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R ²

F : ¯ D(O, R) → R ²

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x

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∂D(O, R)

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F (x)

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O / ∈ C

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n +

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n −

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deg (F, D, O) = n + − n −

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deg (F, D, O) 6= 0

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x 0 ∈ D(O, R)

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F (x 0 ) = 0

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R ⁿ

F ∈ C ¹ (Ω; R ⁿ )

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y / ∈ F (∂Ω) ∪ F (S)

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S = {x ∈ Ω; F ⁰ (x) = 0}

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F ⁰

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deg (F, Ω, y) = X

x∈F

⁻¹

(y)

sign F ⁰ (x)

}>

F : [a, b] → R

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F(a)

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F (b) 6= 0

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deg (F, ]a, b] , 0) =







0 if F (a) F (b) > 0

1 if F (a) < 0 and F (b) > 0

−1 if F (a) > 0 and F (b) < 0

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deg (F, ]a, b] , O) 6= 0

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F (x) = 0

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R R k =1 n =1 n =0 k =2 n + k u R R f g f g

1

1 , 2

2

1

{

}

2

{

}

R 2

R 3

u

R n+k

R k

n + k

k = 2

n = 0

k = 1

n = 1

Ω ⊂ R 2

R 3

h ex

J (u, A) = 1 2

Z

Ω

( |∇u − i Au| 2

+ 1

2ε 2 (1 − |u| 2 ) 2 ) dx + Z

R

|h − h ex | 2 ) dx

A

h = curl A

ε

u

| u(x) | ' 1

| u(x) | ' 0

h 1 = O( |log ε| )

h 2 = O( ε 1

)

h ex < h 1

| u(x) | ' 1

h ex = h 1

ε

| u | ' 0

h ex = h 2

| u | ' 0

h = h ex

R 2

R 3

E ε (u) = 1 2

Z

Ω

( |∇u| 2 + 1

2ε 2 (1 − |u| 2 ) 2 ) dx

∂u

∂t = ∆u + 1

ε 2 u ( 1 − |u| 2 )

u(x) = u 0 (x)

∂Ω

R

Ω

|u − u 0 | 2 dx

t = 0

u(x, 0) = u 0 (x)

d = deg(u 0 , ∂Ω)

u 0

∂Ω

H 1 (Ω ; C )

H 1 (Ω ; R 2 ) = {u : Ω → R 2 ; u ∈ L 2 (Ω) 2

∇u ∈ L 2 (Ω) 4 }

∇u

u

H u 1

(Ω ; R 2 ))

u

H 1 (Ω ; C )

u = u 0

∂Ω

Ω ⊂ R 2

R ²

R ³

R ^n+k

R ^k

Ω ⊂ R ²

R ³

( |∇u − i Au| ²

2ε ² (1 − |u| ² ) ² ) dx + Z

|h − h ex | ² ) dx

h 2 = O( _ε ¹

R ²

R ³

( |∇u| ² + 1

2ε ² (1 − |u| ² ) ² ) dx

ε ² u ( 1 − |u| ² )

|u − u 0 | ² dx

H ¹ (Ω ; C )

H ¹ (Ω ; R ² ) = {u : Ω → R ² ; u ∈ L ² (Ω) ²

∇u ∈ L ² (Ω) ⁴ }

H _u ¹

(Ω ; R ² ))

H ¹ (Ω ; C )

Ω ⊂ R ²

H _u ¹

(Ω ; R ² ))

− ∆u ∗ = u ∗ | ∇u ∗ | ²

C _loc ^k (Ω\ {a 1 , a 2 , ..., a d }), ∀k

R ⁿ

R ²

g : ∂Ω → S ¹

S ¹ =

x ∈ R ² ; |x| = 1

u ∈ H ¹ (Ω; C )

H _g ¹ (Ω; C ) = ∅

1 − f (x) ²

u ε ∈ H ¹ (Ω; C )

a(x) |∇u| ² + 1 ²

1 − |u| ² 2

|u − u 0 | ²

² u 1 − |u| ²