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Boundary Value Problems for Holomorphic Functions in R2

Boundary Value Problems for the Dirac Operator in Rm

Recent results toward higher order Dirac equations

Boundary value problems for the Dirac equationin fractal domains

Ricardo Abreu Blaya

CIMPA 2016

Santiago de Cuba

June, 2016

Ricardo Abreu Blaya Boundary value problems in fractal domains




Jump problem

Let Ω ⊂ R2 be a Jordan domain withboundary Γ and let be g a continuouscomplex valued function defined on Γ.

To find a function Φ, holomorphic inR2 \ Γ, with continuous boundary values upto Γ satisfying

Φ+(t)− Φ−(t) = g(t) , t ∈ Γ, (1)





Jump problem



Φ+(t)− Φ−(t) = g(t) , t ∈ Γ, (1)





Jump problem



Φ+(t)− Φ−(t) = g(t) , t ∈ Γ, (1)





Riemann-Hilbert problem

Riemann Condition

Φ+(t)− G (t)Φ−(t) = g(t) , t ∈ Γ, (2)

Hilbert Condition

<[F (t)Φ+(t)] = f (t) , t ∈ Γ. (3)

Gajov F. D., Boundary value problems, 3rd ed; Nauka,Moskow; English transl. of 2nd ed, Pergamon Press, Oxford,(1977); and Addison-Wesley, Reading, MA (1966).

J. K. Lu. Boundary value problems for analytic functions.World Scientific Publish., 1993.





Riemann-Hilbert problem

Riemann Condition

Φ+(t)− G (t)Φ−(t) = g(t) , t ∈ Γ, (2)

Hilbert Condition

<[F (t)Φ+(t)] = f (t) , t ∈ Γ. (3)

Gajov F. D., Boundary value problems, 3rd ed; Nauka,Moskow; English transl. of 2nd ed, Pergamon Press, Oxford,(1977); and Addison-Wesley, Reading, MA (1966).

J. K. Lu. Boundary value problems for analytic functions.World Scientific Publish., 1993.





Cauchy transform: the main tool

CΓg(z) =1

2πi

∫Γ

g(τ)dτ

τ − z, z /∈ Γ (4)

Plemelj-Sokhotski Formulae

C+Γ g(t)− C−Γ g(t) = g(t)

C+Γ g(t) + C−Γ g(t) =

1

πi

∫Γ

g(τ)dτ

τ − t





Cauchy transform: the main tool

CΓg(z) =1

2πi

∫Γ

g(τ)dτ

τ − z, z /∈ Γ (4)

Plemelj-Sokhotski Formulae

C+Γ g(t)− C−Γ g(t) = g(t)

C+Γ g(t) + C−Γ g(t) =

1

πi

∫Γ

g(τ)dτ

τ − t





Holomorphic functions in Rm

Cauchy Riemann Operator

∂z =1

2(∂x + i∂y ), z = x + iy .

f : Ω→ C is holomorphic in Ω ⊂ R2 ⇐⇒ ∂z f = 0 in Ω

Dirac Operator in Rm

∂ =m∑i=1

ei∂xi

f : Ω→ R0,m is holomorphic in Ω ⊂ Rm ⇐⇒ ∂f = 0 in Ω

holomorphic⇐⇒ monogenic





Holomorphic functions in Rm

Cauchy Riemann Operator

∂z =1

2(∂x + i∂y ), z = x + iy .

f : Ω→ C is holomorphic in Ω ⊂ R2 ⇐⇒ ∂z f = 0 in Ω

Dirac Operator in Rm

∂ =m∑i=1

ei∂xi

f : Ω→ R0,m is holomorphic in Ω ⊂ Rm ⇐⇒ ∂f = 0 in Ω

holomorphic⇐⇒ monogenic





Special Instances of Dirac equation ∂f = 0

Solenoidal and irrotational vector fields

~f : R3 7→ R3, ∂~f = 0⇐⇒

div~f = 0

curl~f = 0

Harmonic Vector Fields in Rm

~f : Rm 7→ Rm, ∂~f = 0⇐⇒

div~f = 0

∂ ∧ ~f = 0

(∧ denotes the usual outer product)Time-harmonic electromagnetic fields

∂−α~ϕ = div~j + α~j

∂α ~ψ = −div~j + α~j⇐⇒

curl ~H = −iωε~E + ~j

curl~E = iωµ ~H

div~E = ρε

div ~H = 0







~f : R3 7→ R3, ∂~f = 0⇐⇒

div~f = 0

curl~f = 0


~f : Rm 7→ Rm, ∂~f = 0⇐⇒

div~f = 0

∂ ∧ ~f = 0

(∧ denotes the usual outer product)

Time-harmonic electromagnetic fields


∂α ~ψ = −div~j + α~j⇐⇒


curl~E = iωµ ~H

div~E = ρε

div ~H = 0







~f : R3 7→ R3, ∂~f = 0⇐⇒

div~f = 0

curl~f = 0


~f : Rm 7→ Rm, ∂~f = 0⇐⇒

div~f = 0

∂ ∧ ~f = 0

(∧ denotes the usual outer product)Time-harmonic electromagnetic fields


∂α ~ψ = −div~j + α~j⇐⇒


curl~E = iωµ ~H

div~E = ρε

div ~H = 0Ricardo Abreu Blaya Boundary value problems in fractal domains




Higher Order Dirac Equations

∂2f = 0(Bimonogenic)⇐⇒4f = 0(Harmonic)

∂4f = 0(Tetramonogenic)⇐⇒44f = 0(Biharmonic)

∂k f = 0(polymonogenic)⇐⇒4k f = 0(Polyharmonic)





The jump problem for ∂ in Rm

Let Ω+ := Ω, and Ω− := Rm \ (Ω ∪ Γ) denote the complementaryconnected domains separated by Γ on Rm.

The jump problem associated to ∂ is the problem of reconstructinga R0,m-valued function Ψ satisfying in Rm \ Γ the Dirac equation∂Ψ = 0, vanishing at infinity and having a prescribed jump gacross Γ, i.e.,

Ψ+(x)−Ψ−(x) = g(x), x ∈ Γ, Ψ(∞) = 0, (5)

where Ψ±(x) = limΩ±3y→x

Ψ(y).





The jump problem for ∂ in Rm

Let Ω+ := Ω, and Ω− := Rm \ (Ω ∪ Γ) denote the complementaryconnected domains separated by Γ on Rm.The jump problem associated to ∂ is the problem of reconstructinga R0,m-valued function Ψ satisfying in Rm \ Γ the Dirac equation∂Ψ = 0, vanishing at infinity and having a prescribed jump gacross Γ, i.e.,

Ψ+(x)−Ψ−(x) = g(x), x ∈ Γ, Ψ(∞) = 0, (5)

where Ψ±(x) = limΩ±3y→x

Ψ(y).





Clifford Cauchy transform: the main tool

CΓg(x) :=

∫Γ

E0(y − x)κ(y)g(y)dy , x /∈ Γ,

where

E0(x) = − 1

σm

x

|x |m.

κ(y) denotes the exterior normal vector at y ∈ Γ and σm thesurface area of the the unit sphere in Rm.





Classical Setting

(5) is solvable⇐⇒ CΓg(x) has continuous extension

I Clifford-Cauchy transform for compact Liapunov surfaces.

I Plemelj-Sokhotski Formula

(CΓg)+(x)− (CΓg)−(x) = g(x) x ∈ Γ

Iftimie V. Fonctions hypercomplexes. Bulletin Mathematiquede la Societe des Sciences Mathematiques de la RepublicqueSocialiste de Romanie; 9(57): 279-332, 1965.





Non-smooth setting

I Continuous extension of the Clifford-Cauchy transform CΓg(x)on a rectifiable and Ahlfors-David regular surface Γ in Rm isoptimally answered under assumption that∫

Γ\|y−x |≤εE0(y − x)κ(y)(g(y)− g(x))dy , x ∈ Γ

converges uniformly on Γ as ε→ 0.

J. Bory Reyes and R. Abreu Blaya. Cauchy transform andrectifiability in Clifford Analysis. Z. Anal. Anwend, Vol.24, No 1, 167-178, 2005.





Non-smooth setting

I Continuous extension of the Clifford-Cauchy transform CΓg(x)on a rectifiable and Ahlfors-David regular surface Γ in Rm isoptimally answered under assumption that∫

Γ\|y−x |≤εE0(y − x)κ(y)(g(y)− g(x))dy , x ∈ Γ

converges uniformly on Γ as ε→ 0.

J. Bory Reyes and R. Abreu Blaya. Cauchy transform andrectifiability in Clifford Analysis. Z. Anal. Anwend, Vol.24, No 1, 167-178, 2005.





Non-rectifiable setting

I Let Hm−1(Γ) <∞. If g satisfies a Holder condition on Γ withexponent ν, 0 < ν ≤ 1, then under condition

ν >m − 1

m, (6)

jump problem (5) permits a solution given by CΓg .

R. Abreu Blaya; J. Bory Reyes and T. Moreno Garcıa.Cauchy Transform on non-rectifiable surfaces in CliffordAnalysis. J. Math. Anal. Appl., Vol. 339, 31-44, 2008.





Non-rectifiable setting

I Let Hm−1(Γ) <∞. If g satisfies a Holder condition on Γ withexponent ν, 0 < ν ≤ 1, then under condition

ν >m − 1

m, (6)

jump problem (5) permits a solution given by CΓg .

R. Abreu Blaya; J. Bory Reyes and T. Moreno Garcıa.Cauchy Transform on non-rectifiable surfaces in CliffordAnalysis. J. Math. Anal. Appl., Vol. 339, 31-44, 2008.





Domains with fractal boundary

Alternative way of defining theClifford-Cauchy transform, where a centralrole is played by the Teodorescu transforminvolving fractal dimensions, is described.

(C∗Γf )(x) := g(x)χΩ(x)+

∫Ω

E0(y−x)∂g(y)dy

I g → a Whitney extension of g

I χΩ(x) → characteristic function of Ω.





Fractal dimension setting

I If f satisfies a Holder condition on Γ with exponent ν,0 < ν ≤ 1, then a solution of (1) can be obtained by usingC∗Γg under assumption

ν >M(Γ)

m. (7)

R. Abreu Blaya; J. Bory Reyes and T. Moreno Garcıa.Minkowski dimension and Cauchy transform in Cliffordanalysis. Compl. Anal. Oper. Theory, Vol. 1, No. 3, 301-315,2007.





Fractal dimension setting

I If f satisfies a Holder condition on Γ with exponent ν,0 < ν ≤ 1, then a solution of (1) can be obtained by usingC∗Γg under assumption

ν >M(Γ)

m. (7)

R. Abreu Blaya; J. Bory Reyes and T. Moreno Garcıa.Minkowski dimension and Cauchy transform in Cliffordanalysis. Compl. Anal. Oper. Theory, Vol. 1, No. 3, 301-315,2007.





Fractal dimension setting (cont.)

The condition (7) cannot be improved on the whole class ofsurfaces with fixed upper Minkowski fractal dimension.

TheoremFor any m ∈ [m − 1,m) and 0 < ν ≤ m

m there exists a surfaceΓ∗ ⊂ Rm such that M(Γ) = m and a ν-Holder continuous functiong∗ in Γ, such that the jump problem (5) has no solution.

R. Abreu Blaya; J. Bory Reyes and T. Moreno Garcıa.Teodorescu transform decomposition of multivector fields onfractal hypersurfaces, Oper. Theory Adv. Appl., Vol. 167,1-16, 2006.























d-summable setting

I If g satisfies the ν-Holder condition, 0 < ν ≤ 1

I Γ is d-summable

1∫0

NΓ(τ)τd−1dτ converges

where NΓ(τ) denotes the number of τ -balls needed to cover Γ.

I Then, under condition

ν >d

m(8)

the jump problem (1) permits a solution given by C∗Γg .





d-summable setting


I Γ is d-summable

1∫0




ν >d

m(8)






d-summable setting


I Γ is d-summable

1∫0




ν >d

m(8)






d-summable setting

I The condition (8) cannot be weakened on the whole class ofd-summable surfaces.

R. Abreu Blaya and J. Bory Reyes. Criteria for monogenicity ofClifford algebra-valued functions on fractal domain, Arch.Math. (Basel), 95, No. 1, 45-51, 2010.

R. Abreu Blaya; J. Bory Reyes; T. Moreno Garcıa. Thesharpness of condition for solving the jump problem. Comm.Math. Anal. Vol. 12, No. 2, 26-33, 2012.





d-summable setting

I The condition (8) cannot be weakened on the whole class ofd-summable surfaces.

R. Abreu Blaya and J. Bory Reyes. Criteria for monogenicity ofClifford algebra-valued functions on fractal domain, Arch.Math. (Basel), 95, No. 1, 45-51, 2010.

R. Abreu Blaya; J. Bory Reyes; T. Moreno Garcıa. Thesharpness of condition for solving the jump problem. Comm.Math. Anal. Vol. 12, No. 2, 26-33, 2012.





Time-Harmonic Maxwell equation in fractal domains

Let ρ and ~j belong to Lp(Ω) (p > 3). Let ~e and ~h be complexvector valued functions in C 0,ν(Γ), ν > d

3 .

If there exists a pair of vector fields ~E and ~H, both in C 0,ν(Ω ∪ Γ),satisfying in Ω the time-harmonic Maxwell equations and on Γ theboundary conditions

~E |Γ = ~e, ~H|Γ = ~h, (9)

ThenT−α ∂−α(−iωε~e + α~h)|Γ = T−α(div~j + α~j)|Γ, (10)

Sc(T−α ∂−α(−iωε~e + α~h)) = Sc(T−α(div~j + α~j)) in Ω (11)





Time-Harmonic Maxwell equation in fractal domains.Cont.

andTα ∂α(iωε~e + α~h)|Γ = Tα(−div~j + α~j)|Γ (12)

Sc(Tα ∂α(iωε~e + α~h)) = Sc(Tα(−div~j + α~j)) in Ω (13)

where α = ω√εµ.






Conversely, if (10)-(13) are satisfied, then the vector fields

~E = ~e − 1

2iωεiωε(Tα ∂α + T−α ∂−α)(~e) + α(T−α ∂−α −

−Tα ∂α)(~h)− (Tα + T−α)(div~j) + α(Tα − T−α)(~j)

and

~H = ~h − 1

2αiωε(T−α ∂α − Tα ∂−α)(~e)− α(T−α ∂−α +

+Tα ∂α)(~h)− (Tα − T−α)(div~j) + α(Tα + T−α)(~j)

satisfy Maxwell equations together with the boundary conditions(9)






Abreu Blaya, R.; Avila Avila, R., Bory Reyes, J. (2015).Boundary value problems for Dirac operators and Maxwell’sequations in fractal domains. Math. Meth. Appl. Sci., vol. 38,393-402.





Higher order Lipschitz data

Let k be a non-negative integer and 0 < α ≤ 1. We shall say thata function g , defined in Γ, belongs to the higher order Lipschitzclass Lip(Γ, k + ν) if there exist bounded functions g (j),0 < |j | ≤ k , defined on Γ, with g (0) = g , and so that

Rj(x , y) = g (j)(x)−∑

|l |≤k−|j |

g (j+l)(y)

l!(x − y)l , x , y ∈ Γ (14)

satisfies

|Rj(x , y)| = O(|x − y |k+ν−|j |), x , y ∈ Γ, |j | ≤ k . (15)





Higher order Lipschitz data: Whitney extension theorem

Such a function g can be extended to a C k,ν function g on all ofRm with

∂(j)g

∂x j:=

∂|j |g

∂x j11 ∂xj22 . . . ∂x

jmm

= g (j) in Γ,

for all |j | := j1 + j2 + · · ·+ jm ≤ k .





Jump Problem for polymonogenic functions

Let be g ∈ Lip(Γ, k − 1 + ν). We are interested in the followingboundary value problem:Find a polymonogenic function Φ, i.e., ∂kΦ = 0 in Rm \ Γsatisfying the boundary conditions

(∂ iΦ)+(x)− (∂ iΦ)−(x) = ∂ ig(x), x ∈ Γ for 0 ≤ i ≤ k − 1(∂ iΦ)−(∞) = 0, for 0 ≤ i ≤ k − 1.

(16)





Solution. Higher order Teodorescu operator

Theorem

If g ∈ Lip(Γ, k − 1 + ν), with ν >d

m, then the jump problem (16)

has a solution given by

Φ(x) =

g(x)− Tk [∂k g ](x), x ∈ Ω+

−Tk [∂k g ](x), x ∈ Ω−(17)

where

Tk f (x) :=(−1)k

σm

∫Ω

(ξ − x)(ξ − x + ξ − x)k−1

2k−1(k − 1)!|ξ − x |mf (ξ)dξ

is a higher order Teodorescu operator.





Uniqueness

Theorem

Let be g ∈ Lip(Γ, k − 1 + ν), with ν >d

mand let

dimH(Γ)− (m − 1) < β <mν − d

m − d.

Then the function given by (17) is the unique solution of the jumpproblem (16) which belongs to the class

Lipkβ := φ ∈ Lip(Ω+, k+β)∩Lip(Ω−, k+β), ∂ iφ(∞) = 0, 0 < i ≤ k−1

Abreu Blaya, R., Avila Avila, R., Bory Reyes, J. (2015). Boundaryvalue problems with higher order Lipschitz boundary data forpolymonogenic functions in fractal domains. Appl. Math. Comput.,vol. 269, no. 10, 802-808.





Problems for the sandwich equation ∂Φ∂ = 0

Inframonogenic Functions

∂Φ∂ = 0 =⇒

Φ is biharmonicΦ two-sided 3-monogenic

Inframonogenic Vector Fields

∂ ~F∂ = 0⇐⇒

∂ • (∂ • ~F ) = 0

∂ ∧ (∂ • ~F )− ∂ • (∂ ∧ ~F ) = 0

∂ ∧ (∂ ∧ ~F ) = 0

(• denotes the usual inner product (up to a minus sign)).





Problems for the sandwich equation ∂Φ∂ = 0

Inframonogenic Functions

∂Φ∂ = 0 =⇒

Φ is biharmonicΦ two-sided 3-monogenic

Inframonogenic Vector Fields

∂ ~F∂ = 0⇐⇒

∂ • (∂ • ~F ) = 0

∂ ∧ (∂ • ~F )− ∂ • (∂ ∧ ~F ) = 0

∂ ∧ (∂ ∧ ~F ) = 0

(• denotes the usual inner product (up to a minus sign)).





Inframonogenic Teodorescu operator

T φ(x) =1

2[

∫Ω

E0(y − x)φ(y)(y − x)dy +

+n∑

i=1

ei

∫Ω

E1(y − x)φ(y)dyei ],

where

E1(x) =1

(m − 2)σm|x |m−2, x 6= 0

is the fundamental solution of the Laplace operator 4 in Rm.





THANK YOU


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