november 9, 2012 lectio praecursoriausers.jyu.fi/~jatujaas/files/lektio_handout.pdf · on linear...
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On Linear and Nonlinear Beltrami Systems
Jarmo Jääskeläinen
University of Helsinki
November 9, 2012
Lectio praecursoria
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Grötzch ProblemHerbert Grötzsch asked in 1928 the most nearly conformal mappingbetween a square and a rectangle.
Conformal mapping preserves angles locally.
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Quasiconformal Mappings
radius of the smallest disk outsideradius of the biggest disk inside
=L(z, r)l(z, r)
A homeomorphism f : C→ C is quasiconformal if for every point z
lim supr→∞
L(z, r)l(z, r)
6 K
one point to one point, mapping and its inverse are continuous (ifpoints are close then the image of the points are close)
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Quasiconformal MappingsInfinitesimal behaviour can be characterized by the derivative.
disks 7→ ellipsoids
major axisminor axis
=|∂z f |+ |∂z̄ f ||∂z f |− |∂z̄ f |
6 K(z) 6 K
Conformal functions (preserve angles locally) map infinitesimallydisks to disks.
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Back to GrötzchHerbert Grötzsch asked in 1928 the most nearly conformal mappingbetween a square and a rectangle.
major axisminor axis
=|∂z f |+ |∂z̄ f ||∂z f |− |∂z̄ f |
6 K(z) 6 K most nearly = smallest K
f (z) =12
(aR
aS+
bR
bS
)z +
12
(aR
aS−
bR
bS
)z̄
This natural generalization of conformal maps is a superficial reason tostudy quasiconformal functions. Quasiconformal mappings arise inmany questions of geometry and analysis (holomorphic dynamics,elliptic PDEs, etc.)
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Beltrami Inequality
major axisminor axis
=|∂z f |+ |∂z̄ f ||∂z f |− |∂z̄ f |
6 K(z) 6 K
|∂z̄f | 6 k|∂zf |, k =K − 1K + 1
< 1
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Beltrami EquationBeltrami inequality
|∂z̄ f | 6 k|∂z f |, k =K − 1K + 1
< 1.
Beltrami equation
∂z̄ f (z) = µ(z)∂z f (z), |µ(z)| 6 k,
named after Eugenio Beltrami(1835–1899); studied differentialgeometry and mathematicalphysics.
Already Carl Friedrich Gauss(1777–1855) used the equationwhen he studied isothermalcoordinates.
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Examples (Quasicircles)Koch snowflake
Douady Rabbit
the rabbit on right: RBerenguelJarmo Jääskeläinen On Linear and Nonlinear Beltrami Systems Nov 9, 2012 7 / 14
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Examples (Radial Stretching)
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Reduced Beltrami EquationBeltrami equation
∂z̄ f (z) = µ(z)∂z f (z), |µ(z)| 6 k.
Reduced Beltrami Equation
|∂z̄ f | = λ(z) Im∂z f (z), |λ(z)| 6 k < 1
Reduced Beltrami inequality
|∂z̄ f | 6 k| Im∂z f |.
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Reduced Beltrami EquationReduced Beltrami Equation
|∂z̄ f | = λ(z) Im∂z f (z), |λ(z)| 6 k < 1
Identity is always a solution,f (z) = z.
The identity is the uniquehomeomorphic solution fixingpoints z = 0 and z = 1,i.e., f (0) = 0 and f (1) = 1.
Im∂z f is non-vanishing almost everywhere.
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Radial Stretching
When rotated by 90◦, the mapping is reduced quasiregular.Jarmo Jääskeläinen On Linear and Nonlinear Beltrami Systems Nov 9, 2012 11 / 14
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Linear FamiliesLinear families of quasiconformal mappings can be studied with thehelp of a reduced Beltrami equation. That is families ofquasiconformal mappings a f + b g, where a and b are real numbers,e.g., f + g, f − g, 0.1f − 6g, etc.
A linear family of quasiconformal mappings has a unique associatedlinear Beltrami equation,
∂z̄ f (z) = µ(z)∂z f (z) + ν(z)∂z f (z).
∂z̄ f (z) = µ(z)∂z f (z) and ∂z̄ f (z) = λ(z) Im∂z f (z) are linear equations,i.e., if f , g are solutions, then a f + b g is a solution.
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Nonlinear Situation
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Uniqueness in Nonlinear Equation
Beltrami Systems = we have uniform ellipticity bound k < 1.
The identity was the unique homeomorphic solution to the reducedBeltrami equation such that 0 7→ 0 and 1 7→ 1. There is a uniquesolution in the case of the Beltrami equation.
The same is true in the nonlinear case under an explicit bound of theellipticity. Namely, if we fix two points, say f (0) = 0 and f (1) = 1, thenthere is a unique solution to nonlinear Beltrami equation ifk(z) < 3 − 2
√2 = 0.17157 . . . when z is near ∞.
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