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Bi-Lipschitz Solutions to the Prescribed Jacobian Inequality in the - - PowerPoint PPT Presentation

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The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs Bi-Lipschitz Solutions to the Prescribed Jacobian Inequality in the Plane and Applications to Nonlinear Elasticity Olivier Kneuss joint work with

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SLIDE 1

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Bi-Lipschitz Solutions to the Prescribed Jacobian Inequality in the Plane and Applications to Nonlinear Elasticity

Olivier Kneuss

joint work with Julian Fischer (MPI Leibzig)

Fields Institute Toronto

30.9.2014

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SLIDE 2

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation

Let Ω ⊂ Rn be a smooth bounded domain, f : Ω → R, n ≥ 2. Can we find a map φ : Ω → Rn satisfying

  • det∇φ = f

in Ω

φ = id

  • n ∂Ω?

(1) Obvious necessary condition:

  • Ω

f = |Ω|.

slide-3
SLIDE 3

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation

Let Ω ⊂ Rn be a smooth bounded domain, f : Ω → R, n ≥ 2. Can we find a map φ : Ω → Rn satisfying

  • det∇φ = f

in Ω

φ = id

  • n ∂Ω?

(1) Obvious necessary condition:

  • Ω

f = |Ω|.

slide-4
SLIDE 4

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation

Let Ω ⊂ Rn be a smooth bounded domain, f : Ω → R, n ≥ 2. Can we find a map φ : Ω → Rn satisfying

  • det∇φ = f

in Ω

φ = id

  • n ∂Ω?

(1) Obvious necessary condition:

  • Ω

f = |Ω|.

slide-5
SLIDE 5

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

Existence results when f is regular enough (Hölder continuous):

  • f ∈ Cr,α(Ω), f > 0, r ≥ 0, 0 < α < 1

⇒ Existence of φ ∈ Cr+1,α(Ω;Ω) satisfying (1): Dacorogna-Moser

’90, also Rivière-Ye ’96 and Carlier-Dacorogna ’13.

  • f ∈ W m,p(Ω), inff > 0, with m ≥ 1 and p > max{1,n/m}

⇒ Existence of φ ∈ W m+1,p(Ω;Ω) satisfying (1): Ye ’94.

  • f ∈ Cr,α(Ω), no sign hypothesis on f, r ≥ 1, 0 ≤ α ≤ 1

⇒ Existence of φ ∈ Cr,α(Ω;Rn) satisfying (1): Cupini-Dacorogna-K

’09.

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SLIDE 6

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

Existence results when f is regular enough (Hölder continuous):

  • f ∈ Cr,α(Ω), f > 0, r ≥ 0, 0 < α < 1

⇒ Existence of φ ∈ Cr+1,α(Ω;Ω) satisfying (1): Dacorogna-Moser

’90, also Rivière-Ye ’96 and Carlier-Dacorogna ’13.

  • f ∈ W m,p(Ω), inff > 0, with m ≥ 1 and p > max{1,n/m}

⇒ Existence of φ ∈ W m+1,p(Ω;Ω) satisfying (1): Ye ’94.

  • f ∈ Cr,α(Ω), no sign hypothesis on f, r ≥ 1, 0 ≤ α ≤ 1

⇒ Existence of φ ∈ Cr,α(Ω;Rn) satisfying (1): Cupini-Dacorogna-K

’09.

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SLIDE 7

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

Existence results when f is regular enough (Hölder continuous):

  • f ∈ Cr,α(Ω), f > 0, r ≥ 0, 0 < α < 1

⇒ Existence of φ ∈ Cr+1,α(Ω;Ω) satisfying (1): Dacorogna-Moser

’90, also Rivière-Ye ’96 and Carlier-Dacorogna ’13.

  • f ∈ W m,p(Ω), inff > 0, with m ≥ 1 and p > max{1,n/m}

⇒ Existence of φ ∈ W m+1,p(Ω;Ω) satisfying (1): Ye ’94.

  • f ∈ Cr,α(Ω), no sign hypothesis on f, r ≥ 1, 0 ≤ α ≤ 1

⇒ Existence of φ ∈ Cr,α(Ω;Rn) satisfying (1): Cupini-Dacorogna-K

’09.

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SLIDE 8

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

Existence results when f is regular enough (Hölder continuous):

  • f ∈ Cr,α(Ω), f > 0, r ≥ 0, 0 < α < 1

⇒ Existence of φ ∈ Cr+1,α(Ω;Ω) satisfying (1): Dacorogna-Moser

’90, also Rivière-Ye ’96 and Carlier-Dacorogna ’13.

  • f ∈ W m,p(Ω), inff > 0, with m ≥ 1 and p > max{1,n/m}

⇒ Existence of φ ∈ W m+1,p(Ω;Ω) satisfying (1): Ye ’94.

  • f ∈ Cr,α(Ω), no sign hypothesis on f, r ≥ 1, 0 ≤ α ≤ 1

⇒ Existence of φ ∈ Cr,α(Ω;Rn) satisfying (1): Cupini-Dacorogna-K

’09.

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SLIDE 9

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

Existence results when f is regular enough (Hölder continuous):

  • f ∈ Cr,α(Ω), f > 0, r ≥ 0, 0 < α < 1

⇒ Existence of φ ∈ Cr+1,α(Ω;Ω) satisfying (1): Dacorogna-Moser

’90, also Rivière-Ye ’96 and Carlier-Dacorogna ’13.

  • f ∈ W m,p(Ω), inff > 0, with m ≥ 1 and p > max{1,n/m}

⇒ Existence of φ ∈ W m+1,p(Ω;Ω) satisfying (1): Ye ’94.

  • f ∈ Cr,α(Ω), no sign hypothesis on f, r ≥ 1, 0 ≤ α ≤ 1

⇒ Existence of φ ∈ Cr,α(Ω;Rn) satisfying (1): Cupini-Dacorogna-K

’09.

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SLIDE 10

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

Existence results when f is regular enough (Hölder continuous):

  • f ∈ Cr,α(Ω), f > 0, r ≥ 0, 0 < α < 1

⇒ Existence of φ ∈ Cr+1,α(Ω;Ω) satisfying (1): Dacorogna-Moser

’90, also Rivière-Ye ’96 and Carlier-Dacorogna ’13.

  • f ∈ W m,p(Ω), inff > 0, with m ≥ 1 and p > max{1,n/m}

⇒ Existence of φ ∈ W m+1,p(Ω;Ω) satisfying (1): Ye ’94.

  • f ∈ Cr,α(Ω), no sign hypothesis on f, r ≥ 1, 0 ≤ α ≤ 1

⇒ Existence of φ ∈ Cr,α(Ω;Rn) satisfying (1): Cupini-Dacorogna-K

’09.

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SLIDE 11

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

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SLIDE 12

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

slide-13
SLIDE 13

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

slide-14
SLIDE 14

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

slide-15
SLIDE 15

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

slide-16
SLIDE 16

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

slide-17
SLIDE 17

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Equation: Exsitence Theory

When f is not regular enough (continuous or less):

  • Non existence result (Burago-Kleiner ’98, McMullen ’98): for all

ε > 0 there exists f ∈ C0(Ω) with f − 1L∞ ≤ ε for which there

exists no Lipschitz solution to (1).

  • Rivière-Ye ’96: f ∈ C0(Ω), f > 0, ⇒ ∃ φ ∈ ∩α<1C0,α(Ω;Ω)

f ∈ L∞(Ω), inff > 0, ⇒ ∃ φ ∈ ∩α<β C0,α(Ω;Ω) for some β ≤ 1 depending on f − 1L∞

  • Monge-Ampère theory: f ∈ C0 ⇒, f > 0 ∃ u ∈ ∩p<∞W 2,p

loc with

det∇2u = f (Caffarelli) f ∈ L∞, inff > 0 ⇒ ∃ u ∈ W 2,1+ε

loc

with det∇2u = f (De-Phillipis-Figalli).

  • Open problem: does there exist a W 1,p solution of (1) for some p

when f is only C0 (and positive)?

slide-18
SLIDE 18

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality

  • det∇φ ≥ f

(a.e.) in Ω

φ = id

  • n ∂Ω.

(2)

  • Natural necessary condition:
  • Ω

f < |Ω|.

  • Note that if
  • Ω f = |Ω| then (2) reduced to (1).
slide-19
SLIDE 19

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality

  • det∇φ ≥ f

(a.e.) in Ω

φ = id

  • n ∂Ω.

(2)

  • Natural necessary condition:
  • Ω

f < |Ω|.

  • Note that if
  • Ω f = |Ω| then (2) reduced to (1).
slide-20
SLIDE 20

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality

  • det∇φ ≥ f

(a.e.) in Ω

φ = id

  • n ∂Ω.

(2)

  • Natural necessary condition:
  • Ω

f < |Ω|.

  • Note that if
  • Ω f = |Ω| then (2) reduced to (1).
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SLIDE 21

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the L∞ case)

Assume

  • Ω ⊂ R2 connected bounded and smooth
  • f ∈ L∞(Ω), f ≥ 0,
  • Ω f < |Ω|.

Then there exists φ : Ω → Ω bi-Lipschitz satisfying (2). Moreover the regularity is sharp in general.

  • The case f ∈ C0 is trivial: by convolution find

f ∈ C∞(Ω) with f ≥ f and

  • Ω

f = |Ω| then apply one of the previous mentioned results.

  • When f ∈ L∞ not longer easy: take f = 2χA where A ⊂ Ω is open

and dense with |A| small enough. Then there is no f continuous with f ≥ f and

  • Ω

f = |Ω| (if it were the case then f ≥ 2 in Ω).

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SLIDE 22

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the L∞ case)

Assume

  • Ω ⊂ R2 connected bounded and smooth
  • f ∈ L∞(Ω), f ≥ 0,
  • Ω f < |Ω|.

Then there exists φ : Ω → Ω bi-Lipschitz satisfying (2). Moreover the regularity is sharp in general.

  • The case f ∈ C0 is trivial: by convolution find

f ∈ C∞(Ω) with f ≥ f and

  • Ω

f = |Ω| then apply one of the previous mentioned results.

  • When f ∈ L∞ not longer easy: take f = 2χA where A ⊂ Ω is open

and dense with |A| small enough. Then there is no f continuous with f ≥ f and

  • Ω

f = |Ω| (if it were the case then f ≥ 2 in Ω).

slide-23
SLIDE 23

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the L∞ case)

Assume

  • Ω ⊂ R2 connected bounded and smooth
  • f ∈ L∞(Ω), f ≥ 0,
  • Ω f < |Ω|.

Then there exists φ : Ω → Ω bi-Lipschitz satisfying (2). Moreover the regularity is sharp in general.

  • The case f ∈ C0 is trivial: by convolution find

f ∈ C∞(Ω) with f ≥ f and

  • Ω

f = |Ω| then apply one of the previous mentioned results.

  • When f ∈ L∞ not longer easy: take f = 2χA where A ⊂ Ω is open

and dense with |A| small enough. Then there is no f continuous with f ≥ f and

  • Ω

f = |Ω| (if it were the case then f ≥ 2 in Ω).

slide-24
SLIDE 24

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the L∞ case)

Assume

  • Ω ⊂ R2 connected bounded and smooth
  • f ∈ L∞(Ω), f ≥ 0,
  • Ω f < |Ω|.

Then there exists φ : Ω → Ω bi-Lipschitz satisfying (2). Moreover the regularity is sharp in general.

  • The case f ∈ C0 is trivial: by convolution find

f ∈ C∞(Ω) with f ≥ f and

  • Ω

f = |Ω| then apply one of the previous mentioned results.

  • When f ∈ L∞ not longer easy: take f = 2χA where A ⊂ Ω is open

and dense with |A| small enough. Then there is no f continuous with f ≥ f and

  • Ω

f = |Ω| (if it were the case then f ≥ 2 in Ω).

slide-25
SLIDE 25

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the L∞ case)

Assume

  • Ω ⊂ R2 connected bounded and smooth
  • f ∈ L∞(Ω), f ≥ 0,
  • Ω f < |Ω|.

Then there exists φ : Ω → Ω bi-Lipschitz satisfying (2). Moreover the regularity is sharp in general.

  • The case f ∈ C0 is trivial: by convolution find

f ∈ C∞(Ω) with f ≥ f and

  • Ω

f = |Ω| then apply one of the previous mentioned results.

  • When f ∈ L∞ not longer easy: take f = 2χA where A ⊂ Ω is open

and dense with |A| small enough. Then there is no f continuous with f ≥ f and

  • Ω

f = |Ω| (if it were the case then f ≥ 2 in Ω).

slide-26
SLIDE 26

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the L∞ case)

Assume

  • Ω ⊂ R2 connected bounded and smooth
  • f ∈ L∞(Ω), f ≥ 0,
  • Ω f < |Ω|.

Then there exists φ : Ω → Ω bi-Lipschitz satisfying (2). Moreover the regularity is sharp in general.

  • The case f ∈ C0 is trivial: by convolution find

f ∈ C∞(Ω) with f ≥ f and

  • Ω

f = |Ω| then apply one of the previous mentioned results.

  • When f ∈ L∞ not longer easy: take f = 2χA where A ⊂ Ω is open

and dense with |A| small enough. Then there is no f continuous with f ≥ f and

  • Ω

f = |Ω| (if it were the case then f ≥ 2 in Ω).

slide-27
SLIDE 27

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the Lp case)

Let Ω ⊂ R2 connected bounded and smooth open set. Then there exists a constant D > 2 with the following property:

  • for every p > 2D
  • for every f ∈ Lp(Ω) with f ≥ 0 and
  • Ω f < |Ω|

there exists a bi-Sobolev map φ with φ,φ −1 ∈ W 1,p/D(Ω;Ω) satisfying (2). Moreover the constant D is computable.

slide-28
SLIDE 28

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the Lp case)

Let Ω ⊂ R2 connected bounded and smooth open set. Then there exists a constant D > 2 with the following property:

  • for every p > 2D
  • for every f ∈ Lp(Ω) with f ≥ 0 and
  • Ω f < |Ω|

there exists a bi-Sobolev map φ with φ,φ −1 ∈ W 1,p/D(Ω;Ω) satisfying (2). Moreover the constant D is computable.

slide-29
SLIDE 29

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the Lp case)

Let Ω ⊂ R2 connected bounded and smooth open set. Then there exists a constant D > 2 with the following property:

  • for every p > 2D
  • for every f ∈ Lp(Ω) with f ≥ 0 and
  • Ω f < |Ω|

there exists a bi-Sobolev map φ with φ,φ −1 ∈ W 1,p/D(Ω;Ω) satisfying (2). Moreover the constant D is computable.

slide-30
SLIDE 30

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

The Prescribed Jacobian Inequality: Existence of Solutions

Theorem (J. Fischer and K. ’14: the Lp case)

Let Ω ⊂ R2 connected bounded and smooth open set. Then there exists a constant D > 2 with the following property:

  • for every p > 2D
  • for every f ∈ Lp(Ω) with f ≥ 0 and
  • Ω f < |Ω|

there exists a bi-Sobolev map φ with φ,φ −1 ∈ W 1,p/D(Ω;Ω) satisfying (2). Moreover the constant D is computable.

slide-31
SLIDE 31

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Functionals in Nonlinear Elasticity

Consider model functionals of the form

F[u] :=

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx where Ω ⊂ R2 smooth and bounded, β > 0 and µ ≥ 0.

  • Classical functionals: µ = 0 (blow up when det∇u = 0)
  • However µ > 0 reasonable: in practice compression beyond a

certain limit (almost) impossible

  • Necessary conditions for minimizers with a Dirichlet condition?
slide-32
SLIDE 32

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Functionals in Nonlinear Elasticity

Consider model functionals of the form

F[u] :=

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx where Ω ⊂ R2 smooth and bounded, β > 0 and µ ≥ 0.

  • Classical functionals: µ = 0 (blow up when det∇u = 0)
  • However µ > 0 reasonable: in practice compression beyond a

certain limit (almost) impossible

  • Necessary conditions for minimizers with a Dirichlet condition?
slide-33
SLIDE 33

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Functionals in Nonlinear Elasticity

Consider model functionals of the form

F[u] :=

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx where Ω ⊂ R2 smooth and bounded, β > 0 and µ ≥ 0.

  • Classical functionals: µ = 0 (blow up when det∇u = 0)
  • However µ > 0 reasonable: in practice compression beyond a

certain limit (almost) impossible

  • Necessary conditions for minimizers with a Dirichlet condition?
slide-34
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The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Functionals in Nonlinear Elasticity

Consider model functionals of the form

F[u] :=

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx where Ω ⊂ R2 smooth and bounded, β > 0 and µ ≥ 0.

  • Classical functionals: µ = 0 (blow up when det∇u = 0)
  • However µ > 0 reasonable: in practice compression beyond a

certain limit (almost) impossible

  • Necessary conditions for minimizers with a Dirichlet condition?
slide-35
SLIDE 35

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ = 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u)β

+

dx

  • Equilibrium equation
  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

1

(det∇u)β

+

divξ(u) dx = 0 for all ξ ∈ C∞

cpt(Ω) (Ball ’76/77)

  • Derivation by ansatz

liminf

ε→0

F[(id+εξ)◦ u]−F[u] ε ≥ 0

slide-36
SLIDE 36

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ = 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u)β

+

dx

  • Equilibrium equation
  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

1

(det∇u)β

+

divξ(u) dx = 0 for all ξ ∈ C∞

cpt(Ω) (Ball ’76/77)

  • Derivation by ansatz

liminf

ε→0

F[(id+εξ)◦ u]−F[u] ε ≥ 0

slide-37
SLIDE 37

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ > 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx

Theorem (J. Fischer and K. ’14)

The Equilibrium equation holds: i.e. for all ξ ∈ C∞

cpt(Ω).

  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

det∇u

(det∇u − µ)β+1

+

divξ(u) dx = 0

  • First difficulty: need to show det∇u ·(det∇u − µ)−β−1

+

∈ L1(Ω)

  • Second difficulty:

F[(id+εξ)◦ u] =

  • Ω ...+

1

  • det(Id+ε∇ξ(u))det∇u − µ

β

+

dx

⇒ regularization required on {det∇u µ}

slide-38
SLIDE 38

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ > 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx

Theorem (J. Fischer and K. ’14)

The Equilibrium equation holds: i.e. for all ξ ∈ C∞

cpt(Ω).

  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

det∇u

(det∇u − µ)β+1

+

divξ(u) dx = 0

  • First difficulty: need to show det∇u ·(det∇u − µ)−β−1

+

∈ L1(Ω)

  • Second difficulty:

F[(id+εξ)◦ u] =

  • Ω ...+

1

  • det(Id+ε∇ξ(u))det∇u − µ

β

+

dx

⇒ regularization required on {det∇u µ}

slide-39
SLIDE 39

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ > 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx

Theorem (J. Fischer and K. ’14)

The Equilibrium equation holds: i.e. for all ξ ∈ C∞

cpt(Ω).

  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

det∇u

(det∇u − µ)β+1

+

divξ(u) dx = 0

  • First difficulty: need to show det∇u ·(det∇u − µ)−β−1

+

∈ L1(Ω)

  • Second difficulty:

F[(id+εξ)◦ u] =

  • Ω ...+

1

  • det(Id+ε∇ξ(u))det∇u − µ

β

+

dx

⇒ regularization required on {det∇u µ}

slide-40
SLIDE 40

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ > 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx

Theorem (J. Fischer and K. ’14)

The Equilibrium equation holds: i.e. for all ξ ∈ C∞

cpt(Ω).

  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

det∇u

(det∇u − µ)β+1

+

divξ(u) dx = 0

  • First difficulty: need to show det∇u ·(det∇u − µ)−β−1

+

∈ L1(Ω)

  • Second difficulty:

F[(id+εξ)◦ u] =

  • Ω ...+

1

  • det(Id+ε∇ξ(u))det∇u − µ

β

+

dx

⇒ regularization required on {det∇u µ}

slide-41
SLIDE 41

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Necessary Conditions for Minimizers: µ > 0

F[u] =

  • Ω |∇u|2 +

1

(det∇u − µ)β

+

dx

Theorem (J. Fischer and K. ’14)

The Equilibrium equation holds: i.e. for all ξ ∈ C∞

cpt(Ω).

  • Ω(2∇ξ(u) ∇u) : ∇u −β ·

det∇u

(det∇u − µ)β+1

+

divξ(u) dx = 0

  • First difficulty: need to show det∇u ·(det∇u − µ)−β−1

+

∈ L1(Ω)

  • Second difficulty:

F[(id+εξ)◦ u] =

  • Ω ...+

1

  • det(Id+ε∇ξ(u))det∇u − µ

β

+

dx

⇒ regularization required on {det∇u µ}

slide-42
SLIDE 42

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Main tool: Bi-Lipschitz Maps Stretching a Measurable Planar Set

Proposition (J. Fischer and K. ’14)

For every τ > 0 and for every measurable set M ⊂ Ω ⊂ R2 (with small enough measure with respect to τ) there exists a bi-Lipschitz map

φ = φτ,M : Ω → Ω preserving the boundary pointwise with

det∇φ ≥ 1+τ a.e. in M, det∇φ ≥ 1− C

  • |M|τ

a.e. in Ω\ M,

||∇φ − Id||Lp(Ω) ≤ C|M|1/(2p)τ

for 1 ≤ p ≤ ∞, where C is a constant depending only on Ω.

slide-43
SLIDE 43

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Main tool: Bi-Lipschitz Maps Stretching a Measurable Planar Set

Proposition (J. Fischer and K. ’14)

For every τ > 0 and for every measurable set M ⊂ Ω ⊂ R2 (with small enough measure with respect to τ) there exists a bi-Lipschitz map

φ = φτ,M : Ω → Ω preserving the boundary pointwise with

det∇φ ≥ 1+τ a.e. in M, det∇φ ≥ 1− C

  • |M|τ

a.e. in Ω\ M,

||∇φ − Id||Lp(Ω) ≤ C|M|1/(2p)τ

for 1 ≤ p ≤ ∞, where C is a constant depending only on Ω.

slide-44
SLIDE 44

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a bi-Lipschitz map φ satisfying (for f ∈ L∞, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω
  • first find

f ∈ C∞(Ω) with

  • Ω

f = |Ω| and |{ f < f}| << 1

  • find ϕ ∈ C∞(Ω;Ω) satisfying
  • det∇ϕ =

f in Ω

ϕ = id

  • n ∂Ω
  • postcompose ϕ by a map streching (by a sufficiently big factor τ)

the set ϕ({ f < f}).

slide-45
SLIDE 45

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a bi-Lipschitz map φ satisfying (for f ∈ L∞, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω
  • first find

f ∈ C∞(Ω) with

  • Ω

f = |Ω| and |{ f < f}| << 1

  • find ϕ ∈ C∞(Ω;Ω) satisfying
  • det∇ϕ =

f in Ω

ϕ = id

  • n ∂Ω
  • postcompose ϕ by a map streching (by a sufficiently big factor τ)

the set ϕ({ f < f}).

slide-46
SLIDE 46

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a bi-Lipschitz map φ satisfying (for f ∈ L∞, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω
  • first find

f ∈ C∞(Ω) with

  • Ω

f = |Ω| and |{ f < f}| << 1

  • find ϕ ∈ C∞(Ω;Ω) satisfying
  • det∇ϕ =

f in Ω

ϕ = id

  • n ∂Ω
  • postcompose ϕ by a map streching (by a sufficiently big factor τ)

the set ϕ({ f < f}).

slide-47
SLIDE 47

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a bi-Lipschitz map φ satisfying (for f ∈ L∞, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω
  • first find

f ∈ C∞(Ω) with

  • Ω

f = |Ω| and |{ f < f}| << 1

  • find ϕ ∈ C∞(Ω;Ω) satisfying
  • det∇ϕ =

f in Ω

ϕ = id

  • n ∂Ω
  • postcompose ϕ by a map streching (by a sufficiently big factor τ)

the set ϕ({ f < f}).

slide-48
SLIDE 48

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a W 1,p/D map φ satisfying (for f ∈ Lp, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω:
  • basic idea: stretch the superlevel sets of f
  • more precisely: by induction construct the map φi stretching the

set φi−1 ◦···◦φ1({f ≥ 2i}) by a factor of 2

  • compose those maps and obtain φ = limi→∞ φi ◦···◦φ1.
slide-49
SLIDE 49

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a W 1,p/D map φ satisfying (for f ∈ Lp, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω:
  • basic idea: stretch the superlevel sets of f
  • more precisely: by induction construct the map φi stretching the

set φi−1 ◦···◦φ1({f ≥ 2i}) by a factor of 2

  • compose those maps and obtain φ = limi→∞ φi ◦···◦φ1.
slide-50
SLIDE 50

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a W 1,p/D map φ satisfying (for f ∈ Lp, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω:
  • basic idea: stretch the superlevel sets of f
  • more precisely: by induction construct the map φi stretching the

set φi−1 ◦···◦φ1({f ≥ 2i}) by a factor of 2

  • compose those maps and obtain φ = limi→∞ φi ◦···◦φ1.
slide-51
SLIDE 51

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Making use of those stretching maps

  • For finding a W 1,p/D map φ satisfying (for f ∈ Lp, f ≥ 0 and
  • Ω f < |Ω|)
  • det∇φ ≥ f

in Ω

φ = id

  • n ∂Ω:
  • basic idea: stretch the superlevel sets of f
  • more precisely: by induction construct the map φi stretching the

set φi−1 ◦···◦φ1({f ≥ 2i}) by a factor of 2

  • compose those maps and obtain φ = limi→∞ φi ◦···◦φ1.
slide-52
SLIDE 52

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-53
SLIDE 53

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-54
SLIDE 54

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-55
SLIDE 55

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-56
SLIDE 56

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-57
SLIDE 57

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-58
SLIDE 58

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-59
SLIDE 59

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

  • Simplification: with no loss of generality we can assume
  • Ω = (0,1)2
  • the set M is compact
  • φ presearves the boundary globally (and not pointwise)
  • make use of Alberti-Csörnyei-Preiss covering: any compact set

M ⊂ (0,1)2 can be covered with by horizontal and vertical 1-Lipschitz strips with:

  • Total area of strips ≤ C
  • |M|
  • number of intersections of horizontal (and respectively vertical)

strips controlled uniformly

  • Stretch the strips by a factor τ (explicit formula).
slide-60
SLIDE 60

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

slide-61
SLIDE 61

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

slide-62
SLIDE 62

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Construction of the Stretching Maps

slide-63
SLIDE 63

The Prescribed Jacobian Inequality Applications to Nonlinear Elasticity Sketch of the Proofs

Thank you for your attention

Literature:

  • J.F

. and Olivier Kneuss, Bi-Lipschitz Solutions to the Prescribed Jacobian Inequality in the Plane and Applications to Nonlinear Elasticity, submitted, 2014