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Ellipticity and Fredholm boundary value problems

Katya Krupchyk and Jukka Tuomela

Department of Mathematics, University of Joensuu, Finland

January 4, 2006

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 2/22

Elliptic PDEs

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 3/22

Elliptic Systems

General linear q’th order PDE: Ay =

• |µ|≤q

aµ(x)∂µy = f where x ∈ Ω ⊂ Rn, aµ(x) ∈ Rk×m, µ ∈ Nn

0 and k ≥ m.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 3/22

Elliptic Systems

General linear q’th order PDE: Ay =

• |µ|≤q

aµ(x)∂µy = f where x ∈ Ω ⊂ Rn, aµ(x) ∈ Rk×m, µ ∈ Nn

0 and k ≥ m.

Principal symbol of A is

A =

• |µ|=q

aµ(x)ξµ ξ ∈ T ∗Ω, ξµ = ξµ1

1 . . . ξµn n

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 3/22

Elliptic Systems

General linear q’th order PDE: Ay =

• |µ|≤q

aµ(x)∂µy = f where x ∈ Ω ⊂ Rn, aµ(x) ∈ Rk×m, µ ∈ Nn

0 and k ≥ m.

Principal symbol of A is

A =

• |µ|=q

aµ(x)ξµ ξ ∈ T ∗Ω, ξµ = ξµ1

1 . . . ξµn n

• Definition. The differential operator A is called elliptic in Ω, if A is

injective for all real ξ = 0 and all x ∈ Ω.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 4/22

Example

• Example. Consider the transformation of the two-dimensional

Laplace equation ∆u = u20 + u02 = 0 to the first order system Ly =      y1

10 − y2 = 0,

y1

01 − y3 = 0,

y2

10 + y3 01 = 0.

L =    ξ1 ξ2 ξ1 ξ2    . The transformed system is not elliptic, although it is equivalent to Laplace’s equation.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 4/22

Example

• Example. Consider the transformation of the two-dimensional

Laplace equation ∆u = u20 + u02 = 0 to the first order system Ly =      y1

10 − y2 = 0,

y1

01 − y3 = 0,

y2

10 + y3 01 = 0.

L =    ξ1 ξ2 ξ1 ξ2    . The transformed system is not elliptic, although it is equivalent to Laplace’s equation. 1st approach to resolve this issue consists in the introduction of a weighted symbol (Douglis and Nirenberg, 1955)

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 4/22

Example

• Example. Consider the transformation of the two-dimensional

Laplace equation ∆u = u20 + u02 = 0 to the first order system Ly =      y1

10 − y2 = 0,

y1

01 − y3 = 0,

y2

10 + y3 01 = 0.

L =    ξ1 ξ2 ξ1 ξ2    . The transformed system is not elliptic, although it is equivalent to Laplace’s equation. 1st approach to resolve this issue consists in the introduction of a weighted symbol (Douglis and Nirenberg, 1955) si: weight for the ith equation tj: weight for the jth dependent variable si + tj ≥ qij

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs

The weighted (principal) symbol of A is

• Aw
• i,j =
• |µ|=si+tj
• aµ(x)
• i,jξµ .

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs

The weighted (principal) symbol of A is

• Aw
• i,j =
• |µ|=si+tj
• aµ(x)
• i,jξµ .
• Definition. A is DN–elliptic, if Aw is injective for all real ξ = 0 and

for all x ∈ Ω.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs

The weighted (principal) symbol of A is

• Aw
• i,j =
• |µ|=si+tj
• aµ(x)
• i,jξµ .
• Definition. A is DN–elliptic, if Aw is injective for all real ξ = 0 and

for all x ∈ Ω. Ordinary ellipticity is a special case of DN-ellipticity with weights si = 0 and tj = q.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs

The weighted (principal) symbol of A is

• Aw
• i,j =
• |µ|=si+tj
• aµ(x)
• i,jξµ .
• Definition. A is DN–elliptic, if Aw is injective for all real ξ = 0 and

for all x ∈ Ω. Ordinary ellipticity is a special case of DN-ellipticity with weights si = 0 and tj = q.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 5/22

DN-elliptic systems of PDEs

The weighted (principal) symbol of A is

• Aw
• i,j =
• |µ|=si+tj
• aµ(x)
• i,jξµ .
• Definition. A is DN–elliptic, if Aw is injective for all real ξ = 0 and

for all x ∈ Ω. Ordinary ellipticity is a special case of DN-ellipticity with weights si = 0 and tj = q. L is DN-elliptic: s1 = s2 = −1, s3 = 0, t1 = 2, t2 = t3 = 1 Lw =    ξ1 −1 ξ2 −1 ξ1 ξ2    .

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs

2nd approach: to complete a system to the involutive form and check ordinary ellipticity

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs

2nd approach: to complete a system to the involutive form and check ordinary ellipticity If Ly = 0, then y2

01 − y3 10 = 0. This new equation is called a

differential consequence or integrability condition of the initial system.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs

2nd approach: to complete a system to the involutive form and check ordinary ellipticity If Ly = 0, then y2

01 − y3 10 = 0. This new equation is called a

differential consequence or integrability condition of the initial system. L′y =          y1

10 − y2 = 0,

y1

01 − y3 = 0,

y2

10 + y3 01 = 0,

y2

01 − y3 10 = 0.

, L′ =      ξ1 ξ2 ξ1 ξ2 ξ2 −ξ1      .

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs

2nd approach: to complete a system to the involutive form and check ordinary ellipticity If Ly = 0, then y2

01 − y3 10 = 0. This new equation is called a

differential consequence or integrability condition of the initial system. L′y =          y1

10 − y2 = 0,

y1

01 − y3 = 0,

y2

10 + y3 01 = 0,

y2

01 − y3 10 = 0.

, L′ =      ξ1 ξ2 ξ1 ξ2 ξ2 −ξ1      . L′ is the involutive form of L because no more new first order differential consequences can be found.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 6/22

Involutive Systems of PDEs

2nd approach: to complete a system to the involutive form and check ordinary ellipticity If Ly = 0, then y2

01 − y3 10 = 0. This new equation is called a

differential consequence or integrability condition of the initial system. L′y =          y1

10 − y2 = 0,

y1

01 − y3 = 0,

y2

10 + y3 01 = 0,

y2

01 − y3 10 = 0.

, L′ =      ξ1 ξ2 ξ1 ξ2 ξ2 −ξ1      . L′ is the involutive form of L because no more new first order differential consequences can be found. A system is involutive, if it contains all its differential conse- quences (up to given order).

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 7/22

Main Result

Theorem (Krupchyk, Seiler, Tuomela). If a system is DN-elliptic, then involutive form is elliptic.

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 7/22

Main Result

Theorem (Krupchyk, Seiler, Tuomela). If a system is DN-elliptic, then involutive form is elliptic. MuPad (DETools) - to compute the involutive form

Elliptic PDEs Classical Definition Problem DN-elliptic PDEs Involutive PDEs Main Result The SL-condition

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 7/22

Main Result

Theorem (Krupchyk, Seiler, Tuomela). If a system is DN-elliptic, then involutive form is elliptic. MuPad (DETools) - to compute the involutive form

• Example. Ay = ∇ × y + y = 0.

A is neither elliptic nor DN–elliptic. However, adding the integrability condition ∇ · y = 0 gives the symbol

A′ =

     ξ3 −ξ2 −ξ3 ξ1 ξ2 −ξ1 ξ1 ξ2 ξ3      which is elliptic.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 8/22

The SL-condition

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 9/22

Well-posed problems

Consider a BVP

• Ay = f ,

x ∈ Ω ⊂ Rn By = g , x ∈ Γ

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 9/22

Well-posed problems

Consider a BVP

• Ay = f ,

x ∈ Ω ⊂ Rn By = g , x ∈ Γ Is the problem well-posed?

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 9/22

Well-posed problems

Consider a BVP

• Ay = f ,

x ∈ Ω ⊂ Rn By = g , x ∈ Γ Is the problem well-posed? Well-posedness means either the solution exists and is unique in some space

• r more generally

(A, B) is Fredholm in some spaces (kernel and cokernel are finite dimensional)

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 9/22

Well-posed problems

Consider a BVP

• Ay = f ,

x ∈ Ω ⊂ Rn By = g , x ∈ Γ Is the problem well-posed? Well-posedness means either the solution exists and is unique in some space

• r more generally

(A, B) is Fredholm in some spaces (kernel and cokernel are finite dimensional) If A is elliptic, then a BVP is well-posed in some Sobolev spaces, if the boundary conditions satisfy the Shapiro-Lopatinskij condition.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 10/22

Definition of the SL-condition

Let A1, . . . , Ar be all m × m submatrices of A; A is of size k × m, k ≥ m pi = det(Ai) ∈ R[ξ].

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 10/22

Definition of the SL-condition

Let A1, . . . , Ar be all m × m submatrices of A; A is of size k × m, k ≥ m pi = det(Ai) ∈ R[ξ].

• Definition. The characteristic polynomial pA of an elliptic operator

A is pA = gcd

• p1, . . . , pr
• .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 10/22

Definition of the SL-condition

Let A1, . . . , Ar be all m × m submatrices of A; A is of size k × m, k ≥ m pi = det(Ai) ∈ R[ξ].

• Definition. The characteristic polynomial pA of an elliptic operator

A is pA = gcd

• p1, . . . , pr
• .

A is elliptic & deg(pA) = 2ν = ⇒ we take ν boundary conditions and consider a boundary

• perator

By = g, x ∈ Γ, where B is of size ν × m.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 11/22

Definition of the SL-condition

Local coordinates Let us choose local coordinates x = (x′, xn) where x′ = (x1, . . . , xn−1) on Ω such that Γ is given by xn = 0 and similarly let ξ = (ξ′, ζ).

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 11/22

Definition of the SL-condition

Local coordinates Let us choose local coordinates x = (x′, xn) where x′ = (x1, . . . , xn−1) on Ω such that Γ is given by xn = 0 and similarly let ξ = (ξ′, ζ). M+ =

• u : [0, ∞) → Cm
• lim

xn→∞ |u(xn)| = 0

• .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 11/22

Definition of the SL-condition

Local coordinates Let us choose local coordinates x = (x′, xn) where x′ = (x1, . . . , xn−1) on Ω such that Γ is given by xn = 0 and similarly let ξ = (ξ′, ζ). M+ =

• u : [0, ∞) → Cm
• lim

xn→∞ |u(xn)| = 0

• .
• Definition. (ν boundary conditions). The BV–operator (A, B) sat-

isfies the SL–condition, if the IVP

• A(x′, 0, ξ′, Dn/i)u(xn) = 0 , xn > 0,

Bw(x′, 0, ξ′, Dn/i)u(xn)

• xn=0 = d,

has a unique solution in M+ for all d and for all ξ′ = 0.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 12/22

Criterion for the SL-condition

Fix 0 = ξ′ ∈ Rn−1. Let ζ1, . . . , ζν be the roots of pA in the upper half of the complex plane. We set p+

A = (ζ − ζ1) · · · (ζ − ζν)

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 12/22

Criterion for the SL-condition

Fix 0 = ξ′ ∈ Rn−1. Let ζ1, . . . , ζν be the roots of pA in the upper half of the complex plane. We set p+

A = (ζ − ζ1) · · · (ζ − ζν)

• Lemma. There is a column v of adj(Aj) which is not divisible by

the polynomial p+

A .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 12/22

Criterion for the SL-condition

Fix 0 = ξ′ ∈ Rn−1. Let ζ1, . . . , ζν be the roots of pA in the upper half of the complex plane. We set p+

A = (ζ − ζ1) · · · (ζ − ζν)

• Lemma. There is a column v of adj(Aj) which is not divisible by

the polynomial p+

A .

Set h = Bwv. Dividing each element of h by p+

A we get

h = q p+

A + h

where

h =

ν−1

• τ=0

hτζτ H =

• h0, . . . , hν−1

∈ Cν×ν.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 12/22

Criterion for the SL-condition

Fix 0 = ξ′ ∈ Rn−1. Let ζ1, . . . , ζν be the roots of pA in the upper half of the complex plane. We set p+

A = (ζ − ζ1) · · · (ζ − ζν)

• Lemma. There is a column v of adj(Aj) which is not divisible by

the polynomial p+

A .

Set h = Bwv. Dividing each element of h by p+

A we get

h = q p+

A + h

where

h =

ν−1

• τ=0

hτζτ H =

• h0, . . . , hν−1

∈ Cν×ν.

• Theorem. (Krupchyk, Tuomela). Following statements are

equivalent:

■ an operator (A, B) satisfies the SL-condition; ■ for any ξ′ = 0, rank(H) = ν.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 13/22

Computational test in case of two variables

The case of two variables:

■ pA is a homogeneous in ξ = (ξ1, ζ) ■ setting ξ1 = 1 we get a polynomial ˆ

pA ∈ K[ζ] and symbol matrices ˆ A and ˆ Bw.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 13/22

Computational test in case of two variables

The case of two variables:

■ pA is a homogeneous in ξ = (ξ1, ζ) ■ setting ξ1 = 1 we get a polynomial ˆ

pA ∈ K[ζ] and symbol matrices ˆ A and ˆ Bw. Let K(α) be a splitting field for ˆ pA with minimal polynomial

• pmin. Let ˆ

α ∈ C be a root of pmin. We have injective homomorphisms: ι : K(α) → C

and

˜ ι : K(α)[ζ] → C[ζ] induced by the map ι(α) = ˆ α

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 13/22

Computational test in case of two variables

The case of two variables:

■ pA is a homogeneous in ξ = (ξ1, ζ) ■ setting ξ1 = 1 we get a polynomial ˆ

pA ∈ K[ζ] and symbol matrices ˆ A and ˆ Bw. Let K(α) be a splitting field for ˆ pA with minimal polynomial

• pmin. Let ˆ

α ∈ C be a root of pmin. We have injective homomorphisms: ι : K(α) → C

and

˜ ι : K(α)[ζ] → C[ζ] induced by the map ι(α) = ˆ α Let ρ1, . . . , ρν ∈ K(α) be the roots of ˆ pA such that ι(ρ1), . . . , ι(ρν) are in the upper half of the complex plane and let ˆ p+

A = (ζ − ρ1) · · · (ζ − ρν) ∈ K(α)[ζ] .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 14/22

Computational test in case of two variables

Then we compute ˆ h = ˆ Bwˆ v = ˆ qˆ p+

A + ˆ

h and set ˆ h =

ν−1

• τ=0

ˆ hτζτ

and

ˆ H = ˆ h0, . . . , ˆ hν−1 ∈

• K(α)

ν×ν

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 14/22

Computational test in case of two variables

Then we compute ˆ h = ˆ Bwˆ v = ˆ qˆ p+

A + ˆ

h and set ˆ h =

ν−1

• τ=0

ˆ hτζτ

and

ˆ H = ˆ h0, . . . , ˆ hν−1 ∈

• K(α)

ν×ν

• Theorem. (Krupchyk, Tuomela). An operator (A, B) satisfies the

SL-condition if and only if det( ˆ H) = 0.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 14/22

Computational test in case of two variables

Then we compute ˆ h = ˆ Bwˆ v = ˆ qˆ p+

A + ˆ

h and set ˆ h =

ν−1

• τ=0

ˆ hτζτ

and

ˆ H = ˆ h0, . . . , ˆ hν−1 ∈

• K(α)

ν×ν

• Theorem. (Krupchyk, Tuomela). An operator (A, B) satisfies the

SL-condition if and only if det( ˆ H) = 0.

• Proof. It can be shown that rank(H) = ν for all ξ1 = 0 is

equivalent to det( ˆ H) = 0.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 15/22

Example

• Example. Let us consider the following system

Ay =              y1

20 + y1 02 + y3 20 = 0,

2y1

20 + y2 11 = 0,

2y1

11 + y2 02 = 0,

y2

20 + y3 11 = 0,

y2

11 + y3 02 = 0

in R2

+ = {x ∈ R2 : x2 > 0}.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 15/22

Example

• Example. Let us consider the following system

Ay =              y1

20 + y1 02 + y3 20 = 0,

2y1

20 + y2 11 = 0,

2y1

11 + y2 02 = 0,

y2

20 + y3 11 = 0,

y2

11 + y3 02 = 0

in R2

+ = {x ∈ R2 : x2 > 0}.

The principal symbol: A =        ζ2 + ξ2

1

ξ2

1

2ξ2

1

ξ1ζ 2ξ1ζ ζ2 ξ2

1

ξ1ζ ξ1ζ ζ2        .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 16/22

Example

A is elliptic because pA = gcd

• p1, . . . , p10
• = ζ4 + ζ2ξ2

1 + 2ξ4 1

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 16/22

Example

A is elliptic because pA = gcd

• p1, . . . , p10
• = ζ4 + ζ2ξ2

1 + 2ξ4 1

Set ˆ pA = ζ4 + ζ2 + 2.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 16/22

Example

A is elliptic because pA = gcd

• p1, . . . , p10
• = ζ4 + ζ2ξ2

1 + 2ξ4 1

Set ˆ pA = ζ4 + ζ2 + 2. Computing with Singular we get:

■ the splitting field of ˆ

pA is Q(α) with a minimal polynomial pmin = ζ8 + 10ζ6 + 5ζ4 − 100ζ2 + 2116

■ the roots of ˆ

pA: ρ2 = −ρ1 =

7 105432α7 + 265 52716α5 + 3485 105432α3 + 6883 17572α

ρ3 = −ρ4 =

7 52716α7 + 265 26358α5 + 3485 52716α3 − 1903 8786α

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 17/22

Example

Computing numerically the roots of pmin gives: α1 = −α2 = −α3 = α4 ≈ −0.676 + i 2.935 , α5 = −α6 = α7 = −α8 ≈ −2.028 + i 0.978 .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 17/22

Example

Computing numerically the roots of pmin gives: α1 = −α2 = −α3 = α4 ≈ −0.676 + i 2.935 , α5 = −α6 = α7 = −α8 ≈ −2.028 + i 0.978 . Choosing α1 we get ρ1(α1) = −ρ2(α1) = −ρ3(α1) = ρ4(α1) ≈ 0.676 − i 0.978 .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 17/22

Example

Computing numerically the roots of pmin gives: α1 = −α2 = −α3 = α4 ≈ −0.676 + i 2.935 , α5 = −α6 = α7 = −α8 ≈ −2.028 + i 0.978 . Choosing α1 we get ρ1(α1) = −ρ2(α1) = −ρ3(α1) = ρ4(α1) ≈ 0.676 − i 0.978 . Hence ˆ p+

A = (ζ − ρ2)(ζ − ρ4) ∈ Q(α)[ζ]

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 17/22

Example

Computing numerically the roots of pmin gives: α1 = −α2 = −α3 = α4 ≈ −0.676 + i 2.935 , α5 = −α6 = α7 = −α8 ≈ −2.028 + i 0.978 . Choosing α1 we get ρ1(α1) = −ρ2(α1) = −ρ3(α1) = ρ4(α1) ≈ 0.676 − i 0.978 . Hence ˆ p+

A = (ζ − ρ2)(ζ − ρ4) ∈ Q(α)[ζ]

Let us compute the adjoint matrix of ˆ A124.

adj(ˆ

A124) =    ζ2 1 −ζ −2ζ ζ3 + ζ 2 2 −ζ2 − 1 ζ3 + ζ   

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 18/22

Example

Let ˆ v be the first column of adj(ˆ A124). ˆ v is nonzero modulo ˆ p+

A .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 18/22

Example

Let ˆ v be the first column of adj(ˆ A124). ˆ v is nonzero modulo ˆ p+

A .

Since ν = 2, we need to impose 2 boundary conditions.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 18/22

Example

Let ˆ v be the first column of adj(ˆ A124). ˆ v is nonzero modulo ˆ p+

A .

Since ν = 2, we need to impose 2 boundary conditions. First consider B1y =

• y1 + y3 = 0

y2 = 0

• n

∂R2

+.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 18/22

Example

Let ˆ v be the first column of adj(ˆ A124). ˆ v is nonzero modulo ˆ p+

A .

Since ν = 2, we need to impose 2 boundary conditions. First consider B1y =

• y1 + y3 = 0

y2 = 0

• n

∂R2

+.

Hence B1 =

• 1

1 1

• and

ˆ h = B1ˆ v =

• ζ2 + 2

−2ζ

• .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 18/22

Example

Let ˆ v be the first column of adj(ˆ A124). ˆ v is nonzero modulo ˆ p+

A .

Since ν = 2, we need to impose 2 boundary conditions. First consider B1y =

• y1 + y3 = 0

y2 = 0

• n

∂R2

+.

Hence B1 =

• 1

1 1

• and

ˆ h = B1ˆ v =

• ζ2 + 2

−2ζ

• .

Dividing by ˆ p+

A we get

h =

• β1ζ + β0

−2ζ

• and

H =

• β0

β1 −2

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 19/22

Example

where β0 =

1 382α6 + 15 764α4 − 123 764α2 + 579 382

β1 = −

7 105432α7 − 265 52716α5 − 3485 105432α3 + 10689 17572α

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 19/22

Example

where β0 =

1 382α6 + 15 764α4 − 123 764α2 + 579 382

β1 = −

7 105432α7 − 265 52716α5 − 3485 105432α3 + 10689 17572α

Hence det(H) = −2β0 = 0 which implies that B1 satisfies the SL–condition.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 19/22

Example

where β0 =

1 382α6 + 15 764α4 − 123 764α2 + 579 382

β1 = −

7 105432α7 − 265 52716α5 − 3485 105432α3 + 10689 17572α

Hence det(H) = −2β0 = 0 which implies that B1 satisfies the SL–condition. Let us now choose B2y =

• y1 = 0

y2

01 = 0

• n

∂R2

+.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 19/22

Example

where β0 =

1 382α6 + 15 764α4 − 123 764α2 + 579 382

β1 = −

7 105432α7 − 265 52716α5 − 3485 105432α3 + 10689 17572α

Hence det(H) = −2β0 = 0 which implies that B1 satisfies the SL–condition. Let us now choose B2y =

• y1 = 0

y2

01 = 0

• n

∂R2

+.

This gives (B2)w =

• 1

ζ

• and

ˆ h = (B2)wˆ v =

• ζ2

−2ζ2

• .

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 20/22

Example

Dividing again by ˆ p+

A we get

h =

• β1ζ + b0

−2β1ζ − 2b0

• and

H =

• b0

β1 −2b0 −2β1

• where

b0 =

1 382α6 + 15 764α4 − 123 764α2 − 185 382

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 20/22

Example

Dividing again by ˆ p+

A we get

h =

• β1ζ + b0

−2β1ζ − 2b0

• and

H =

• b0

β1 −2b0 −2β1

• where

b0 =

1 382α6 + 15 764α4 − 123 764α2 − 185 382

Now det(H) = 0 and hence B2 does not satisfy the SL–condition.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 21/22

The SL-condition (general case)

Let us impose ˜ ν ≥ ν boundary conditions.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 21/22

The SL-condition (general case)

Let us impose ˜ ν ≥ ν boundary conditions.

• Theorem. (Krupchyk, Tuomela). Following statements are

equivalent:

■ an operator (A, B) satisfies the SL-condition; ■ for any ξ′ = 0: ◆ rank(H, d) = ν for all d ∈ ker((Φ22 1 )w); ◆ dim

• ker((Φ22

1 )w)

• = ν.

Φ1(f, g) =

• Φ11

1 f , Φ21 1 f + Φ22 1 g

• is the compatibility operator

for (A, B)

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 21/22

The SL-condition (general case)

Let us impose ˜ ν ≥ ν boundary conditions.

• Theorem. (Krupchyk, Tuomela). Following statements are

equivalent:

■ an operator (A, B) satisfies the SL-condition; ■ for any ξ′ = 0: ◆ rank(H, d) = ν for all d ∈ ker((Φ22 1 )w); ◆ dim

• ker((Φ22

1 )w)

• = ν.

Φ1(f, g) =

• Φ11

1 f , Φ21 1 f + Φ22 1 g

• is the compatibility operator

for (A, B) We propose a constructive method to compute compatibility

• perators for BV operators based on Gröbner bases

techniques.

Elliptic PDEs The SL-condition Well-posedness Definition Criterion Computational test Example General case

Ellipticity and Fredholm boundary value problems Katya Krupchyk and Jukka Tuomela - p. 22/22

And finally...

Thank you for your attention!