[PPT] - The Existence theorem of the Stokes-Neumann Problem Nasrin Arab CASA PowerPoint Presentation

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The Existence theorem of the Stokes-Neumann Problem

Nasrin Arab

CASA Tu/e

28 April 2010

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 1 / 21

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Overview

1

Review

2

Stokes-Neumann problem

3

Solvability of the Stokes equations

4

conformal bijection

5

Main theorem

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 2 / 21

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Outline

1

Review

2

Stokes-Neumann problem

3

Solvability of the Stokes equations

4

conformal bijection

5

Main theorem

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 3 / 21

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Review

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 4 / 21

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The Stokes equation in R2

Stokes equations ∆v − ∇p = 0 ∇ · v = 0 x ∈ G.

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 5 / 21

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The Stokes equation in R2

Stokes equations ∆v − ∇p = 0 ∇ · v = 0 x ∈ G. we can rewrite it as ∇ · T = 0 ∇ · v = 0 x ∈ G.

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 5 / 21

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The Stokes equation in R2

Stokes equations ∆v − ∇p = 0 ∇ · v = 0 x ∈ G. we can rewrite it as ∇ · T = 0 ∇ · v = 0 x ∈ G. where T := −pI + ∇v + (∇v)T

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 5 / 21

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General solutions of Stokes equations without regarding boundary conditions

Holomorphic representation

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

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General solutions of Stokes equations without regarding boundary conditions

Holomorphic representation

Theorem

If p(x), v(x) solves the Stokes equation on G, then there exists a pair of analytic functions z −→ ϕ(z), χ(z) on G, such that

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

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General solutions of Stokes equations without regarding boundary conditions

Holomorphic representation

Theorem

If p(x), v(x) solves the Stokes equation on G, then there exists a pair of analytic functions z −→ ϕ(z), χ(z) on G, such that                      v1 + iv2 = −ϕ + zϕ′ + χ′ −4p = T11 + T22 = −8Reϕ′ ⇒ p = 2Reϕ′ T · n = 2i d

ds(zϕ′ + ϕ + χ′)

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

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General solutions of Stokes equations without regarding boundary conditions

Holomorphic representation

Theorem

If p(x), v(x) solves the Stokes equation on G, then there exists a pair of analytic functions z −→ ϕ(z), χ(z) on G, such that                      v1 + iv2 = −ϕ + zϕ′ + χ′ −4p = T11 + T22 = −8Reϕ′ ⇒ p = 2Reϕ′ T · n = 2i d

ds(zϕ′ + ϕ + χ′)

Vice versa

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

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General solutions of Stokes equations without regarding boundary conditions

Holomorphic representation

Theorem

If p(x), v(x) solves the Stokes equation on G, then there exists a pair of analytic functions z −→ ϕ(z), χ(z) on G, such that                      v1 + iv2 = −ϕ + zϕ′ + χ′ −4p = T11 + T22 = −8Reϕ′ ⇒ p = 2Reϕ′ T · n = 2i d

ds(zϕ′ + ϕ + χ′)

Vice versa The holomorphic representation of a solution by {ϕ, χ} is unique if ϕ(0) = χ(0) = 0

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 6 / 21

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Outline

1

Review

2

Stokes-Neumann problem

3

Solvability of the Stokes equations

4

conformal bijection

5

Main theorem

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 7 / 21

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Stokes-Neumann problem

Stokes equation with Neumann boundary condition            ∇ · T (x) = 0 , x ∈ G ∇ · v(x) = 0 , x ∈ G T (x) · n(x) = f(x) , x ∈ ∂G

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

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Stokes-Neumann problem

Stokes equation with Neumann boundary condition            ∇ · T (x) = 0 , x ∈ G ∇ · v(x) = 0 , x ∈ G T (x) · n(x) = f(x) , x ∈ ∂G

n the prescribed boundary stress field x −→ f(x) ∈ R2 we put condition on f

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

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Stokes-Neumann problem

Stokes equation with Neumann boundary condition            ∇ · T (x) = 0 , x ∈ G ∇ · v(x) = 0 , x ∈ G T (x) · n(x) = f(x) , x ∈ ∂G

n the prescribed boundary stress field x −→ f(x) ∈ R2 we put condition on f

f(x(s)) = d ds{K1(s)n(s) + K2(s)t(s)}

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

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Stokes-Neumann problem

Stokes equation with Neumann boundary condition            ∇ · T (x) = 0 , x ∈ G ∇ · v(x) = 0 , x ∈ G T (x) · n(x) = f(x) , x ∈ ∂G

n the prescribed boundary stress field x −→ f(x) ∈ R2 we put condition on f

f(x(s)) = d ds{K1(s)n(s) + K2(s)t(s)}

∂G

K1(s)ds = 0

∂G

{K1(s)n(s) + K2(s)t(s)}ds = 0

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

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Stokes-Neumann problem

Stokes equation with Neumann boundary condition            ∇ · T (x) = 0 , x ∈ G ∇ · v(x) = 0 , x ∈ G T (x) · n(x) = f(x) , x ∈ ∂G

n the prescribed boundary stress field x −→ f(x) ∈ R2 we put condition on f

f(x(s)) = d ds{K1(s)n(s) + K2(s)t(s)}

∂G

K1(s)ds = 0

∂G

{K1(s)n(s) + K2(s)t(s)}ds = 0 ”What about (non)-uniqueness of the Stokes-Neumann problem”

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 8 / 21

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Outline

1

Review

2

Stokes-Neumann problem

3

Solvability of the Stokes equations

4

conformal bijection

5

Main theorem

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 9 / 21

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Solvability

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 10 / 21

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necessary condition in terms of {ϕ, χ}

we want to find analytic ϕ, χ : G −→ C, such that at the boundary ∂G

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

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necessary condition in terms of {ϕ, χ}

we want to find analytic ϕ, χ : G −→ C, such that at the boundary ∂G T · n(s) = 2i d ds(z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s))) = −i d dsK(s)˙ z(s). (1)

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

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necessary condition in terms of {ϕ, χ}

we want to find analytic ϕ, χ : G −→ C, such that at the boundary ∂G T · n(s) = 2i d ds(z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s))) = −i d dsK(s)˙ z(s). (1)            z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s)) = −1

2K(s)˙

z ϕ(0) = χ(0) = 0 Imϕ′(0) = 0

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

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necessary condition in terms of {ϕ, χ}

we want to find analytic ϕ, χ : G −→ C, such that at the boundary ∂G T · n(s) = 2i d ds(z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s))) = −i d dsK(s)˙ z(s). (1)            z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s)) = −1

2K(s)˙

z ϕ(0) = χ(0) = 0 Imϕ′(0) = 0 d ds

z(s)ϕ(z(s)) + χ(z(s))
+ ϕ(z(s))˙

¯ z(s) − ϕ(z(s))˙ z(s) = −1 2K(s)

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

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necessary condition in terms of {ϕ, χ}

we want to find analytic ϕ, χ : G −→ C, such that at the boundary ∂G T · n(s) = 2i d ds(z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s))) = −i d dsK(s)˙ z(s). (1)            z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s)) = −1

2K(s)˙

z ϕ(0) = χ(0) = 0 Imϕ′(0) = 0 d ds

z(s)ϕ(z(s)) + χ(z(s))
+ ϕ(z(s))˙

¯ z(s) − ϕ(z(s))˙ z(s) = −1 2K(s)

∂G

K1(s)ds = 0

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 11 / 21

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Outline

1

Review

2

Stokes-Neumann problem

3

Solvability of the Stokes equations

4

conformal bijection

5

Main theorem

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 12 / 21

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conformal bijection

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 13 / 21

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Conformal bijection

Φ(ζ) = ϕ(Ω(ζ)), X(ζ) = χ(Ω(ζ)), ζ = eiθ

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 14 / 21

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Conformal bijection

Φ(ζ) = ϕ(Ω(ζ)), X(ζ) = χ(Ω(ζ)), ζ = eiθ            z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s)) = −1

2K(s)˙

z ϕ(0) = χ(0) = 0 Imϕ′(0) = 0 ⇓

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 14 / 21

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Conformal bijection

Φ(ζ) = ϕ(Ω(ζ)), X(ζ) = χ(Ω(ζ)), ζ = eiθ            z(s)ϕ′(z(s)) + ϕ(z(s)) + χ′(z(s)) = −1

2K(s)˙

z ϕ(0) = χ(0) = 0 Imϕ′(0) = 0 ⇓            Ω(ζ)∂θΦ(ζ) + ∂θΩ(ζ)Φ(ζ) + ∂θX(ζ) = − 1

2|∂θΩ(ζ)|K(s(θ)).

Φ(0) = X(0) = 0 ImΦ′(0) = 0

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 14 / 21

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Operator J

The mapping J : L2(S1; R1; {1}⊥) → L2(S1; R1; {1}⊥) f1 → Jf1 = f2

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 15 / 21

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Operator J

The mapping J : L2(S1; R1; {1}⊥) → L2(S1; R1; {1}⊥) f1 → Jf1 = f2 J2 = −I

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 15 / 21

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Operator J

The mapping J : L2(S1; R1; {1}⊥) → L2(S1; R1; {1}⊥) f1 → Jf1 = f2 J2 = −I Jf1(θ) = f2(θ) = 1

2π

π

−π cot(1 2(θ − θ1))f1(θ1)dθ1

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 15 / 21

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Operator J

The mapping J : L2(S1; R1; {1}⊥) → L2(S1; R1; {1}⊥) f1 → Jf1 = f2 J2 = −I Jf1(θ) = f2(θ) = 1

2π

π

−π cot(1 2(θ − θ1))f1(θ1)dθ1

∂θJ = J∂θ, J(f1g1) = J ((Jf1)(Jg1)) + (Jf1)g1 + f1(Jg1).

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 15 / 21

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Outline

1

Review

2

Stokes-Neumann problem

3

Solvability of the Stokes equations

4

conformal bijection

5

Main theorem

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 16 / 21

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The Existence Theorem

Theorem

Let K1, K2 : ∂G −→ R be given. a. θ −→ |∂θΩ(eiθ)|K1(s(θ)) ∈ L2(S1; R1; {1}⊥). b. θ −→ |∂θΩ(eiθ)|K2(s(θ)) ∈ L2(S1; R1).

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 17 / 21

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The Existence Theorem

Theorem

Let K1, K2 : ∂G −→ R be given. a. θ −→ |∂θΩ(eiθ)|K1(s(θ)) ∈ L2(S1; R1; {1}⊥). b. θ −→ |∂θΩ(eiθ)|K2(s(θ)) ∈ L2(S1; R1). c. θ −→ |∂θΩ(eiθ)| is bounded on S1. d. θ −→ |∂θ∂θΩ(eiθ)| is bounded on S1. Then there exist unique Φ, X : D −→ C , such that

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 17 / 21

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The Existence Theorem

Theorem

Let K1, K2 : ∂G −→ R be given. a. θ −→ |∂θΩ(eiθ)|K1(s(θ)) ∈ L2(S1; R1; {1}⊥). b. θ −→ |∂θΩ(eiθ)|K2(s(θ)) ∈ L2(S1; R1). c. θ −→ |∂θΩ(eiθ)| is bounded on S1. d. θ −→ |∂θ∂θΩ(eiθ)| is bounded on S1. Then there exist unique Φ, X : D −→ C , such that            Ω(ζ)∂θΦ(ζ) + ∂θΩ(ζ)Φ(ζ) + ∂θX(ζ) = − 1

2|∂θΩ(ζ)|K(s(θ)).

Φ(0) = X(0) = 0 ImΦ′(0) = 0 and at the boundary Φ, X ∈ L2(S1; C)

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 17 / 21

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Proof

Proof.

Ω1∂θΦ1 + Ω2∂θΦ2 + (∂θΩ1)Φ1 + (∂θΩ2)Φ2 + ∂θX1 = − 1

2|∂θΩ|K1

Ω2∂θΦ1 − Ω1∂θΦ2 − (∂θΩ2)Φ1 + (∂θΩ1)Φ2 − ∂θX2 = − 1

2|∂θΩ|K2

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 18 / 21

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Proof

Proof.

Ω1∂θΦ1 + Ω2∂θΦ2 + (∂θΩ1)Φ1 + (∂θΩ2)Φ2 + ∂θX1 = − 1

2|∂θΩ|K1

Ω2∂θΦ1 − Ω1∂θΦ2 − (∂θΩ2)Φ1 + (∂θΩ1)Φ2 − ∂θX2 = − 1

2|∂θΩ|K2

use the operator J Ω1∂θΦ1 + Ω2J∂θΦ1 + ˙ Ω1Φ1 + ˙ Ω2JΦ1 + ∂θX1 = − 1

2|Ω′|K1

Ω2∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + ˙ Ω1JΦ1 − J∂θX1 = −1

2|Ω′|K2

Nasrin Arab (CASA Tu/e)

The Existence theorem of the Stokes-Neumann Problem 28 April 2010 18 / 21

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Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

Nasrin Arab (CASA Tu/e)

The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

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Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

[JΩ1 − Ω1J] ∂θΦ1 +
J ˙

Ω1 − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

Nasrin Arab (CASA Tu/e)

The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

SLIDE 43

Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

[JΩ1 − Ω1J] ∂θΦ1 +
J ˙

Ω1 − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

∂θ
[JΩ1 − Ω1J] Φ1
+

˙ Ω1J − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

Nasrin Arab (CASA Tu/e)

The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

SLIDE 44

Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

[JΩ1 − Ω1J] ∂θΦ1 +
J ˙

Ω1 − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

∂θ
[JΩ1 − Ω1J] Φ1
+

˙ Ω1J − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

k(Φ1) + L(Φ1) = −1

4 J(|Ω′|K1) + |Ω′|K2

Nasrin Arab (CASA Tu/e)

The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

SLIDE 45

Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

[JΩ1 − Ω1J] ∂θΦ1 +
J ˙

Ω1 − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

∂θ
[JΩ1 − Ω1J] Φ1
+

˙ Ω1J − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

k(Φ1) + L(Φ1) = −1

4 J(|Ω′|K1) + |Ω′|K2

k can be written

k(Φ1) = − 1 2π∂θ π

−π

cos(θ − θ1 2 )Ω1(θ) − Ω1(θ1) sin( θ−θ1

2 )

Φ1(θ1)dθ1

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

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Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

[JΩ1 − Ω1J] ∂θΦ1 +
J ˙

Ω1 − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

∂θ
[JΩ1 − Ω1J] Φ1
+

˙ Ω1J − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

k(Φ1) + L(Φ1) = −1

4 J(|Ω′|K1) + |Ω′|K2

k can be written

k(Φ1) = − 1 2π∂θ π

−π

cos(θ − θ1 2 )Ω1(θ) − Ω1(θ1) sin( θ−θ1

2 )

Φ1(θ1)dθ1 Then condition d. implies that k is Hilbert-Schmidt. so it is compact.

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

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Proof.

JΩ1∂θΦ1 − Ω1J∂θΦ1 − ˙ Ω2Φ1 + J ˙ Ω1Φ1 = −1 4 J(|Ω′|K1) + |Ω′|K2

[JΩ1 − Ω1J] ∂θΦ1 +
J ˙

Ω1 − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

∂θ
[JΩ1 − Ω1J] Φ1
+

˙ Ω1J − ˙ Ω2

Φ1 = −1

4 J(|Ω′|K1) + |Ω′|K2

k(Φ1) + L(Φ1) = −1

4 J(|Ω′|K1) + |Ω′|K2

k can be written

k(Φ1) = − 1 2π∂θ π

−π

cos(θ − θ1 2 )Ω1(θ) − Ω1(θ1) sin( θ−θ1

2 )

Φ1(θ1)dθ1 Then condition d. implies that k is Hilbert-Schmidt. so it is compact. The operator L is bijection!

Nasrin Arab (CASA Tu/e)

The Existence theorem of the Stokes-Neumann Problem 28 April 2010 19 / 21

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Fredholm alternative

Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 20 / 21

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Fredholm alternative

Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

r

Let K(x, y) be an integral kernel, and consider the homogeneous equation, λφ(x) − b

a K(x, y)φ(y) dy = 0

and the inhomogeneous equation λφ(x) − b

a K(x, y)φ(y) dy = f(x).

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 20 / 21

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Fredholm alternative

Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

r

Let K(x, y) be an integral kernel, and consider the homogeneous equation, λφ(x) − b

a K(x, y)φ(y) dy = 0

and the inhomogeneous equation λφ(x) − b

a K(x, y)φ(y) dy = f(x).

The Fredholm alternative states that, ∀ 0 λ ∈ C, either the first equation has a non-trivial solution, or the second equation has a solution for all f(x).

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 20 / 21

SLIDE 51

Fredholm alternative

Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.

r

Let K(x, y) be an integral kernel, and consider the homogeneous equation, λφ(x) − b

a K(x, y)φ(y) dy = 0

and the inhomogeneous equation λφ(x) − b

a K(x, y)φ(y) dy = f(x).

The Fredholm alternative states that, ∀ 0 λ ∈ C, either the first equation has a non-trivial solution, or the second equation has a solution for all f(x). A sufficient condition for this theorem to hold is for K(x, y) to be square integrable on the rectangle [a, b] × [a, b].

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 20 / 21

SLIDE 52

Thank you

Nasrin Arab (CASA Tu/e) The Existence theorem of the Stokes-Neumann Problem 28 April 2010 21 / 21