Global stability Giovanni Alessandrini for coupled physics inverse - - PowerPoint PPT Presentation

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Global stability Giovanni Alessandrini for coupled physics inverse - - PowerPoint PPT Presentation

Aug 13, 2023 298 likes •906 views

STABILITY FOR COUPLED PHYSICS IPs Global stability Giovanni Alessandrini for coupled physics inverse problems. Introduction A case study An example A priori assumptions Main Theorem Giovanni Alessandrini Stability for |u|

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Global stability for coupled physics inverse problems. A case study

Giovanni Alessandrini

Università degli Studi di Trieste

Problémes Inverses et Imagerie 12-13/02/2014 Institut Henri Poincaré

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Since the ’80s, a dominant theme in Inverse Problems has been: To image the interior of an object from measurements taken in its exterior

  • overdetermined boundary data,
  • scattering data.

With coupled physics IPs there is a shift to data associated to interior information.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Since the ’80s, a dominant theme in Inverse Problems has been: To image the interior of an object from measurements taken in its exterior

  • overdetermined boundary data,
  • scattering data.

With coupled physics IPs there is a shift to data associated to interior information.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Interior data may provide much better stability than inverse boundary problems, or inverse scattering problems. Available results require nondegeneracy conditions on the solutions of the involved direct problems:

  • Nonvanishing of solution.
  • Nonvanishing of gradients.
  • Nonvanishing of Jacobians.
  • Nonvanishing of augmented Jacobians.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Interior data may provide much better stability than inverse boundary problems, or inverse scattering problems. Available results require nondegeneracy conditions on the solutions of the involved direct problems:

  • Nonvanishing of solution.
  • Nonvanishing of gradients.
  • Nonvanishing of Jacobians.
  • Nonvanishing of augmented Jacobians.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Interior data may provide much better stability than inverse boundary problems, or inverse scattering problems. Available results require nondegeneracy conditions on the solutions of the involved direct problems:

  • Nonvanishing of solution.
  • Nonvanishing of gradients.
  • Nonvanishing of Jacobians.
  • Nonvanishing of augmented Jacobians.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Interior data may provide much better stability than inverse boundary problems, or inverse scattering problems. Available results require nondegeneracy conditions on the solutions of the involved direct problems:

  • Nonvanishing of solution.
  • Nonvanishing of gradients.
  • Nonvanishing of Jacobians.
  • Nonvanishing of augmented Jacobians.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Introduction

Interior data may provide much better stability than inverse boundary problems, or inverse scattering problems. Available results require nondegeneracy conditions on the solutions of the involved direct problems:

  • Nonvanishing of solution.
  • Nonvanishing of gradients.
  • Nonvanishing of Jacobians.
  • Nonvanishing of augmented Jacobians.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Global stability

  • Question. Is it possible to obtain global stability from

measurements arising from arbitrary (nontrivial) solutions of the direct problem? The model problem A problem arising in microwave imaging coupled with ultrasound, Triki (2010). ∆u + qu = 0 in Ω Find q ≥ constant > 0 given the local energy qu2 and the boundary data u|∂Ω.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Global stability

  • Question. Is it possible to obtain global stability from

measurements arising from arbitrary (nontrivial) solutions of the direct problem? The model problem A problem arising in microwave imaging coupled with ultrasound, Triki (2010). ∆u + qu = 0 in Ω Find q ≥ constant > 0 given the local energy qu2 and the boundary data u|∂Ω.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Global stability

  • Question. Is it possible to obtain global stability from

measurements arising from arbitrary (nontrivial) solutions of the direct problem? The model problem A problem arising in microwave imaging coupled with ultrasound, Triki (2010). ∆u + qu = 0 in Ω Find q ≥ constant > 0 given the local energy qu2 and the boundary data u|∂Ω.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Global stability

  • Question. Is it possible to obtain global stability from

measurements arising from arbitrary (nontrivial) solutions of the direct problem? The model problem A problem arising in microwave imaging coupled with ultrasound, Triki (2010). ∆u + qu = 0 in Ω Find q ≥ constant > 0 given the local energy qu2 and the boundary data u|∂Ω.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The full problem

Ammari, Capdeboscq, De Gournay, Rozanova-Pierrat, Triki (2011): div(a∇u) + k2qu = 0 in Ω Find a, q ≥ constant > 0 given the local energies qu2 , a|∇u|2 (with several u’s and k’s!). Here: u = electric field, q = electric permittivity, a−1 = magnetic permeability.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The full problem

Ammari, Capdeboscq, De Gournay, Rozanova-Pierrat, Triki (2011): div(a∇u) + k2qu = 0 in Ω Find a, q ≥ constant > 0 given the local energies qu2 , a|∇u|2 (with several u’s and k’s!). Here: u = electric field, q = electric permittivity, a−1 = magnetic permeability.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Goals

  • Stability of global type.
  • Measurements for a single (nontrivial) solution u

possibly sign changing.

  • No spectral assumptions on ∆ + q.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Goals

  • Stability of global type.
  • Measurements for a single (nontrivial) solution u

possibly sign changing.

  • No spectral assumptions on ∆ + q.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Goals

  • Stability of global type.
  • Measurements for a single (nontrivial) solution u

possibly sign changing.

  • No spectral assumptions on ∆ + q.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Goals

  • Stability of global type.
  • Measurements for a single (nontrivial) solution u

possibly sign changing.

  • No spectral assumptions on ∆ + q.
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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

An example

In dimension n = 1, fix 0 < r < R and, for every k = 1, 2, . . ., set qk(x) = Ak if |x| < r , 1 if r ≤ |x| ≤ R , where Ak = π 2 + 2kπ 2 r −2 . A solution to uxx + qku = 0 in (−R, R) is uk(x) =

  • 1

√

Ak cos(√Akx)

if |x| < r , − sin(|x| − r) if r ≤ |x| ≤ R .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

An example

In dimension n = 1, fix 0 < r < R and, for every k = 1, 2, . . ., set qk(x) = Ak if |x| < r , 1 if r ≤ |x| ≤ R , where Ak = π 2 + 2kπ 2 r −2 . A solution to uxx + qku = 0 in (−R, R) is uk(x) =

  • 1

√

Ak cos(√Akx)

if |x| < r , − sin(|x| − r) if r ≤ |x| ≤ R .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

An example

we have q2ku2

2k − qku2 k∞ ≤ 2

whereas, for any p , 1 ≤ p ≤ ∞ q2k − qkp → ∞ .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

An example

we have q2ku2

2k − qku2 k∞ ≤ 2

whereas, for any p , 1 ≤ p ≤ ∞ q2k − qkp → ∞ .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

A priori assumptions

Given a bounded domain Ω ⊂ Rn with Lipschitz boundary (in quantitative form!), we consider one solution u ∈ W 1,2(Ω) ∩ C(Ω) to ∆u + qu = 0 in Ω where q ∈ L∞(Ω) is assumed to satisfy 0 < K −1 ≤ q ≤ K a.e. in Ω for a given K ≥ 1.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

A priori assumptions

Energy bound. E > 0 is given such that:

  • Ω

|∇u|2 + u2 ≤ E2 . Nontriviality of the data. H > 0 is given such that:

  • Ω

qu2 ≥ H2 > 0 . A priori data: K, E, H and Ω (diam(Ω), constants of its Lipschitz character). Notation: For every d > 0: Ωd = {x ∈ Ω|dist(x, ∂Ω) > d}.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

A priori assumptions

Energy bound. E > 0 is given such that:

  • Ω

|∇u|2 + u2 ≤ E2 . Nontriviality of the data. H > 0 is given such that:

  • Ω

qu2 ≥ H2 > 0 . A priori data: K, E, H and Ω (diam(Ω), constants of its Lipschitz character). Notation: For every d > 0: Ωd = {x ∈ Ω|dist(x, ∂Ω) > d}.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

A priori assumptions

Energy bound. E > 0 is given such that:

  • Ω

|∇u|2 + u2 ≤ E2 . Nontriviality of the data. H > 0 is given such that:

  • Ω

qu2 ≥ H2 > 0 . A priori data: K, E, H and Ω (diam(Ω), constants of its Lipschitz character). Notation: For every d > 0: Ωd = {x ∈ Ω|dist(x, ∂Ω) > d}.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

A priori assumptions

Energy bound. E > 0 is given such that:

  • Ω

|∇u|2 + u2 ≤ E2 . Nontriviality of the data. H > 0 is given such that:

  • Ω

qu2 ≥ H2 > 0 . A priori data: K, E, H and Ω (diam(Ω), constants of its Lipschitz character). Notation: For every d > 0: Ωd = {x ∈ Ω|dist(x, ∂Ω) > d}.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The main Theorem

  • Theorem. Let q1, q2 and the corresponding solutions u1, u2

satisfy the a priori assumptions and suppose that q1u2

1 − q2u2 2L∞(Ω) ≤ ε ,

(1) for a given ε > 0 , and also |u1| − |u2|L∞(∂Ω) ≤ √ Kε . (2) Then, for every d > 0, there exists η ∈ (0, 1) and C > 0,

  • nly depending on d and on the a priori data such that

q1 − q2L2(Ωd) ≤ C

  • ε1/3 + ε

η , Note: If we know q1 = q2 near ∂Ω, then (1) ⇒ (2).

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The main Theorem

  • Theorem. Let q1, q2 and the corresponding solutions u1, u2

satisfy the a priori assumptions and suppose that q1u2

1 − q2u2 2L∞(Ω) ≤ ε ,

(1) for a given ε > 0 , and also |u1| − |u2|L∞(∂Ω) ≤ √ Kε . (2) Then, for every d > 0, there exists η ∈ (0, 1) and C > 0,

  • nly depending on d and on the a priori data such that

q1 − q2L2(Ωd) ≤ C

  • ε1/3 + ε

η , Note: If we know q1 = q2 near ∂Ω, then (1) ⇒ (2).

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The main Theorem

  • Theorem. Let q1, q2 and the corresponding solutions u1, u2

satisfy the a priori assumptions and suppose that q1u2

1 − q2u2 2L∞(Ω) ≤ ε ,

(1) for a given ε > 0 , and also |u1| − |u2|L∞(∂Ω) ≤ √ Kε . (2) Then, for every d > 0, there exists η ∈ (0, 1) and C > 0,

  • nly depending on d and on the a priori data such that

q1 − q2L2(Ωd) ≤ C

  • ε1/3 + ε

η , Note: If we know q1 = q2 near ∂Ω, then (1) ⇒ (2).

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The main Theorem

  • Theorem. Let q1, q2 and the corresponding solutions u1, u2

satisfy the a priori assumptions and suppose that q1u2

1 − q2u2 2L∞(Ω) ≤ ε ,

(1) for a given ε > 0 , and also |u1| − |u2|L∞(∂Ω) ≤ √ Kε . (2) Then, for every d > 0, there exists η ∈ (0, 1) and C > 0,

  • nly depending on d and on the a priori data such that

q1 − q2L2(Ωd) ≤ C

  • ε1/3 + ε

η , Note: If we know q1 = q2 near ∂Ω, then (1) ⇒ (2).

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Main tools

Theorem A (Stability for |u|) There exists C > 0, only depending on K, E and Ω, such that

  • Ω

||u1| − |u2||3 ≤ Cε . Theorem B (Integrability of |u|−δ) For every d > 0, there exists p > 1, C > 0, only depending on K, E, H and Ω, such that

  • Ωd

|u1|−

2 p−1 ≤ C .

Note: This is a form of quantitative estimate of unique continuation.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Main tools

Theorem A (Stability for |u|) There exists C > 0, only depending on K, E and Ω, such that

  • Ω

||u1| − |u2||3 ≤ Cε . Theorem B (Integrability of |u|−δ) For every d > 0, there exists p > 1, C > 0, only depending on K, E, H and Ω, such that

  • Ωd

|u1|−

2 p−1 ≤ C .

Note: This is a form of quantitative estimate of unique continuation.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Main tools

Theorem A (Stability for |u|) There exists C > 0, only depending on K, E and Ω, such that

  • Ω

||u1| − |u2||3 ≤ Cε . Theorem B (Integrability of |u|−δ) For every d > 0, there exists p > 1, C > 0, only depending on K, E, H and Ω, such that

  • Ωd

|u1|−

2 p−1 ≤ C .

Note: This is a form of quantitative estimate of unique continuation.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Proof of the main theorem

(q1 − q2)u2

1 = q2(u2 2 − u2 1) + (q1u2 1 − q2u2 2) =

= q2(|u2| + |u1|)(|u2| − |u1|) + (q1u2

1 − q2u2 2)

hence

  • Ω

|q1 − q2|u2

1 ≤ K|u1| + |u2|L3/2(Ω)|u1| − |u2|L3(Ω) + |Ω|ε

and, by Theorem A,

  • Ω

|q1 − q2|u2

1 ≤ C(ε1/3 + ε) .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Proof of the main theorem

(q1 − q2)u2

1 = q2(u2 2 − u2 1) + (q1u2 1 − q2u2 2) =

= q2(|u2| + |u1|)(|u2| − |u1|) + (q1u2

1 − q2u2 2)

hence

  • Ω

|q1 − q2|u2

1 ≤ K|u1| + |u2|L3/2(Ω)|u1| − |u2|L3(Ω) + |Ω|ε

and, by Theorem A,

  • Ω

|q1 − q2|u2

1 ≤ C(ε1/3 + ε) .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Proof of the main theorem

Now, by Hölder’s inequality, for any δ > 0

  • Ωd

|q1 − q2|

δ δ+2 ≤

  • Ωd

|u1|−δ

  • 2

δ+2

Ω

|q1 − q2|u2

1

  • δ

δ+2

and choosing δ =

2 p−1, by Theorem B

  • Ωd

|q1 − q2|

δ δ+2 ≤ C

  • Ω

|q1 − q2|u2

1

  • δ

δ+2

. Recalling K −1 ≤ qi ≤ K we arrive at

  • Ωd

|q1 − q2|2 ≤ C

  • ε1/3 + ε
  • δ

δ+2 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Proof of the main theorem

Now, by Hölder’s inequality, for any δ > 0

  • Ωd

|q1 − q2|

δ δ+2 ≤

  • Ωd

|u1|−δ

  • 2

δ+2

Ω

|q1 − q2|u2

1

  • δ

δ+2

and choosing δ =

2 p−1, by Theorem B

  • Ωd

|q1 − q2|

δ δ+2 ≤ C

  • Ω

|q1 − q2|u2

1

  • δ

δ+2

. Recalling K −1 ≤ qi ≤ K we arrive at

  • Ωd

|q1 − q2|2 ≤ C

  • ε1/3 + ε
  • δ

δ+2 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Proof of the main theorem

Now, by Hölder’s inequality, for any δ > 0

  • Ωd

|q1 − q2|

δ δ+2 ≤

  • Ωd

|u1|−δ

  • 2

δ+2

Ω

|q1 − q2|u2

1

  • δ

δ+2

and choosing δ =

2 p−1, by Theorem B

  • Ωd

|q1 − q2|

δ δ+2 ≤ C

  • Ω

|q1 − q2|u2

1

  • δ

δ+2

. Recalling K −1 ≤ qi ≤ K we arrive at

  • Ωd

|q1 − q2|2 ≤ C

  • ε1/3 + ε
  • δ

δ+2 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Stability for |u|

Denote Ni = {ui = 0} , i = 1, 2 . Let Ωj , j = 1, 2, . . . be the connected components of Ω \ (N1 ∪ N2). For each j we may split ∂Ωj = Γ0 ∪ Γ1 ∪ Γ2 , where Γ0 = ∂Ωj ∩ ∂Ω , Γ1 = ∂Ωj ∩ N1 , Γ2 = ∂Ωj ∩ N2 . By assumption, on Γ0 we have ||u1| − |u2|| ≤ √ Kε, while, on Γ1, q2u2

2 ≤ ε and on Γ2, q1u2 1 ≤ ε. Hence, on ∂Ωj we have

||u1| − |u2|| ≤ √ Kε ,

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Stability for |u|

Denote Ni = {ui = 0} , i = 1, 2 . Let Ωj , j = 1, 2, . . . be the connected components of Ω \ (N1 ∪ N2). For each j we may split ∂Ωj = Γ0 ∪ Γ1 ∪ Γ2 , where Γ0 = ∂Ωj ∩ ∂Ω , Γ1 = ∂Ωj ∩ N1 , Γ2 = ∂Ωj ∩ N2 . By assumption, on Γ0 we have ||u1| − |u2|| ≤ √ Kε, while, on Γ1, q2u2

2 ≤ ε and on Γ2, q1u2 1 ≤ ε. Hence, on ∂Ωj we have

||u1| − |u2|| ≤ √ Kε ,

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Stability for |u|

Denote Ni = {ui = 0} , i = 1, 2 . Let Ωj , j = 1, 2, . . . be the connected components of Ω \ (N1 ∪ N2). For each j we may split ∂Ωj = Γ0 ∪ Γ1 ∪ Γ2 , where Γ0 = ∂Ωj ∩ ∂Ω , Γ1 = ∂Ωj ∩ N1 , Γ2 = ∂Ωj ∩ N2 . By assumption, on Γ0 we have ||u1| − |u2|| ≤ √ Kε, while, on Γ1, q2u2

2 ≤ ε and on Γ2, q1u2 1 ≤ ε. Hence, on ∂Ωj we have

||u1| − |u2|| ≤ √ Kε ,

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Stability for |u|

W.l.o.g. we may assume u1, u2 > 0 in Ωj. Set ϕ+ =

  • u1 − u2 − 2

√ Kε + , ϕ− =

  • u2 − u1 − 2

√ Kε + . Note that ϕ± ∈ W 1,2 (Ωj) ∩ C(Ωj) and use ψ±

i

= ϕ±ui as test functions in the weak formulation of ∆ui + qiui = 0. We arrive at

  • Ωj

(|u1| + |u2|)(|u1| − |u2|)2 ≤ C

  • Ωj

(|u1| + |u2|)ε , Adding up w.r.t. j, and using the energy bound,

  • Ω

||u1| − |u2||3 ≤

  • Ω

(|u1| + |u2|)(|u1| − |u2|)2 ≤ Cε ,

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Stability for |u|

W.l.o.g. we may assume u1, u2 > 0 in Ωj. Set ϕ+ =

  • u1 − u2 − 2

√ Kε + , ϕ− =

  • u2 − u1 − 2

√ Kε + . Note that ϕ± ∈ W 1,2 (Ωj) ∩ C(Ωj) and use ψ±

i

= ϕ±ui as test functions in the weak formulation of ∆ui + qiui = 0. We arrive at

  • Ωj

(|u1| + |u2|)(|u1| − |u2|)2 ≤ C

  • Ωj

(|u1| + |u2|)ε , Adding up w.r.t. j, and using the energy bound,

  • Ω

||u1| − |u2||3 ≤

  • Ω

(|u1| + |u2|)(|u1| − |u2|)2 ≤ Cε ,

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Stability for |u|

W.l.o.g. we may assume u1, u2 > 0 in Ωj. Set ϕ+ =

  • u1 − u2 − 2

√ Kε + , ϕ− =

  • u2 − u1 − 2

√ Kε + . Note that ϕ± ∈ W 1,2 (Ωj) ∩ C(Ωj) and use ψ±

i

= ϕ±ui as test functions in the weak formulation of ∆ui + qiui = 0. We arrive at

  • Ωj

(|u1| + |u2|)(|u1| − |u2|)2 ≤ C

  • Ωj

(|u1| + |u2|)ε , Adding up w.r.t. j, and using the energy bound,

  • Ω

||u1| − |u2||3 ≤

  • Ω

(|u1| + |u2|)(|u1| − |u2|)2 ≤ Cε ,

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Integrability of |u|−δ

Lipschitz propagation of smallness (A. and Rosset ’98, A., Rondi, Rosset and Vessella 2009) If

  • Ω |∇u|2 + u2
  • Ω u2

≤ F then for any Bρ(x0) ⊂ Ω we have

  • Bρ(x0)

u2 ≥ C

  • Ω

|∇u|2 + u2 where C > 0 only depends on ρ, K, Ω and on F. Note: Under our a priori assumptions: F = KE2 H2 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Integrability of |u|−δ

Lipschitz propagation of smallness (A. and Rosset ’98, A., Rondi, Rosset and Vessella 2009) If

  • Ω |∇u|2 + u2
  • Ω u2

≤ F then for any Bρ(x0) ⊂ Ω we have

  • Bρ(x0)

u2 ≥ C

  • Ω

|∇u|2 + u2 where C > 0 only depends on ρ, K, Ω and on F. Note: Under our a priori assumptions: F = KE2 H2 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Integrability of |u|−δ

Doubling inequality (Garofalo and Lin ’86) There exists R = R(K) such that if r0 ≤ R and Br0(x0) ⊂ Ω then

  • B2r(x0)

u2 ≤ C

  • Br(x0)

u2 , ∀r < r0 4 where C > 0 only depends on r0 and on the a priori data.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Integrability of |u|−δ

Ap property (Garofalo and Lin ’86) For any d > 0 there exist p > 1, C > 0, only depending on d and on the a priori data, such that for every x0 ∈ Ωd and every r ≤ d/4

  • 1

|Br|

  • Br(x0)

u2 1 |Br|

  • Br(x0)

u−

2 p−1

p−1 ≤ C .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

Quantitative estimates of unique continuation seem to be a necessary ingredient for stability estimates for IP with interior data, when available data depend on few solutions

  • f the direct problem, and few restrictions can be imposed
  • n such solutions.

Open issues may arise in investigating the vanishing rate of gradients or Jacobians.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

Quantitative estimates of unique continuation seem to be a necessary ingredient for stability estimates for IP with interior data, when available data depend on few solutions

  • f the direct problem, and few restrictions can be imposed
  • n such solutions.

Open issues may arise in investigating the vanishing rate of gradients or Jacobians.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

For equations with no zero order term, such as div(σ∇u) = 0 , with λ−1I ≤ σ ≤ λI, and if n ≥ 3, σ ∈ C0,1, estimates on the vanishing rate of |∇u|2 are available. If also a zero order term is present, e.g.: div(σ∇u) + qu = 0 , with |q| ≤ K, estimates on the vanishing rate are known for |∇u|2 + |u|2. The situation is not clear for |∇u|2 alone.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

For equations with no zero order term, such as div(σ∇u) = 0 , with λ−1I ≤ σ ≤ λI, and if n ≥ 3, σ ∈ C0,1, estimates on the vanishing rate of |∇u|2 are available. If also a zero order term is present, e.g.: div(σ∇u) + qu = 0 , with |q| ≤ K, estimates on the vanishing rate are known for |∇u|2 + |u|2. The situation is not clear for |∇u|2 alone.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

For equations with no zero order term, such as div(σ∇u) = 0 , with λ−1I ≤ σ ≤ λI, and if n ≥ 3, σ ∈ C0,1, estimates on the vanishing rate of |∇u|2 are available. If also a zero order term is present, e.g.: div(σ∇u) + qu = 0 , with |q| ≤ K, estimates on the vanishing rate are known for |∇u|2 + |u|2. The situation is not clear for |∇u|2 alone.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

An example

uxx + qu = 0 on (0, +∞) , with q(x) = χ(0, π

2 )(x) ,

and u(x) = sin x if 0 < x ≤ π

2 ,

1 if x ≥ π

2 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

Further problems arise with Jacobians. Example (Laugesen ’96). ∀ε > 0 ∃Φ : ∂B → ∂B homeomorphism, such that |Φ(x) − x| < ε, ∀x ∈ ∂B and the solution U = (u1, u2, u3) to ∆U = 0, in B, U = Φ,

  • n

∂B. is not one-to-one.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

Further problems arise with Jacobians. Example (Laugesen ’96). ∀ε > 0 ∃Φ : ∂B → ∂B homeomorphism, such that |Φ(x) − x| < ε, ∀x ∈ ∂B and the solution U = (u1, u2, u3) to ∆U = 0, in B, U = Φ,

  • n

∂B. is not one-to-one.

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

Concluding remarks

Example (Jin and Kazdan ’91). ∃σ ∈ C∞(R3) , λ−1I ≤ σ ≤ λI and a solution U = (u1, u2, u3) to div(σ∇U) = 0 in R3 , such that detDU = 0, for x3 ≤ 0 , detDU > 0, for x3 > 0 .

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STABILITY FOR COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions Main Theorem Stability for |u| Quantitative UCP Concluding remarks End

The end.

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