1

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Schr ¨odinger operators with singular potentials

In quantum mechanics, the hamiltonian of a non-relativistic charged particle in an electromagnetic field has the form

(−i∇ + A)² + V

A : R^N → R^N magnetic potential associated to the magnetic field B = curl A . V : R^N → R electric potential.

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Schr ¨odinger operators with singular potentials

In quantum mechanics, the hamiltonian of a non-relativistic charged particle in an electromagnetic field has the form

(−i∇ + A)² + V

A : R^N → R^N magnetic potential associated to the magnetic field B = curl A . V : R^N → R electric potential.

For ^{N > 2}, we consider singular homogeneous electromagnetic

potentials which make the operator invariant by scaling A(x) =

A ^x

|x|

A ∈ C¹(S^{N −1}, R^N)

and ^{V (x) =} ⁻

a _|x|^x

|x|² a ∈ L^∞(S^{N −1}, R)

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A prototype in dimension 2

Aharonov-Bohm magnetic potentials are associated to thin solenoids: if the radius of the solenoid tends to zero while the flux through it remains constant, then the particle is subject to a δ-type magnetic field, which is called Aharonov-Bohm field. An associated vector potential in ^R² is

A(x₁, x₂) = α

− x₂

|x|², x₁

|x|²

, (x₁, x₂) ∈ R², with ^α = circulation of ^A around the solenoid.

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A prototype in dimension 2

A(x₁, x₂) = α

− x₂

|x|², x₁

|x|²

, (x₁, x₂) ∈ R²,

with ^α = circulation of ^A around the solenoid.

Aharonov-Bohm vector potentials are

• singular at 0,

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A prototype in dimension 2

A(x₁, x₂) = α

− x₂

|x|², x₁

|x|²

, (x₁, x₂) ∈ R²,

• singular at 0,

• homogeneous of degree ₋₁

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A prototype in dimension 2

A(x₁, x₂) = α

− x₂

|x|², x₁

|x|²

, (x₁, x₂) ∈ R²,

• singular at 0,

• homogeneous of degree ₋₁

• transversal, i.e. ^A_(θ) _{· θ = 0} for all θ ∈ S^{N −1} ^(TC)

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Singular homogeneous electric potentials

which scale as the laplacian arise in nonrelativistic molecular physics, see [J. M. L ´evy-Leblond, Phys. Rev. (1967)]. The potential describing the interaction between an electric charge and the dipole moment ^D _{∈ R}^N of a molecule has the form

V (x) = −λ (x · d)

|x|³ in ^R^N, where λ ∝ magnitude of the dipole moment ^D

d = D/|D| = orientation of ^D.

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Singular homogeneous electric potentials

which scale as the laplacian arise in nonrelativistic molecular physics, see [J. M. L ´evy-Leblond, Phys. Rev. (1967)]. The potential describing the interaction between an electric charge and the dipole moment ^D _{∈ R}^N of a molecule has the form

V (x) = −λ (x · d)

|x|³ in ^R^N, where λ ∝ magnitude of the dipole moment ^D

d = D/|D| = orientation of ^D.

Schr ¨odinger operators with dipole-type potentials ^a(x/|x|)_|x|2 are studied in

[Terracini, Adv. Diff. Equations (1996)]

[F.-Marchini-Terracini, Discr. Contin. Dyn. Syst. (2008)]

[F.-Marchini-Terracini, Indiana Univ. Math. J., to appear]

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Problem:

describe the asymptotic behavior at the singularity of solutions to equations associated to Schr ¨odinger operators with singular electromagnetic potentials of type

L^A^,a := −i ∇ +

A ^x

|x|

!2

− a _|x|^x

|x|²

in a domain Ω ⊂ R^N containing either the origin or a neighborhood of ∞^.

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Problem:

L^A^,a := −i ∇ +

A ^x

|x|

!2

− a _|x|^x

|x|²

• Linear perturbation of

L

Â^,a^: ^LÂ^,a^{u =} ^h(x) û

with h ∈ L^∞_loc(Ω \ {0}) negligible with respect to _|x|¹2

near the singularity

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Problem:

L^A^,a := −i ∇ +

A ^x

|x|

!2

− a _|x|^x

|x|²

• Linear perturbation of

L

Â^,a^: ^LÂ^,a^{u =} ^h(x) û

with h ∈ L^∞_loc(Ω \ {0}) negligible with respect to _|x|¹2

near the singularity

• Semilinear equations of type L^A^,au = f (x,u(x))

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References

Regularity properties of solutions to Schrödinger equations with less singular magnetic and electric potentials:

[Kurata, Math. Z., (1997)]: N > 3^, local boundedness (and continuity) if the electric potential and the square of the magnetic one belong to the Kato class.

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References

[Kurata, Proc. AMS, 125 (1997)]: N > 3^, unique continuation (Kato class).

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References

[Kurata, Proc. AMS, 125 (1997)]: N > 3^, unique continuation (Kato class).

[Chabrowski-Szulkin, Topol. Methods Nonlinear Anal., (2005)]:

boundedness and decay at ∞ of solutions for N > 3^, L²_loc ^magnetic

potentials, and electric potentials with L^N/2 negative part.

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References

Behavior of solutions to Schrödinger equations with singular inverse square electric potentials for ^A = 0 (i.e. no magnetic vector potential):

[F.-Schneider, Adv. Nonl. Studies (2003)] : H ¨older continuity results for degenerate elliptic equations with singular weights; include asymptotics of solutions near the pole for potentials _|x|^λ₂.

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References

[F.-Marchini-Terracini, Discrete Contin. Dynam. Systems (2008)]: exact asymptotics of solutions near the pole for anisotropic inverse-square singular potentials ^a(x/|x|)_|x|₂ , through separation of variables and comparison methods.

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References

[F.-Marchini-Terracini, Discrete Contin. Dynam. Systems (2008)]: exact asymptotics of solutions near the pole for anisotropic inverse-square singular potentials ^a(x/|x|)_|x|₂ , through separation of variables and comparison methods.

[Pinchover, Ann. IHP Anal. Nonlinaire (1994)]: existence of the limit at the singularity of any quotient of two positive solutions in some linear and

semilinear cases.

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Remark

Comparison and maximum principles play a crucial role both in

[F.-Marchini-Terracini (2008)] and [Pinchover (1994)].

In the presence of a singular magnetic potential, comparison methods are no more available, preventing us from a direct extension of the aforementioned results.

We overcome this difficulty by a Almgren type monotonicity formula and blow-up methods.

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The angular operator

We aim to describe the rate and the shape of the singularity of solutions, by relating them to the eigenvalues and the

eigenfunctions of a Schrödinger operator on the sphere ^S^{N −1} corresponding to the angular part of _LA,a:

L^A_,a := − i ∇^S^N−1 + A2

− a

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The angular operator

L^A_,a := − i ∇^S^N−1 + A2

− a

For a ∈ L^∞(S^{N −1}, R) and ^A _{∈ C}¹(S^{N −1}, R^N), the operator L^A_,a on ^S^{N −1} admits a diverging sequence of real eigenvalues

µ₁(A, a) 6 µ₂(A, a) 6 · · · 6 µ^k(A, a) 6 · · ·.

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The angular operator

L^A_,a := − i ∇^S^N−1 + A2

− a

For a ∈ L^∞(S^{N −1}, R) and ^A _{∈ C}¹(S^{N −1}, R^N), the operator L^A_,a on ^S^{N −1} admits a diverging sequence of real eigenvalues

µ₁(A, a) 6 µ₂(A, a) 6 · · · 6 µ^k(A, a) 6 · · ·.

Positivity of the quadratic form associated to _LA,a is ensured by

N − 22

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The functional setting

D^∗^1,2(R^N, C) := completion of C_c^∞(R^N \ {0}, C) with respect to

kuk_D^1,2_∗ _(R^N_,C) :=

Z

R^N

∇u(x)

² + |u(x)|²

|x|²

dx

1/2

.

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The functional setting

kuk_D^1,2_∗ _(R^N_,C) :=

Z

R^N

∇u(x)

² + |u(x)|²

|x|²

dx

1/2

.

D_∗^1,2(R^N, C) =

u ∈ L¹_loc(R^N \ {0}, C) : u

|x|,∇u ∈ L²(R^N, C^N)

.

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The functional setting

kuk_D^1,2_∗ _(R^N_,C) :=

Z

R^N

∇u(x)

² + |u(x)|²

|x|²

dx

1/2

.

D_∗^1,2(R^N, C) =

u ∈ L¹_loc(R^N \ {0}, C) : u

|x|,∇u ∈ L²(R^N, C^N)

.

Under assumptions (TC) and (PD), D^∗^1,2(R^N, C) = DA^1,2,a(R^N), where DA^1,2,a(R^N) is the completion of C_c^∞(R^N \ {0}, C) with respect to

kuk_D^1,2

A,a(R^N) :=

Z

R^N

∇+i A x/|x|

|x|

u(x)

2

−a x/|x|

|x|² |u(x)|²

dx

1/2

Moreover the norms k · k_D_∗^1,2_(R^N_,C) and k · k_D^1,2

A,a(R^N) are equivalent.

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The functional setting

N > 3

Hardy’s inequality

N −2 2

2 R

R^N

|u|²

|x|² dx 6 R

R^N |∇u|² dx

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The functional setting

N > 3

Hardy’s inequality

N −2 2

2 R

R^N

|u|²

|x|² dx 6 R

R^N |∇u|² dx

⇓

D∗^1,2(R^N, C) = D^1,2(R^N) = C_c^∞(R^N, C)^k·k^{D1,2(RN )}

kukD^1,2(R^N) :=

Z

R^N

∇u(x)

² dx

1/2

and the norms k · k_D^1,2_∗ _(R^N_,C) and k · kD^1,2(R^N) are equivalent.

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The functional setting

^If _{N = 2} ^, ^(TC) holds (i.e. A(θ)·θ = 0), and

ΦA := 1 2π

Z 2π 0

α(t) dt 6∈ Z ^(ND) where α(t) := A(cos t, sin t) · (− sin t, cos t)

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The functional setting

ΦA := 1 2π

Z 2π 0

α(t) dt 6∈ Z ^(ND)

where α(t) := A(cos t, sin t) · (− sin t, cos t)

⇓ [Laptev-Weidl (1999)]

C_c^∞(R^N \ {0}, C) functions satisfy the following Hardy inequality:

min

k∈Z |k − Φ^A|2 Z

R²

|u(x)|²

|x|² dx 6 Z

R²

∇u(x) + i A x/|x|

|x| u(x)

2

dx

| {z }

optimal

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The functional setting

ΦA := 1 2π

Z 2π 0

α(t) dt 6∈ Z ^(ND)

where α(t) := A(cos t, sin t) · (− sin t, cos t)

⇓ [Laptev-Weidl (1999)]

C_c^∞(R^N \ {0}, C) functions satisfy the following Hardy inequality:

min

k∈Z |k − Φ^A|2 Z

R²

|u(x)|²

|x|² dx 6 Z

R²

∇u(x) + i A x/|x|

|x| u(x)

2

dx

| {z }

optimal

D^1,2(R^N, C) = D^1,2(R²) = completion w.r.t.

u+i A x/|x| u

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The functional setting

In an open bounded domain Ω ⊂ R^N, 0 ∈ Ω, let

H_∗¹(Ω, C) = completion of







u ∈ H¹(Ω, C) ∩ C^∞(Ω, C) :

u vanishes in a neighborhood of 0







w.r.t.

kukH_∗¹(Ω,C) =

k∇uk²_L²_(Ω,C^N₎ + kuk²_L²_(Ω,C) +

u

|x|

2

L²(Ω,C)

1/2

.

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The functional setting







u ∈ H¹(Ω, C) ∩ C^∞(Ω, C) :







w.r.t.

kukH_∗¹(Ω,C) =

k∇uk²_L²_(Ω,C^N₎ + kuk²_L²_(Ω,C) +

u

|x|

2

L²(Ω,C)

1/2

.

H_∗¹(Ω, C) =

u ∈ H¹(Ω, C) : u

|x| ∈ L²(Ω, C)

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The functional setting







u ∈ H¹(Ω, C) ∩ C^∞(Ω, C) :







w.r.t.

kukH_∗¹(Ω,C) =

k∇uk²_L²_(Ω,C^N₎ + kuk²_L²_(Ω,C) +

u

|x|

2

L²(Ω,C)

1/2

.

H_∗¹(Ω, C) =

u ∈ H¹(Ω, C) : u

|x| ∈ L²(Ω, C)

• If N > 3, H_∗¹(Ω) = H¹(Ω, C) and their norms are equivalent.

• If N = 2, H_∗¹(Ω) is strictly smaller than H¹(Ω, C).

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The Almgren frequency function

Studying regularity of area-minimizing surfaces of codimension

> 1, in 1979 Almgren introduced the frequency function N (r) = r R

Br |∇u|² dx R

∂Br u²

and observed that, if u is harmonic, then _N _ր in _r .

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The Almgren frequency function

Br |∇u|² dx R

∂Br u²

“frequency ”: if u is a harmonic function in ^R² homogeneous of degree k u_k(r, θ) = a_kr^k sin(kθ)

, then N (r) = k.

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The Almgren frequency function

Br |∇u|² dx R

∂Br u²

The Almgren monotonicity formula was used in

• [Garofalo-Lin, Indiana Univ. Math. J. (1986)]: generalization to variable coefficient elliptic operators in divergence form (unique continuation)

• [Athanasopoulos-Caffarelli-Salsa, Amer. J. Math. (2008)]: regularity of the free boundary in obstacle problems.

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The Almgren type frequency function

In an open bounded Ω ∋ 0, let u be a H_∗¹(Ω)-weak solution to L^A,au = h(x) u, with h satisfying

h ∈ L^∞loc(Ω \ {0}, C), |h(x)| = O(|x|^−2+ε) as |x| → 0 ⁽H₀₎

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The Almgren type frequency function

For small r > 0 define D(r) = _rN¹−2

Z

Br

h∇u + i^A⁽

x

|x|)

|x| u

² − ^a(

x

|x|)

|x|² |u|² − (ℜh)|u|²i dx,

H(r) = _rN−1¹

Z

∂Br

|u|² dS.

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The Almgren type frequency function

For small r > 0 define D(r) = _rN¹−2

Z

Br

h∇u + i^A⁽

x

|x|)

|x| u

² − ^a(

x

|x|)

|x|² |u|² − (ℜh)|u|²i dx,

H(r) = _rN−1¹

Z

∂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(r) = D(r) H(r) is well defined in a suitably small interval (0, ¯r).

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The Almgren type frequency function

N ∈ W_loc^1,1(0, r) and, in a distributional sense and for a.e. r ∈ (0, r),

N^′(r) =

2r R

∂Br

^∂u_∂ν

² dS R

∂Br|u|²dS

−R

∂Brℜ u^∂u_∂ν

dS2

R

∂Br |u|²dS2 + α_h(r)

R

∂Br |u|²dS

where α_h(r)= 2

Z

Br

ℜ(h(x) u(x) (x · ∇u(x))) dx

+N − 2 2

Z

Br

(ℜh(x))|u(x)|² dx − r 2

Z

∂Br

(ℜh(x))|u(x)|² dS

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The Almgren type frequency function

N^′(r) =

2r R

∂Br

^∂u_∂ν

² dS R

∂Br|u|²dS

−R

dS2

R

∂Br |u|²dS

| {z }

Schwarz’s inequality

6

0

where α_h(r)= 2

Z

Br

ℜ(h(x) u(x) (x · ∇u(x))) dx +N − 2

2

Z

Br

(ℜh(x))|u(x)|² dx − r 2

Z

∂Br

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The Almgren type frequency function

N^′(r) =

2r R

∂Br

^∂u_∂ν

² dS R

∂Br|u|²dS

−R

dS2

R

∂Br |u|²dS

| {z }

6

0

where α_h(r)= 2

Z

Br

ℜ(h(x) u(x) (x · ∇u(x))) dx +N − 2

2

Z

Br

(ℜh(x))|u(x)|² dx − r 2

Z

∂Br

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The Almgren type frequency function

N^′(r) =

2r R

∂Br

^∂u_∂ν

² dS R

∂Br|u|²dS

−R

dS2

R

∂Br |u|²dS

| {z }

6

0

where α_h(r)= 2

Z

Br

ℜ(h(x) u(x) (x · ∇u(x))) dx +N − 2

2

Z

Br

(ℜh(x))|u(x)|² dx − r 2

Z

∂Br

=⇒ the limit γ := lim_r→0⁺ N (r) exists and is finite.

| {z }

integrable

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Blow-up:

_set _w^λ_{(x) =} _p^u(λx)

H(λ), so that Z

∂B1

|w^λ|²dS = 1.

{w^λ}λ∈(0,¯λ) is bounded in H_∗¹(B₁) =⇒ for any λ_n → 0⁺, w^λ^nk ⇀ w in H_∗¹(B₁) along a subsequence λ_n_k → 0⁺, and R

∂B1 |w|²dS = 1.

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Blow-up:

_set _w^λ_{(x) =} _p^u(λx)

H(λ), so that Z

∂B1

|w^λ|²dS = 1.

∂B1 |w|²dS = 1.

(E_k) L^A,aw^λ^nk (x) = λ²_n_kh(λ_n_kx)w^λ^nk (x) ^weak;

limit (E) L^A,aw(x) = 0 in B₁

Bootstrap and classical regularity theory ⇒

w^λ^nk → w in C_loc^1,τ(B₁ \ {0}), τ ∈ (0, 1), H¹(B_r, C), H_∗¹(B_r), r ∈ (0, 1).

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Blow-up:

_set _w^λ_{(x) =} _p^u(λx)

H(λ), so that Z

∂B1

|w^λ|²dS = 1.

∂B1 |w|²dS = 1.

(E_k) L^A,aw^λ^nk (x) = λ²_n_kh(λ_n_kx)w^λ^nk (x) ^weak;

limit (E) L^A,aw(x) = 0 in B₁

Bootstrap and classical regularity theory ⇒

w^λ^nk → w in C_loc^1,τ(B₁ \ {0}), τ ∈ (0, 1), H¹(B_r, C), H_∗¹(B_r), r ∈ (0, 1).

If N^k(r) the Almgren frequency function associated to (E_k) and Nw(r) is the Almgren frequency function associated to (E), then

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Blow-up:

_{By scaling} _N_k_{(r) =} _{N (λ}_n

kr)

⇓ Nw(r) = lim

k→∞N (λnkr)= γ ∀r ∈ (0, 1)

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Blow-up:

kr)

⇓ Nw(r) = lim

k→∞N (λnkr)= γ ∀r ∈ (0, 1)

Then N^w is constant in (0, 1) and hence Nw^′ (r) = 0 for any r ∈ (0, 1) Z ⇓

∂Br

∂w

∂ν

2

dS

!

·

Z

∂Br

|w|²dS

−

Z

∂Br

ℜ

w∂w

∂ν

dS

2

= 0

Therefore w and ^∂w_∂ν are parallel as vectors in L²(∂B_r, C), 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:

kr)

⇓ Nw(r) = lim

k→∞N (λnkr)= γ ∀r ∈ (0, 1)

Then N^w is constant in (0, 1) and hence Nw^′ (r) = 0 for any r ∈ (0, 1) Z ⇓

∂Br

∂w

∂ν

2

dS

!

·

Z

∂Br

|w|²dS

−

Z

∂Br

ℜ

w∂w

∂ν

dS

2

= 0

Therefore w and ^∂w_∂ν are parallel as vectors in L²(∂B_r, C), 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^R¹^r ^η(s)dsw(1, θ) = ϕ(r)ψ(θ), r ∈ (0, 1), θ ∈ S^{N −1}.

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Blow-up:

w(r, θ) = ϕ(r)ψ(θ)

Rewriting equation (E) L^A^,aw(x) = 0 in polar coordinates we obtain

−ϕ^′′(r) − ^{N −1}_r ϕ^′(r)

ψ(θ) + r⁻²ϕ(r)LA,aψ(θ) = 0.

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Blow-up:

w(r, θ) = ϕ(r)ψ(θ)

Rewriting equation (E) L^A^,aw(x) = 0 in polar coordinates we obtain

−ϕ^′′(r) − ^{N −1}_r ϕ^′(r)

ψ(θ) + r⁻²ϕ(r)LA,aψ(θ) = 0.

Then ψ is an eigenfunction of the operator LA,a.

Let µ_k₀(A, a) be the corresponding eigenvalue =⇒ ϕ(r) solves

−ϕ^′′(r) − ^{N −1}_r ϕ(r) + r⁻²µ_k₀(A, a)ϕ(r) = 0.