[PPT] - Elastic deformations on the plane and approximations (lecture VVI) PowerPoint Presentation

SLIDE 1

Elastic deformations on the plane and approximations

(lecture V–VI) Aldo Pratelli Department of Mathematics, University of Pavia (Italy) “Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations: Analysis and Control”, Sissa, June 20–24 2011

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 1 / 6

SLIDE 2

Plan of the course

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 3

Plan of the course

Lecture I: Mappings of finite distorsion and orientation-preserving

homeomorphisms.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 4

Plan of the course

Lecture I: Mappings of finite distorsion and orientation-preserving

homeomorphisms.

Lecture II: Approximation questions: hystory, strategies and results.
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 5

Plan of the course

Lecture I: Mappings of finite distorsion and orientation-preserving

homeomorphisms.

Lecture II: Approximation questions: hystory, strategies and results.
Lecture III: Smooth approximation of (countably) piecewise affine

homeomorphisms.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 6

Plan of the course

Lecture I: Mappings of finite distorsion and orientation-preserving

homeomorphisms.

Lecture II: Approximation questions: hystory, strategies and results.
Lecture III: Smooth approximation of (countably) piecewise affine

homeomorphisms.

Lecture IV: The approximation result.
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 7

Plan of the course

Lecture I: Mappings of finite distorsion and orientation-preserving

homeomorphisms.

Lecture II: Approximation questions: hystory, strategies and results.
Lecture III: Smooth approximation of (countably) piecewise affine

homeomorphisms.

Lecture IV: The approximation result.
Lecture V: Bi-Lipschits extension Theorem (part 1).
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 8

Plan of the course

Lecture I: Mappings of finite distorsion and orientation-preserving

homeomorphisms.

Lecture II: Approximation questions: hystory, strategies and results.
Lecture III: Smooth approximation of (countably) piecewise affine

homeomorphisms.

Lecture IV: The approximation result.
Lecture V: Bi-Lipschits extension Theorem (part 1).
Lecture VI: Bi-Lipschits extension Theorem (part 2).
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 6

SLIDE 9

The bi-Lipschitz extension theorem

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 6

SLIDE 10

The bi-Lipschitz extension theorem

Theorem (Daneri, P.): Let u : ∂D → R2 be piecewise affine and L bi-Lipschitz. Then there exists an extension of u which is CL4 bi-Lipschitz.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 6

SLIDE 11

The bi-Lipschitz extension theorem

Theorem (Daneri, P.): Let u : ∂D → R2 be piecewise affine and L bi-Lipschitz. Then there exists an extension of u which is CL4 bi-Lipschitz.

In particular, there is such a u finitely piecewise affine.
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 6

SLIDE 12

The bi-Lipschitz extension theorem

Theorem (Daneri, P.): Let u : ∂D → R2 be piecewise affine and L bi-Lipschitz. Then there exists an extension of u which is CL4 bi-Lipschitz.

In particular, there is such a u finitely piecewise affine.
You may prefer to have a smooth CL28/3 bi-Lipschitz extension.
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 6

SLIDE 13

The bi-Lipschitz extension theorem

Theorem (Daneri, P.): Let u : ∂D → R2 be piecewise affine and L bi-Lipschitz. Then there exists an extension of u which is CL4 bi-Lipschitz.

In particular, there is such a u finitely piecewise affine.
You may prefer to have a smooth CL28/3 bi-Lipschitz extension.
If u is generic, then there is again a CL4 bi-Lipschitz extension.
A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 6

SLIDE 14

The proof of the result (1/2)

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 6

SLIDE 15

The proof of the result (1/2)

Step I: Selecting the central ball.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 6

SLIDE 16

The proof of the result (1/2)

Step I: Selecting the central ball. Step II: Definition and properties of the primary sectors.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 6

SLIDE 17

The proof of the result (1/2)

Step I: Selecting the central ball. Step II: Definition and properties of the primary sectors. Step III: How to partition a sector in ordered triangles.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 6

SLIDE 18

The proof of the result (1/2)

Step I: Selecting the central ball. Step II: Definition and properties of the primary sectors. Step III: How to partition a sector in ordered triangles. Step IV: Definition of the good paths.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 6

SLIDE 19

The proof of the result (1/2)

Step I: Selecting the central ball. Step II: Definition and properties of the primary sectors. Step III: How to partition a sector in ordered triangles. Step IV: Definition of the good paths. Step V: Estimate on the length of the good paths.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 6

SLIDE 20

The proof of the result (2/2)

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 6

SLIDE 21

The proof of the result (2/2)

Step VI: Definition of the speed function.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 6

SLIDE 22

The proof of the result (2/2)

Step VI: Definition of the speed function. Step VII: The bi-Lipschitz extension on each primary sector.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 6

SLIDE 23

The proof of the result (2/2)

Step VI: Definition of the speed function. Step VII: The bi-Lipschitz extension on each primary sector. Step VIII: The bi-Lipschitz extension in the internal polygon.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 6

SLIDE 24

The proof of the result (2/2)

Step VI: Definition of the speed function. Step VII: The bi-Lipschitz extension on each primary sector. Step VIII: The bi-Lipschitz extension in the internal polygon. Step IX: The smooth extension.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 6

SLIDE 25

The proof of the result (2/2)

Step VI: Definition of the speed function. Step VII: The bi-Lipschitz extension on each primary sector. Step VIII: The bi-Lipschitz extension in the internal polygon. Step IX: The smooth extension. Step X: The non piecewise affine case.

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 6

SLIDE 26

Thank you

A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 6