On an inverse boundary value problem for a nonlinear time-harmonic Maxwell system

Cătălin I. Cârstea

doi:10.1515/jiip-2020-0071

Article

On an inverse boundary value problem for a nonlinear time-harmonic Maxwell system

Cătălin I. Cârstea

Published/Copyright: January 6, 2020

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From the journal Journal of Inverse and Ill-posed Problems Volume 30 Issue 3

Abstract

This paper considers a class of nonlinear time-harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form X ⁢ ( x , | E → ⁢ ( x ) | 2 ) ⁢ E → ⁢ ( x ) , Y ⁢ ( x , | H → ⁢ ( x ) | 2 ) ⁢ H → ⁢ ( x ) such that X ⁢ ( x , s ) , Y ⁢ ( x , s ) are both real analytic in 𝑠. Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions for 𝒳 and 𝒴, it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonlinear terms.

Keywords: Inverse problems; Maxwell system; nonlinear elliptic equations

MSC 2010: 35R30; 35F60

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11931011

Funding statement: The author is supported by NSF of China under grant 11931011.

Appendix A Proof of Proposition 3.2

Here we will sketch the proof of Proposition 3.2. The method given below is due to [26, 27]. We will mostly follow the construction as given in [2], summarizing the results when the argument proceeds identically and giving more detail when not.

Let α = log ⁡ ϵ , β = log ⁡ μ , and denote by I n be the identity matrix in dimension 𝑛. Suppose that

X = ( h H → e E → )

satisfies the equation ( P + V ) ⁢ X = 0 , where

P = 1 i ⁢ ( ∇ ⋅ ∇ - ∇ × ∇ ⋅ ∇ ∇ × ) , V = 1 i ⁢ ( i ⁢ ω ⁢ μ ( ∇ α ) ⋅ i ⁢ ω ⁢ μ ⁢ I 3 ( ∇ ⁡ α ) ( ∇ β ) ⋅ i ⁢ ω ⁢ ϵ ( ∇ ⁡ β ) i ⁢ ω ⁢ ϵ ⁢ I 3 ) .

Observe that if 𝑒 and ℎ vanish, then ( E → , H → ) is a solution of

{ ∇ × E → = i ⁢ ω ⁢ μ ⁢ H → , ∇ × H → = - i ⁢ ω ⁢ ϵ ⁢ E → .

Let

Y = ( μ 1 / 2 ⁢ I 4 ϵ 1 / 2 ⁢ I 4 ) ⁢ X , κ = ω ⁢ μ 1 / 2 ⁢ ϵ 1 / 2 , W = κ ⁢ I 8 + 1 2 ⁢ i ⁢ ( ( ∇ α ) ⋅ ( ∇ ⁡ α ) ( ∇ α ) × ( ∇ β ) ⋅ ( ∇ ⁡ β ) - ( ∇ β ) × ) .

Then

(A.1) ( P + W ) ⁢ Y = 0 .

Note that ( P + W ) ⁢ ( P - W t ) = - △ + Q , where

Q = 1 2 ⁢ ( ( △ ⁢ α ) 2 ⁢ ( ∂ i ⁡ ∂ j ⁡ α ) i ⁢ j - ( △ ⁢ α ) ⁢ I 3 ( △ ⁢ β ) 2 ⁢ ( ∂ i ⁡ ∂ j ⁡ β ) i ⁢ j - ( △ ⁢ β ) ⁢ I 3 ) - ( ( κ 2 - 1 4 ⁢ ( ∇ ⁡ α ⋅ ∇ ⁡ α ) ) ⁢ I 4 - 2 i ( ∇ κ ) ⋅ - 2 ⁢ i ⁢ ( ∇ ⁡ κ ) - 2 i ( ∇ κ ) ⋅ - 2 ⁢ i ⁢ ( ∇ ⁡ κ ) ( κ 2 - 1 4 ⁢ ( ∇ ⁡ β ⋅ ∇ ⁡ β ) ) ⁢ I 4 ) .

If 𝑍 is a solution of

(A.2) ( - △ + Q ) ⁢ Z = 0 ,

then Y = ( P - W t ) ⁢ Z is a solution to (A.1). We would like to construct solutions of (A.2) that are of the form

Z ⁢ ( ζ → , x ) = e i ⁢ ζ → ⋅ x ⁢ ( L ⁢ ( ζ → ) + R ⁢ ( ζ → , x ) ) , ζ → ∈ C 3 ,

where L : C 3 ∖ { 0 } → C 8 will be chosen to be of the form

L = 1 | ζ → | ⁢ ( ζ → ⋅ a → ω ⁢ b → ζ → ⋅ b → ω ⁢ a → ) , a → , b → ∈ C 3 ,

though until Lemma A.2, we may take it to be any vector in C 8 . To do so, first extend the coefficients 𝜖, 𝜇 to R 3 so that ϵ - 1 , μ - 1 ∈ C 0 5 ⁢ ( R 3 ) . Then ω 2 ⁢ I 8 + Q ∈ C 0 3 ⁢ ( R 3 ) . Let ρ > 0 be such that supp ⁡ ( ω 2 ⁢ I 8 + Q ) is contained in the ball of radius 𝜌. We can prove the following.

Lemma A.1

Lemma A.1 (compare to [2, Lemma 8])

There exist a C ⁢ ( ρ ) > 0 such that, for any L ∈ C 8 , ζ → ∈ C 3 with ζ → ⋅ ζ → = ω 2 and

(A.3) | ζ → | > C ⁢ ( ρ ) ⁢ ∥ ω 2 ⁢ I 8 + Q ∥ L ∞ ⁢ ( R 3 ) ,

there exists Z = e i ⁢ ζ → ⋅ x ⁢ ( L + R ) , a solution of (A.2) in R 3 , Z ∈ W 3 , 2 ⁢ ( Ω ) and with

(A.4) ∥ R ∥ W 3 , 2 ⁢ ( Ω ) ≤ 1 | ζ → | ⁢ C ⁢ ( ρ ) ⁢ | L | ⁢ ∥ ω 2 + Q ∥ W 3 , ∞ ⁢ ( R 3 ) .

Proof

We only need to show that such an 𝑅 exists. The equation it needs to satisfy is

( - △ - 2 ⁢ i ⁢ ζ → ⋅ ∇ ) ⁢ R + ( ω 2 ⁢ I 8 + Q ) ⁢ R = - ( ω 2 ⁢ I 8 + Q ) ⁢ L .

We would like to, in a certain sense, invert ( - △ - 2 ⁢ i ⁢ ζ → ⋅ ∇ ) . For some - 1 < δ < 0 , define the spaces

L δ 2 ⁢ ( R 3 ) = { f : ∥ f ∥ L δ 2 = ∥ ( 1 + | x | 2 ) δ / 2 ⁢ f ∥ L 2 ⁢ ( R 3 ) < ∞ } ,

W δ s , 2 ⁢ ( R 3 ) = { f : ∥ f ∥ W δ s , 2 = ∥ ( 1 + | x | 2 ) δ / 2 ⁢ f ∥ W s , 2 ⁢ ( R 3 ) < ∞ } .

There exists (see, for example, [33, Corollary 2.2]) G ζ → : W δ + 1 s , 2 ⁢ ( R 3 ) → W δ s , 2 ⁢ ( R 3 ) such that ( - △ - 2 ⁢ i ⁢ ζ → ⋅ ∇ ) ⁢ G ζ → ⁢ ϕ = f and

∥ G ζ → ⁢ ϕ ∥ W δ s , 2 ⁢ ( R 3 ) ≤ 1 | ζ → | ⁢ C ⁢ ( δ ) ⁢ ∥ f ∥ W δ + 1 s , 2 ⁢ ( R 3 ) .

The equation 𝑅 should satisfy can then be written as ( I 8 + G ζ → ⁢ ( ω 2 ⁢ I 8 + Q ) ) ⁢ R = - G ζ → ⁢ ( ω 2 ⁢ I 8 + Q ) ⁢ L . We can choose the constant C ⁢ ( ρ ) in (A.3) so that

∥ G ζ → ⁢ ( ω 2 ⁢ I 8 + Q ) ⁢ R ∥ W δ 3 , 2 ⁢ ( R 3 ) ≤ 1 2 ⁢ ∥ R ∥ W δ 3 , 2 ⁢ ( R 3 ) ,

in which case there exists a solution R = - ( I 8 + G ζ → ⁢ ( ω 2 ⁢ I 8 + Q ) ) - 1 ⁢ G ζ → ⁢ ( ω 2 ⁢ I 8 + Q ) ⁢ L , and it satisfies estimate (A.4). ∎

The following lemma is a restatement of a result in [2].

Lemma A.2

Lemma A.2 (see [2, Proposition 9])

There exists a constant C ⁢ ( ρ , ∥ ϵ - 1 ∥ W 5 , ∞ ⁢ ( R 3 ) , ∥ μ - 1 ∥ W 5 , ∞ ⁢ ( R 3 ) ) > 0 such that if ζ → ∈ C 3 , ζ → ⋅ ζ → = ω 2 ,

| ζ → | > C ⁢ ( ρ , ∥ ϵ - 1 ∥ W 5 , ∞ ⁢ ( R 3 ) , ∥ μ - 1 ∥ W 5 , ∞ ⁢ ( R 3 ) ) , L = 1 | ζ → | ⁢ ( ζ → ⋅ a → ω ⁢ b → ζ → ⋅ b → ω ⁢ a → ) , a → , b → ∈ C 3 ,

then there exists Z = e i ⁢ ζ → ⋅ x ⁢ ( L + R ) , a solution of (A.2) in R 3 , Z ∈ W 3 , 2 ⁢ ( Ω ) and with

∥ R ∥ W 3 , 2 ⁢ ( Ω ) ≤ 1 | ζ → | ⁢ C ⁢ ( ρ ) ⁢ | L | ⁢ ∥ ω 2 + Q ∥ W 3 , ∞ ⁢ ( R 3 ) .

Additionally, Y = ( P - W t ) ⁢ Z solves ( P + W ) ⁢ Y = 0 and is of the form

Y = ( 0 μ 1 / 2 ⁢ H → 0 ϵ 1 / 2 ⁢ E → ) .

Under the conditions of the previous lemma, we get

E → = e i ⁢ ζ → ⋅ x ⁢ ( ϵ - 1 / 2 ⁢ ζ → ⋅ a → | ζ → | ⁢ ζ → + r → e ) , H → = e i ⁢ ζ → ⋅ x ⁢ ( μ - 1 / 2 ⁢ ζ → ⋅ b → | ζ → | ⁢ ζ → + r → h ) .

For σ e , σ h ∈ { 0 , 1 } , choose

a → = σ e ⁢ ζ → ∗ | ζ → | 2 , b → = σ h ⁢ ζ → ∗ | ζ → | 2 .

Then

E → = e i ⁢ ζ → ⋅ x ⁢ ( σ e ⁢ ϵ - 1 / 2 ⁢ ζ → | ζ → | + r → e ) , H → = e i ⁢ ζ → ⋅ x ⁢ ( σ h ⁢ μ - 1 / 2 ⁢ ζ → | ζ → | + r → h ) ,

and applying the previous lemma and Sobolev embedding, ∥ r → e ∥ L ∞ ⁢ ( Ω ) , ∥ r → h ∥ L ∞ ⁢ ( Ω ) = O ⁢ ( | ζ → | - 1 ) .

Acknowledgements

This work was begun while the author was employed at the Hong Kong University of Science and Technology, Jockey Club Institute for Advanced Study. The author is grateful to Prof. Gunther Uhlmann for proposing this problem and for suggesting improvements to the manuscript.

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Received: 2020-06-20

Revised: 2020-11-05

Accepted: 2020-11-25

Published Online: 2020-01-06

Published in Print: 2022-06-01

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https://doi.org/10.1515/jiip-2020-0071

Keywords for this article

Inverse problems; Maxwell system; nonlinear elliptic equations