Regularity theory for Maxwell’s equations

Giovanni S. Alberti

MaLGa - Machine learning Genova CenterUniversity of Genoa

13 January 2021

Problem formulation

Time-harmonic Maxwell’s equations curlH = −i(ωε+ iσ)E + Je in Ω,curlE = iωµH + Jm in Ω,E × ν = 0 on ∂Ω,

withE,H ∈ H(curl,Ω) = F ∈ L2(Ω;C3) : curlF ∈ L2(Ω;C3).

Main regularity questions:I E,H ∈ H1

I E,H ∈ C0,α

I E,H ∈ Hk , E,H ∈ Ck,α

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References

C. Weber.Regularity theorems for Maxwell’s equations.Math. Methods Appl. Sci., 3(4):523–536, 1981.

Jukka Saranen.On an inequality of Friedrichs.Math. Scand., 51(2):310–322 (1983), 1982.

M. Costabel.A remark on the regularity of solutions of Maxwell’sequations on Lipschitz domains.Mathematical Methods in the Applied Sciences,12(4):365–368, 1990.

Hong-Ming Yin.Regularity of weak solution to Maxwell’s equationsand applications to microwave heating.J. Differential Equations, 200(1):137–161, 2004.

Peter Kuhn and Dirk Pauly.Regularity results for generalized electro-magneticproblems.Analysis (Munich), 30(3):225–252, 2010.

Giovanni S. Alberti and Yves Capdeboscq.Elliptic regularity theory applied to time harmonicanisotropic Maxwell’s equations with less thanLipschitz complex coefficients.SIAM J. Math. Anal., 46(1):998–1016, 2014.

Manas Kar and Mourad Sini.AnHs,p(curl;Ω) estimate for the Maxwell system.Math. Ann., 364(1-2):559–587, 2016.

G. S. Alberti and Y. Capdeboscq.Lectures on elliptic methods for hybrid inverseproblems.Societe Mathematique de France, Paris, 2018.

Giovanni S. Alberti.Holder regularity for Maxwell’s equations underminimal assumptions on the coefficients.Calc. Var. Partial Differential Equations, 57(3), 2018.

Basang Tsering-xiao and Wei Xiang.Regularity of solutions to time-harmonic Maxwell’ssystem with lower than Lipschitz coefficients.Nonlinear Anal., 192:111693, 29, 2020.

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Outline

Interior regularity

Global regularity

What else?

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Interior regularity

We consider the regularity of the solutions

E,H ∈ H(curl,Ω)

to

curlH = −i

γ︷ ︸︸ ︷(ωε+ iσ)E + Je in Ω,

curlE = iωµH + Jm in Ω,

in a compact setK b Ω.

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Warm up

Let’s consider the limit ω → 0:curlE = iωµHcurlH = −i(ωε+ iσ)E + Je

=⇒

curlE = 0curlH = σE + Je

Writing E = ∇qE , this yields the conductivity equation for the electric potential qE

−div(σ∇qE) = div Je

Elliptic regularity:I σ ∈W 1,3 =⇒ qE ∈ H2 =⇒ E ∈ H1

I σ ∈ C0,α =⇒ qE ∈ C1,α =⇒ E ∈ C0,α

I Higher regularityI . . .

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Warm up 2

Let’s study H1 regularity in homogeneous isotropic media:curlH = −iγ0E + Je in Ω,curlE = iωµ0H + Jm in Ω.

with sources

Je, Jm ∈ H(div,Ω) = F ∈ L2(Ω;C3) : divF ∈ L2(Ω;C3).

Key observation: divE = −iγ−1

0 div Je ∈ L2(Ω)curlE = iωµ0H + Jm ∈ L2(Ω)

?=⇒ E ∈ H1

loc(Ω)

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Gaffney-Friedrichs Inequality (without boundary)

TheoremWe have

H(curl,Ω) ∩H(div,Ω) ⊆ H1loc(Ω)

and‖∇F‖L2(K) . ‖ curlF‖L2(Ω) + ‖ divF‖L2(Ω) + ‖F‖L2(Ω).

Proof.I Helmholtz decomposition: F = ∇q + curl Φ with div Φ = 0

I By elliptic regularity applied to

−∆Φ = curl curl Φ = curlF

we obtain Φ ∈ H2loc(Ω), so that

curl Φ ∈ H1loc(Ω).

I By elliptic regularity applied to

−∆q = −div∇q = −divF

we obtain q ∈ H2loc(Ω), so that

∇q ∈ H1loc(Ω).

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Basic assumptions

curlH = −i

γ︷ ︸︸ ︷(ωε+ iσ)E + Je in Ω,

curlE = iωµH + Jm in Ω,

I frequency ω > 0

I the coefficients ε, σ ∈ L∞(Ω;R3×3

)and µ ∈ L∞

(Ω;C3×3

)are elliptic:

Λ−1 |η|2 ≤ ξ · εξ, ξ ∈ R3,

Λ−1 |η|2 ≤ ξ ·(µ+ µT

)ξ, ξ ∈ R3,

I sources Je, Jm ∈ L2(Ω;C3)

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H1 regularity

curlH = −iγE + Je in Ω,curlE = iωµH + Jm in Ω,

TheoremIf ε, σ, µ ∈W 1,3 and Je, Jm ∈ H(div,Ω) then E,H ∈ H1

loc(Ω).

Proof.Assume for simplicity ε, µ ∈W 1,∞.I Helmholtz decomposition: E = ∇qE + curl ΦE , H = ∇qH + curl ΦH

I By elliptic regularity applied to

−∆ΦE = iωµH + Jm−∆ΦH = −iγE + Je

we obtain ΦE ,ΦH ∈ H2loc.

I By elliptic regularity applied to

− div (µ∇qH) = div(µ curl ΦH − iω−1Jm

)∈ L2

−div (γ∇qE) = div (γ curl ΦE + iJe) ∈ L2

we obtain qE , qH ∈ H2loc(Ω).

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C0,α regularity

TheoremIf ε, σ, µ ∈ C0,α and Je, Jm ∈ C0,α with α ∈ (0, 1

2 ], then E,H ∈ C0,αloc (Ω).

Proof.The Helmholtz decomposition E = ∇qE + curl ΦE , H = ∇qH + curl ΦH yields

−∆ΦE = iωµH + Jm − div (µ∇qH) = div(µ curl ΦH − iω−1Jm

)−∆ΦH = −iγE + Je − div (γ∇qE) = div (γ curl ΦE + iJe)

I H2 regularity: ΦE ,ΦH ∈ H2 ⊆W 1,6, so that curl ΦE , curl ΦH ∈ L6

I W 1,p regularity: ∇qE ,∇qH ∈ L6, so that E,H ∈ L6

I W 2,p regularity: ΦE ,ΦH ∈W 2,6, so that curl ΦE , curl ΦH ∈W 1,6 ⊆ C0, 12

I Schauder estimates: ∇qE ,∇qH ∈ C0,α, so that E,H ∈ C0,α

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Higher regularity

Higher regularity resultsfor elliptic equations =⇒ Higher regularity results

for Maxwell’s equations

TheoremIf ε, σ, µ ∈WN,3 and Je, Jm ∈ HN (div,Ω) then E,H ∈ HN

loc(Ω).

TheoremIf ε, σ, µ ∈ CN,α and Je, Jm ∈ CN,α with α ∈ (0, 1

2 ], then E,H ∈ C ,αloc(Ω).

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Outline

Interior regularity

Global regularity

What else?

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Elliptic boundary regularity

Key points:

1. Helmholtz decomposition of E and H :

E = ∇qE + curl ΦE , H = ∇qH + curl ΦH

2. Elliptic regularity applied to:

−∆ΦE = iωµH + Jm − div (µ∇qH) = div(µ curl ΦH − iω−1Jm

)−∆ΦH = −iγE + Je − div (γ∇qE) = div (γ curl ΦE + iJe)

So:I We can use boundary elliptic regularity!I Need boundary conditions for the potentials ΦE , ΦH , qE and qH :

ΦE · ν = 0, ΦH × ν = 0, qE = 0 on ∂Ω

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Boundary conditions for qE

I Elliptic PDE:−div (γ∇qE) = div (γ curl ΦE + iJe) in Ω

I The Helmholtz decomposition gives

qE = 0 on ∂Ω

I Dirichlet problem!

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Boundary conditions for qH

I Elliptic PDE:

−div (µ∇qH) = div(µ curl ΦH − iω−1Jm

)in Ω

I From

0 = div(E × ν) = curlE · ν = iωµH · ν + Jm · ν = iωµ∇qH · ν + iωµ curl ΦH · ν + Jm · ν

we obtain−µ∇qH · ν =

(µ curl ΦH − iω−1Jm

)· ν on ∂Ω

I Neumann problem!

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Boundary conditions for ΦE and ΦH

I 6 PDEs:−∆ΦE = iωµH + Jm, −∆ΦH = −iγE + Je in Ω.

I 3 Boundary conditions:

ΦE · ν = 0, ΦH × ν = 0, on ∂Ω

I What to do?

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The flat case

I Let’s focus on ΦH :

−∆ΦH = −iγE + Je in Ω, ΦH × ν = 0, on ∂Ω.

I Suppose Ω = x3 < 0, so that ν = e3. Thus:

ΦH × ν = 0 =⇒ (ΦH)1 = (ΦH)2 = 0

and

div ΦH = 0 =⇒ ∂1(ΦH)1+∂2(ΦH)2+∂3(ΦH)3 = 0 =⇒ ∂3(ΦH)3 = 0 =⇒ ∂ν(ΦH)3 = 0

I Dirichlet and Neumann problems!

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Flattening out the boundary

I Ω is locally defined byx3 < κ(x1, x2)

I Change of coordinates y = Φ(x):

y1 = x1, y2 = x2, y3 = x3 − κ(x1, x2)

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Piola transformation

I SettingE = (Φ′)−TE, γ = Φ′γ(Φ′)T ,

we have curlE = iωµH + Jm−div(γE) = div(iJe)E × ν = 0

∣∣∣∣∣

curl E = (iωµH + Jm)

− div(γE) = div(iJe)

E × e3 = 0

I Same equations!

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What regularity is needed?

I New PDEs

curl E = (iωµH + Jm) , −div(γE) = div(iJe), E × e3 = 0

with coefficientγ = Φ′γ(Φ′)T

I If ∂Ω is of class C1,1, then Φ′ ∈ C0,1 and– H1 regularity: γ ∈W 1,∞ =⇒ γ ∈W 1,∞

– C0,α regularity: γ ∈ C0,α =⇒ γ ∈ C0,α

I Higher regularity: ∂Ω of class CN,1I Non-smooth domains: many results (Buffa, Costabel, Dauge, Nicaise . . . )

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Outline

Interior regularity

Global regularity

What else?

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Other results

I Regularity only for E or H :

γ ∈ C0,α =⇒ E ∈ C0,α

I W 1,p regularity:µ, γ ∈W 1,p, p > 3 =⇒ E,H ∈W 1,p

I Meyers theorem:

no additional assumptions =⇒ E,H ∈ L2+δ

I Asymptotic expansions in the presence of small inhomogeneities

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Maxwell regularity=

Helmholtz decomposition + elliptic regularity

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