Elastic deformations on the plane and approximations (lecture II) - - PowerPoint PPT Presentation

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Elastic deformations on the plane and approximations (lecture II) - - PowerPoint PPT Presentation

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Elastic deformations on the plane and approximations (lecture II) Aldo Pratelli Department of Mathematics, University of Pavia (Italy) Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations: Analysis and Control, Sissa, June

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SLIDE 1

Elastic deformations on the plane and approximations

(lecture II) 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 / 62

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SLIDE 2

Plan of the course

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62

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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 / 62

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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 / 62

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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 / 62

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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 / 62

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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 / 62

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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 / 62

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SLIDE 9

The problem of approximating

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 10

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 11

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence uε : Ω → R2 made by good functions with d(u, uε) ≤ ε.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 12

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence uε : Ω → R2 made by good functions with d(u, uε) ≤ ε.

  • What does good mean?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 13

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence uε : Ω → R2 made by good functions with d(u, uε) ≤ ε.

  • What does good mean?
  • What is d(·, ·)?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 14

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence uε : Ω → R2 made by good functions with d(u, uε) ≤ ε.

  • What does good mean?
  • What is d(·, ·)?
  • Ah, and of course. . . uε must be orient. pres. homeomorphisms!
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 15

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence uε : Ω → R2 made by good functions with d(u, uε) ≤ ε.

  • What does good mean? (smooth / piecewise affine)
  • What is d(·, ·)?
  • Ah, and of course. . . uε must be orient. pres. homeomorphisms!
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 16

The problem of approximating

Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence uε : Ω → R2 made by good functions with d(u, uε) ≤ ε.

  • What does good mean? (smooth / piecewise affine)
  • What is d(·, ·)?
  • Ah, and of course. . . uε must be orient. pres. homeomorphisms!

BAD NEWS: Convolution does not work! (unless u, u−1 ∈ W 2,∞) (Example by Seregin and Shilkin)

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62

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SLIDE 17

A simple idea to approximate

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 18

A simple idea to approximate

Take a triangulation of Ω.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 19

A simple idea to approximate

Take a triangulation of Ω. Build the affine interpolation.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 20

A simple idea to approximate

Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d∗

L∞?

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 21

A simple idea to approximate

Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d∗

L∞? YES (trivial).

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 22

A simple idea to approximate

Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d∗

L∞? YES (trivial).

Is it an homeomorphism? Or, at least, is it orientation preserving?

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 23

A simple idea to approximate

Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d∗

L∞? YES (trivial).

Is it an homeomorphism? Or, at least, is it orientation preserving? Maybe NOT.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 24

A simple idea to approximate

Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d∗

L∞? YES (trivial).

Is it an homeomorphism? Or, at least, is it orientation preserving? Maybe NOT. BAD NEWS: Even taking “randomly” arbitrarily many points does not work!

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62

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SLIDE 25

Good results

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62

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SLIDE 26

Good results

The strategy of last slide (with a careful choice of points) can be adjusted.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62

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SLIDE 27

Good results

The strategy of last slide (with a careful choice of points) can be adjusted. Positive results by Bing, Connell, Kirby, Moise, counterexample by Donaldson and Sullivan.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62

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SLIDE 28

Good results

The strategy of last slide (with a careful choice of points) can be adjusted. Positive results by Bing, Connell, Kirby, Moise, counterexample by Donaldson and Sullivan. All this works with the distance d(u, v) = d∗

L∞(u, v) =

  • u − v
  • L∞ +
  • u−1 − v−1
  • L∞ .
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62

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SLIDE 29

What would we really like?

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62

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SLIDE 30

What would we really like?

If u is thought as a deformation, then the energy is something of the form

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62

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SLIDE 31

What would we really like?

If u is thought as a deformation, then the energy is something of the form W(u) =

  • Ω

|Du|p + h

  • det Du
  • ,

with h diverging both at 0 and +∞.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62

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SLIDE 32

What would we really like?

If u is thought as a deformation, then the energy is something of the form W(u) =

  • Ω

|Du|p + h

  • det Du
  • ,

with h diverging both at 0 and +∞.

  • Why exploding at 0?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62

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SLIDE 33

What would we really like?

If u is thought as a deformation, then the energy is something of the form W(u) =

  • Ω

|Du|p + h

  • det Du
  • ,

with h diverging both at 0 and +∞.

  • Why exploding at 0?
  • Why the determinant?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62

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SLIDE 34

What would we really like?

If u is thought as a deformation, then the energy is something of the form W(u) =

  • Ω

|Du|p + h

  • det Du
  • ,

with h diverging both at 0 and +∞.

  • Why exploding at 0?
  • Why the determinant?

So our dream result is to take u, u−1 ∈ W 1,p, and approximate with d = d∗

W 1,p.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62

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SLIDE 35

The results by Bellido and Mora-Corral

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 62

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SLIDE 36

The results by Bellido and Mora-Corral

Theorem (Mora-Corral): It is possible to approximate u which is smooth

  • ut of a point.
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 62

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SLIDE 37

The results by Bellido and Mora-Corral

Theorem (Mora-Corral): It is possible to approximate u which is smooth

  • ut of a point.

(not trivial at all!!!)

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 62

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SLIDE 38

The results by Bellido and Mora-Corral

Theorem (Mora-Corral): It is possible to approximate u which is smooth

  • ut of a point.

(not trivial at all!!!) Theorem (Bellido, Mora-Corral): If u, u−1 ∈ C 0,α, then it is possible to approximate with d = dC 0,β (but not d = d∗

C 0,β).

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 62

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SLIDE 39

The results by Bellido and Mora-Corral

Theorem (Mora-Corral): It is possible to approximate u which is smooth

  • ut of a point.

(not trivial at all!!!) Theorem (Bellido, Mora-Corral): If u, u−1 ∈ C 0,α, then it is possible to approximate with d = dC 0,β (but not d = d∗

C 0,β). (finally some

derivatives!!!)

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 62

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SLIDE 40

The result by Iwaniec, Kovalev, Onninen

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 62

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SLIDE 41

The result by Iwaniec, Kovalev, Onninen

Theorem (Iwaniec, Kovalev, Onninen): If u ∈ W 1,p (1 < p < ∞), then it is possible to approximate with d = dW 1,p (but not d = d∗

W 1,p).

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 62

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SLIDE 42

The result by Iwaniec, Kovalev, Onninen

Theorem (Iwaniec, Kovalev, Onninen): If u ∈ W 1,p (1 < p < ∞), then it is possible to approximate with d = dW 1,p (but not d = d∗

W 1,p).

  • Technique.
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 62

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SLIDE 43

The result by Iwaniec, Kovalev, Onninen

Theorem (Iwaniec, Kovalev, Onninen): If u ∈ W 1,p (1 < p < ∞), then it is possible to approximate with d = dW 1,p (but not d = d∗

W 1,p).

  • Technique.
  • Why doesn’t it work for the inverse?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 62

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SLIDE 44

The new results

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 62

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SLIDE 45

The new results

Theorem (Mora-Corral, P.): Let d = dW 1,p (or d = d∗

W 1,p). Then,

approximation with with piecewise affine ⇐ ⇒ with smooth.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 62

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SLIDE 46

The new results

Theorem (Mora-Corral, P.): Let d = dW 1,p (or d = d∗

W 1,p). Then,

approximation with with piecewise affine ⇐ ⇒ with smooth. Theorem (Daneri, P.): Let u be bi-Lipschitz. Then, one has approximation with d = d∗

W 1,p for all 1 ≤ p < ∞.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 62

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SLIDE 47

The new results

Theorem (Mora-Corral, P.): Let d = dW 1,p (or d = d∗

W 1,p). Then,

approximation with with piecewise affine ⇐ ⇒ with smooth. Theorem (Daneri, P.): Let u be bi-Lipschitz. Then, one has approximation with d = d∗

W 1,p for all 1 ≤ p < ∞.

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

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 62