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

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

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Elastic deformations on the plane and approximations (lecture I) 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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Elastic deformations on the plane and approximations

(lecture I) 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 / 52

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Plan of the course

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 52

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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 / 52

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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 / 52

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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 / 52

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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 / 52

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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 / 52

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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 / 52

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Jordan curves

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 52

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Jordan curves

A Jordan curve is any continuous map γ : S1 → R2.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 52

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Jordan curves

A Jordan curve is any continuous map γ : S1 → R2. One has the disjoint union R2 = γ(S1) ∪ I ∪ E, with I ⊂⊂ R2 and ∂I = ∂E = γ.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 52

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Jordan curves

A Jordan curve is any continuous map γ : S1 → R2. One has the disjoint union R2 = γ(S1) ∪ I ∪ E, with I ⊂⊂ R2 and ∂I = ∂E = γ. A Jordan curve can be oriented clockwise or counterclockwise.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 52

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Orientation preserving (reversing) homeomorphisms

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 52

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Orientation preserving (reversing) homeomorphisms

Let u : Ω → ∆ an homeomorphism.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 52

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Orientation preserving (reversing) homeomorphisms

Let u : Ω → ∆ an homeomorphism. We say that u is orientation-preserving if the image of a clockwise curve is clockwise, and orientation-reversing otherwise.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 52

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Orientation preserving (reversing) homeomorphisms

Let u : Ω → ∆ an homeomorphism. We say that u is orientation-preserving if the image of a clockwise curve is clockwise, and orientation-reversing otherwise.

  • This is well-defined as soon as Ω is simply connected.
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 52

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Orientation preserving (reversing) homeomorphisms

Let u : Ω → ∆ an homeomorphism. We say that u is orientation-preserving if the image of a clockwise curve is clockwise, and orientation-reversing otherwise.

  • This is well-defined as soon as Ω is simply connected.

Important consequence: to determine whether u is orientation preserving, it is enough to check u|∂Ω.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 52

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What about Du?

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 52

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What about Du?

If u is smooth enough, then O.P. should mean that det Du(x) > 0 ∀ x .

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 52

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What about Du?

If u is smooth enough, then O.P. should mean that det Du(x) > 0 ∀ x . Ok, but how much is “smooth enough”?

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 52

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Mappings of finite distorsion (1/2)

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 52

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Mappings of finite distorsion (1/2)

If u : Rn ⊇ Ω → Rn admits a differential at x and det Du(x) > 0, then we call distorsion of u at x the number Ku(x) =

  • Du(x)
  • n

det Du(x) .

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 52

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Mappings of finite distorsion (1/2)

If u : Rn ⊇ Ω → Rn admits a differential at x and det Du(x) > 0, then we call distorsion of u at x the number Ku(x) =

  • Du(x)
  • n

det Du(x) . Idea: for an ellipsis of axes a and b, the distorsion is a/b.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 52

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SLIDE 24

Mappings of finite distorsion (1/2)

If u : Rn ⊇ Ω → Rn admits a differential at x and det Du(x) > 0, then we call distorsion of u at x the number Ku(x) =

  • Du(x)
  • n

det Du(x) . Idea: for an ellipsis of axes a and b, the distorsion is a/b. Fact: it is always Ku(x) ≥ 2 (in dimension 2).

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 52

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SLIDE 25

Mappings of finite distorsion (1/2)

If u : Rn ⊇ Ω → Rn admits a differential at x and det Du(x) > 0, then we call distorsion of u at x the number Ku(x) =

  • Du(x)
  • n

det Du(x) . Idea: for an ellipsis of axes a and b, the distorsion is a/b. Fact: it is always Ku(x) ≥ 2 (in dimension 2). Lemma: If both Ku(x) and Ku−1

  • u(x)
  • are defined, then they coincide.
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 52

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Mappings of finite distorsion (1/2)

If u : Rn ⊇ Ω → Rn admits a differential at x and det Du(x) > 0, then we call distorsion of u at x the number Ku(x) =

  • Du(x)
  • n

nn/2det Du(x) . Idea: for an ellipsis of axes a and b, the distorsion is a/b. Fact: it is always Ku(x) ≥ 2 (in dimension 2). Lemma: If both Ku(x) and Ku−1

  • u(x)
  • are defined, then they coincide.
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 52

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Mappings of finite distorsion (2/2)

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 52

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Mappings of finite distorsion (2/2)

We say that u has finite distorsion if u ∈ W 1,1

loc , det Du ≥ 0 a.e., and

det Du ∈ L1

loc.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 52

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Mappings of finite distorsion (2/2)

We say that u has finite distorsion if u ∈ W 1,1

loc , det Du ≥ 0 a.e., and

det Du ∈ L1

loc.

If K is bounded we say that u has bounded distorsion (or that it is quasiregular).

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 52

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Mappings of finite distorsion (2/2)

We say that u has finite distorsion if u ∈ W 1,1

loc , det Du ≥ 0 a.e., and

det Du ∈ L1

loc.

If K is bounded we say that u has bounded distorsion (or that it is quasiregular). If u is an homeomorphism and u, u−1 have bounded distorsion, we say that u is quasiconformal.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 52

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Mappings of finite distorsion (2/2)

We say that u has finite distorsion if u ∈ W 1,1

loc , det Du ≥ 0 a.e., and

det Du ∈ L1

loc.

If K is bounded we say that u has bounded distorsion (or that it is quasiregular). If u is an homeomorphism and u, u−1 have bounded distorsion, we say that u is quasiconformal. There is a huge bibliography on this (e.g. Ball, Csorney, Hencl, Iwaniec, Koskela, Maly, Sbordone. . . ).

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 7 / 52

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Main result

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 52

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Main result

The main positive result is the following Theorem:

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 52

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Main result

The main positive result is the following Theorem: If u ∈ W 1,1

loc has finite distorsion

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 52

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Main result

The main positive result is the following Theorem: If u ∈ W 1,1

loc has finite distorsion

then u is continuous.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 52

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Main result

The main positive result is the following Theorem: If u ∈ W 1,1

loc has finite distorsion and either

  • u ∈ W 1,n

loc

then u is continuous.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 52

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Main result

The main positive result is the following Theorem: If u ∈ W 1,1

loc has finite distorsion and either

  • u ∈ W 1,n

loc , or

  • eλK ∈ L1

loc,

then u is continuous.

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 8 / 52

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More results

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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More results

  • Definition of monotone functions
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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SLIDE 40

More results

  • Definition of monotone functions
  • The Orlicz space Ln log−1 L
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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More results

  • Definition of monotone functions
  • The Orlicz space Ln log−1 L
  • What about being an homeomorphism?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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More results

  • Definition of monotone functions
  • The Orlicz space Ln log−1 L
  • What about being an homeomorphism?
  • Definition of the Lusin (N) property
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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More results

  • Definition of monotone functions
  • The Orlicz space Ln log−1 L
  • What about being an homeomorphism?
  • Definition of the Lusin (N) property
  • Some good results and a bad example
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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More results

  • Definition of monotone functions
  • The Orlicz space Ln log−1 L
  • What about being an homeomorphism?
  • Definition of the Lusin (N) property
  • Some good results and a bad example
  • For an homeomorphism u, u Sobolev

= ⇒ u−1 Sobolev

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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More results

  • Definition of monotone functions
  • The Orlicz space Ln log−1 L
  • What about being an homeomorphism?
  • Definition of the Lusin (N) property
  • Some good results and a bad example
  • For an homeomorphism u, u Sobolev

= ⇒ u−1 Sobolev

  • Can we say that orientation preserving imply det Du > 0 a.e.?
  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 9 / 52

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Thank you

  • A. Pratelli (Pavia)

Homeomorphisms and approximations SISSA, June 20–24 2011 10 / 52