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

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

veronica.felli@unimib.it

joint work with Alberto Ferrero and Susanna Terracini

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

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.

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|

|x|

A ∈ C¹(S^{N −1}, R^N)

and ^{V (x) = −}

a _|x|^x

|x|² a ∈ L^∞(S^{N −1}, R)

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 2

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.

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.

Aharonov-Bohm vector potentials are

• singular at 0,

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 3

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.

Aharonov-Bohm vector potentials are

• singular at 0,

• homogeneous of degree ₋₁

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.

Aharonov-Bohm vector potentials are

• singular at 0,

• homogeneous of degree ₋₁

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

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 3

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 ^RN, where λ ∝ magnitude of the dipole moment ^D

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

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 ^RN, 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]

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 4

Problem:

L^A,a := −i ∇ +

A ^x

|x|

|x|

!2

− a _|x|^x

|x|²

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

Problem:

L^A,a := −i ∇ +

A ^x

|x|

|x|

!2

− a _|x|^x

|x|²

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

• Linear perturbation of

L

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

near the singularity

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 5

Problem:

L^A,a := −i ∇ +

A ^x

|x|

|x|

!2

− a _|x|^x

|x|²

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

• Linear perturbation of

L

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))

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.

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

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.

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

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.

[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.

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

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.

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.

[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.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 7

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.

[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.

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.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 8

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 ^{SN −1} corresponding to the angular part of _LA,a:

L^A_,a := − i ∇^SN−1 + A
2

− a

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 ^{SN −1} corresponding to the angular part of _LA,a:

L^A_,a := − i ∇^SN−1 + A
2

− a

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

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

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 9

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 ^{SN −1} corresponding to the angular part of _LA,a:

L^A_,a := − i ∇^SN−1 + A
2

− a

For a ∈ L^∞(S^{N −1}, R) and ^A _{∈ C}¹(S^{N −1}, R^N), the operator L^A_,a on ^{SN −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

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

.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 10

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

.

D_∗^1,2(R^N, C) =



u ∈ L¹_loc(R^N \ {0}, C) : u

|x|,∇u ∈ L²(R^N, C^N)

 .

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

.

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.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 10

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

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·kD1,2(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.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 11

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)

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

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 12

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

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

.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 13

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

.

H_∗¹(Ω, C) =



u ∈ H¹(Ω, C) : u

|x| ∈ L²(Ω, C)



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

.

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).

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 13

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 .

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 .

“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.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 14

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 .

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.

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₎

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 15

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₎

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.

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₎

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).

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 15

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



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

| {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

(ℜh(x))|u(x)|² dS



“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 16

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

| {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

(ℜh(x))|u(x)|² dS



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

| {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

(ℜh(x))|u(x)|² dS



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

| {z }

integrable

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 16

Blow-up:

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^λnk ⇀ w in H_∗¹(B₁) along a subsequence λ_nk → 0⁺, and R

∂B1 |w|²dS = 1.

Blow-up:

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^λnk ⇀ w in H_∗¹(B₁) along a subsequence λ_nk → 0⁺, and R

∂B1 |w|²dS = 1.

(E_k) L^A,aw^λnk (x) = λ²_nkh(λ_nkx)w^λnk (x) ^weak;

limit (E) L^A,aw(x) = 0 in B₁

Bootstrap and classical regularity theory ⇒

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

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 17

Blow-up:

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^λnk ⇀ w in H_∗¹(B₁) along a subsequence λ_nk → 0⁺, and R

∂B1 |w|²dS = 1.

(E_k) L^A,aw^λnk (x) = λ²_nkh(λ_nkx)w^λnk (x) ^weak;

limit (E) L^A,aw(x) = 0 in B₁

Bootstrap and classical regularity theory ⇒

w^λnk → 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

Blow-up:

kr)

⇓ Nw(r) = lim

k→∞N (λnkr)= γ ∀r ∈ (0, 1)

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 18

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).

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^{R1r η(s)ds}w(1, θ) = ϕ(r)ψ(θ), r ∈ (0, 1), θ ∈ S^{N −1}.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 18

Blow-up:

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.

Blow-up:

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.

“6th European Conference on Elliptic and Parabolic Problems” , Gaeta, May 25, 2009 – p. 19