Non linear elliptic systems and families of quasiregular...

Non linear elliptic systems and families ofquasiregular mappings

K.Astala, A.Clop, D.Faraco, J.Jääskelainen, L.Székelyhidi Jr

Santander , July 2015

D.Faraco (UAM-ICMAT) Non linear Beltrami equation

The main character

The nonlinear Beltrami Equation

∂z f (z) = H(z , ∂z f (z))

Nonlinearity H : C× C→ C s.t(H1) H is k-Lipschitz in the second variable, k < 1

|H(z ,w1)−H(z ,w2)| ≤ |w1 − w2|,

for almost every z ∈ C, and the normalization H(z , 0) ≡ 0holds.

(H2) For every w ∈ C, z 7→ H(z ,w) is measurable on C,

Examples

C linear Beltrami equation.

∂f (z) = µ∂f (z) ‖µ‖L∞ ≤ k < 1

R linear Beltrami

∂f (z) = µ∂f (z) + ν∂f (z), ‖µ‖∞ + ‖ν‖∞ ≤ k < 1

Governs linear elliptic systems.Autonomous equations

∂f (z) = H(∂f )

γ-equations. Let γ be a curve in the complex plane.

∂f (z) = µdist(γ, ∂f (z)) ‖µ‖∞ ≤ k < 1

Key to the solution of Tartar conjecture by F-Szekelyhidi Jr.

History of nonlinear Beltrami equation

Introduced by Bojarksi and Iwaniec 70Astala, Iwaniec and Saksman (Beltrami Operators in the plane)Astala Iwaniec and Martin [Monograph]. Governs all planarelliptic systems. Good existence theory for H(z , f ,w)

Good example of using topology of the solutions to go beyondthe classical ellptic P.D.E techniques. e.g Solution to CalderónProblem

Quasiconformal maps

Normalized solutions = Homeomorphism fixing 0 and 1

‖∂f (z)‖ ≤ k |∂f |︸︷︷︸distortion inequality

Recall: Homeomorphic functions satisfying the distortion inequalityare called quasiconformal.Not homeomorphic solutions of the distortions inequality are calledquasiregular mappings or mappings of bounded distortion.Quasiregular mappings factorize as f = h F where F is anormalized quasiconformal solution and h is holomorphic.

Uniqueness of Normalized solutions

Unique conformal mapping fixing 0 and 1 is z . what happens forBeltrami equations?

C linear Beltrami equation.

∂f (z) = µ∂f (z) |µ‖L∞ ≤ k < 1

Yes from StoilowGeneral R linear Beltrami

∂f (z) = µ∂f (z) + ν∂f (z), ‖µ‖∞ + ‖ν‖∞ ≤ k < 1

Yes Astala-Iwaniec-Martin 2006. Reduction to the reduced equation

∂f (z) = µRe(∂f (z)) |µ‖L∞ ≤ k < 1

Uniqueness for normalized γ quasiconformal maps

Theorem Suppose 0 ∈ γ. If there are zi ∈ C, i = 1, 2, y q0 ∈ γsuch that ϕ(zi ) = q0zi , then ϕ(z) ≡ q0z .Corollary 1 If f ∈W 1,2

loc (C) is a global homeomorphism s.t

|∂f (z)| ≤ kdist(∂f (z), [0, 1])

and f (0) = 0, f (z) = z , for some z ⊂ C, f (z) ≡ z .Corollary 2In particular if 0, Id ∈ Γ and H : C× C→ C k Lipschitz is sucha− = H(z , a+) for almost every z and every A = (a+, a−) ∈ Γ thereis a unique normalized solution to

∂f (z) = H(z , ∂f (z))

Uniqueness of normalized solutions for non linear systems

Is there always uniqueness of normalized solutions to nonlinearBeltrami equations?No! ACFJS IMRN 2011

Uniqueness if lim sup|z|→∞ K (z) = k(z)−1k(z)+1 <

If g(z) = z Uniqueness holds if lim sup|z|→∞ K (z) ≤ 2There is uniqueness for autonomous equations.Different bound for H(f , ∂f ).Uniqueness if there is γ(t) : γ(0) = 0, γ(1) = 1 and such thatH(z , γ(t)) ∈ Lp0 p0 < 1.The first and second point are sharp up to the end point!

Uniqueness of normalized solutions for non linear systems

Is there always uniqueness of normalized solutions to nonlinearBeltrami equations?No! ACFJS IMRN 2011

Uniqueness if lim sup|z|→∞ K (z) = k(z)−1k(z)+1 <

If g(z) = z Uniqueness holds if lim sup|z|→∞ K (z) ≤ 2There is uniqueness for autonomous equations.Different bound for H(f , ∂f ).Uniqueness if there is γ(t) : γ(0) = 0, γ(1) = 1 and such thatH(z , γ(t)) ∈ Lp0 p0 < 1.The first and second point are sharp up to the end point!

Difference of two H solutions is quasiregular

Let, f,g solve the same! H equation. Then

∂(f (z)− g(z)) = H(∂(f ))−H(∂(g))

Hence,

i.e f − g is quasiregular.

Let 1 < K = k−1k+1 <∞. Then

deg(f − g) ≤ K 2

Proof: Follows from writing f − g = P(H) P an holomorphicpolynomial and H a normalized quasiconformal mapping

|z |dK ≤ |f − g | ≤ C |z |K

Corollary if K 2 < 2, d = 1 and thus f = g .In the case, g = z a tolopogical argument show that|f − g | ≤ C |z |which yields the bound d < K and our second claim.

Let 1 < K = k−1k+1 <∞. Then

deg(f − g) ≤ K 2

|z |dK ≤ |f − g | ≤ C |z |K

Let 1 < K = k−1k+1 <∞. Then

deg(f − g) ≤ K 2

|z |dK ≤ |f − g | ≤ C |z |K

We say that a Beltrami equation has the uniqueness property if forevery z0, z1, ω0, ω1 ∈ C there is a unique H-quasiconformal mapsuch that f (z0) = ω0 and f (z1) = ω1.If the equation has the uniqueness property we can considerFH = Φa

1 Φa(0) = 0,2 Φa(1) = a3 Lemma Φb − Φa is quasiconformal!

We say F is a family of QC mpas if 1− 3 holds.

Description of FH

C linear Beltrami equation.Fµ = aΦ1, a ∈ CA complex line in W 1,2

loc (Ω,C)R linear Beltrami equationFµ,ν = aΦ1 + bΦi , a, b ∈ R(Astala-Iwaniec-Martin-Leonetti-Nesi-Jääskelainen)A two dimensional R linear subspace of R2.H linear Beltrami equation.THM ACFJ14If H(z ,w) is C 1(w)FH is an C 1-embedded submanifold of W 1,2

loc(Embedded submanifolds of Frechet spaces are inmersions suchthat the Daϕa splits)

Description of FH

C linear Beltrami equation.Fµ = aΦ1, a ∈ CA complex line in W 1,2

loc (Ω,C)R linear Beltrami equationFµ,ν = aΦ1 + bΦi , a, b ∈ R(Astala-Iwaniec-Martin-Leonetti-Nesi-Jääskelainen)A two dimensional R linear subspace of R2.H linear Beltrami equation.THM ACFJ14If H(z ,w) is C 1(w)FH is an C 1-embedded submanifold of W 1,2

loc(Embedded submanifolds of Frechet spaces are inmersions suchthat the Daϕa splits)

What is the tangent plane?

THM ACFJ14 The tangent plane is given by the solutions to the Rlinear equations.

TΦaFH = Fµa,νa

µa(z) = ∂wH(z , ∂zφa(z)

)and νa(z) = ∂wH

(z , ∂zφa(z)

• ηet = φa+te−φa

t , t ∈ (0,∞), e ∈ C are QC mapsηet (0) = 0, ηe

t (1) = e normal!!• ∂zη

etj (z) == µa(z) ∂zη

etj (z) + νa(z) ∂zηe

tj (z) + htj (z),and

htj (z) =H(z ,w0 + tj ∂zη

etj (z)

(z ,w0

−∂wH

(z ,w0

)tj ∂zη

etj (z) + ∂wH

(z ,w0

)tj ∂zηe

tj (z)

TΦaFH = Fµa,νa

)and νa(z) = ∂wH

(z , ∂zφa(z)

etj (z)

(z ,w0

−∂wH

(z ,w0

)tj ∂zη

etj (z) + ∂wH

(z ,w0

)tj ∂zηe

tj (z)

TΦaFH = Fµa,νa

)and νa(z) = ∂wH

(z , ∂zφa(z)

etj (z)

(z ,w0

−∂wH

(z ,w0

)tj ∂zη

etj (z) + ∂wH

(z ,w0

)tj ∂zηe

tj (z)

TΦaFH = Fµa,νa

)and νa(z) = ∂wH

(z , ∂zφa(z)

etj (z)

(z ,w0

−∂wH

(z ,w0

)tj ∂zη

etj (z) + ∂wH

(z ,w0

)tj ∂zηe

tj (z)

More details

• Choosing tj sparse enough the bad set has measure zero.

B =∞⋃l=1

∞⋂k=1

∞⋃j=k

z ∈ D(0,R) : tj |∂zη

etj (z)| > 2−l

limj→∞

tj |∂zηetj (z)| = 0⇒

|htj (z)||∂zηe

tj (z)|→ 0, j →∞, a.e. on D(0,R).

•The limit map Φe ⊂ Fµa,νa by uniqueness the whole sequenceconverges in L∞(loc)

•Since the sequence ηet is an approximate solution to the linear

equation, approximate versions of Cacciopoli inequality yieldconvergence in the W 1,2 norm

Questions

Theorem [AIM]"There is a one to one corrspondence between R 2 dimensional"planes" of quasiconformal mappings and linear Beltrami equations.FΦ1,Φ2 = Fµ,νTo each H we can associate a family FH.To each family F we can associate an HF if w = ∂zϕa(z) Then

H(z ,w) = ∂zϕa(z)

By quasiregularity of ϕa − ϕb this is always well defined but...howmuch this commutes?• HFH = H?• FHF = F?

Questions

answers

Key:a→ ∂aϕ(z) is a global homeomorphism in C. (for every z ∈ C)O.K under regularity assumptions of HFH does not degenerate• Local homeo in C:

det[Da∂zφa(z)] 6= 0,

(Schauder estimates+New Null Lagrangians)•∂zφ∞(z) =∞

1c≤ |∂zφa(z)|

|a|≤ c , z ∈ D(0,R), a 6= 0.

(Non vanishing of Jacobians)•Topology implies that this is enough to be a globalhomeomorphism.

Non degeneracy 1

f (z) := ∂a1φa(z) = ∂aφa(z) + ∂aφa(z)

g(z) := ∂ai φa(z) = −i

(∂aφa(z)− ∂aφa(z)

Im(∂z f ∂zg

)= |∂a∂zφa|2 − |∂a∂zφa|2 = det[Da∂zφa(z)],

Now f , g ∈ Fµa,νa , thus by (Astala-Jääskelainen,Leonetti-Nesi)

Im(∂z f ∂zg

a.e.z (Problems... the exceptional set depends on a)We need that µa(z) ∈ Cα to have conditions which hold for every zbutµa(z) = ∂wH

(z , ∂zφa(z)

)O.K for H such that H,DH are Hölder.

Non degeneracy 1

f (z) := ∂a1φa(z) = ∂aφa(z) + ∂aφa(z)

g(z) := ∂ai φa(z) = −i

Im(∂z f ∂zg

(z , ∂zφa(z)

Non degeneracy 1

f (z) := ∂a1φa(z) = ∂aφa(z) + ∂aφa(z)

g(z) := ∂ai φa(z) = −i

Im(∂z f ∂zg

(z , ∂zφa(z)

Schauder Estimates for Non Linear Beltrami Equation

[ THM Astala-Clop-F-Jääskeläinen 14]• H(z ,w) is Cα(z), Then [Df ]Cβ < C , β ≤ minα, 1

K HereK = k+1

k−1 .Key Idea on the Proof: Perturbation from the Autonomousequation

∂z f = H(∂z f )

Then fh = f (z+h)−f (z)h is quasiregular, and thus ∂e f is quasiregular

and thus locally 1K Hölder.

Open question: Is this sharp?

Schauder Estimates for Non Linear Beltrami Equation

[ THM Astala-Clop-F-Jääskeläinen 14]• H(z ,w) is Cα(z), Then [Df ]Cβ < C , β ≤ minα, 1

K HereK = k+1

k−1 .Key Idea on the Proof: Perturbation from the Autonomousequation

∂z f = H(∂z f )

Then fh = f (z+h)−f (z)h is quasiregular, and thus ∂e f is quasiregular

and thus locally 1K Hölder.

Open question: Is this sharp?

Schauder Estimates II

Non homogeneus, non linear Beltrami∂z f (z) = H(z , ∂z f (z)) + h(z)

Theorem (ACFJ15)

0 < α < 1 h ∈ Cαloc(C,C), let f ∈W 1,2loc (C,C) be a solution to the

non-homogeneous nonlinear Beltrami equation tw 7→ H(z ,w) ∈ C 1(C,C). Then f ∈ C 1,α

loc (C,C).

Key of the proof: Different Already at the level of the automousequation, which we can honestly differentiate

∂zg(z) = ∂wH(∂zF (z)) ∂zg(z) + ∂wH(∂zF (z)) (1)

and freeze coefficients here to get that g ∈ C 1,β for all β < 1.Then we continue with,(Campanato characterization of Hölder continuity) +Properties oflocal Beurling transform+ Cacciopoli type inequalities

Non degeneracy 2

• ∂zφ∞(z) =∞

1c≤ |∂zφa(z)|

|a|≤ c , z ∈ D(0,R), a 6= 0.

The map ϕa(z)a is a normalized QC map which solves a H for where

H(z ,w) = 1a H(z , aw). Thus the upper bound follows from the

Schauder estimates. Lower bound follows from

Theorem (ACFJ 14)

Let H(z ,w) is Cα(z),. Then H quasiconformal homeomorphism fsatisfy that

J(z , f ) > c

C = C (‖H‖)

C linear equation easy as ∂z f = ew , but no available,....genuinenonlinear proof

Non degeneracy 2

• ∂zφ∞(z) =∞

1c≤ |∂zφa(z)|

|a|≤ c , z ∈ D(0,R), a 6= 0.

Theorem (ACFJ 14)

J(z , f ) > c

C = C (‖H‖)

Non degeneracy 2

• ∂zφ∞(z) =∞

1c≤ |∂zφa(z)|

|a|≤ c , z ∈ D(0,R), a 6= 0.

Theorem (ACFJ 14)

J(z , f ) > c

C = C (‖H‖)

Proof of Non vanishing of Jacobians

Autonomous equation • fh = f (z+h)−f (z)h is quasiregular and never

vanish because f is homeo.•Hurwitz type theorem implies that the limit does not vanish. UseMiniowitz or factorization fh = hh Fh and compactness for Fh +Hurwitz type theorem for hh. General equation.• Perturbation as in the proof of Schauder estimates impliespositivity of J(z , f )• Compactness argument gives the lower bound

final theorems

Theorem

Suppose that H is a regular field. Then FH defines H uniquely,that is, HFH = H.

Theorem

F = φa(z)a∈C non-degenerate family and regular, i.e

‖Dzφa‖C s(DR) ≤ c(s,R) |a|, R <∞, (2)

and DaDzF ∈ (C (z),C (a)) Then there is a unique H = HF ,H(z ,w) ∈ C s

loc(z) , DwH(z ,w) ∈ (C (z),C (w)).

TheoremUnder ellipticity bounds at ∞ FHF = F .

final theorems

Theorem

‖Dzφa‖C s(DR) ≤ c(s,R) |a|, R <∞, (2)

loc(z) , DwH(z ,w) ∈ (C (z),C (w)).

final theorems

Theorem

‖Dzφa‖C s(DR) ≤ c(s,R) |a|, R <∞, (2)

loc(z) , DwH(z ,w) ∈ (C (z),C (w)).

Questions

• What are the advantage of the manifold interpretation? Is there aFrobenius theorem? Can we propagate properties of solutions tothe linear equation to the nonlinear through the exponential map?What is the relation between the curvature of H and that of FH?• How much of the regularity is needed in the last theorem. Inprinciple the exceptional sets depend on a...? Lots of measurabilityproblems....

bullet Higher dimensions?

Non linear elliptic systems and families of quasiregular...

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