Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
3.5 Systems of partial differential equations
The results of sections 3.2-3.4 are valid for the Cauchy problem for principally diagonal systems
Au=fin, (A=A(d)+A1)u=g0, ∂νu=g1on (3.5.1)
whereA(d) is a diagonal second-order matrix partial differential operator with C1()-coefficients andA1is a matrix first-order operator withL∞()-coefficients.
First we observe that Theorem 3.2.1implies
3.5. Systems of partial differential equations 81 Corollary 3.5.1. Let a functionψ be pseudo-convex with respect to all scalar operators formingA(d).
Then there are constants C1(σ)and C2such that forσ >C2,τ >C1(σ)and all vector-functionsu∈H(1)()
e2τϕ(τ3|u|2+τ|u|2)≤C2(
e2τϕ|Au|2+
∂e2τϕ(τ3|u|2+τ|u|2)). Using Corollary 3.5.1 in the proof of Theorem 3.2.2 instead of Theorem 3.2.1 one obtains
Theorem 3.5.2. Let a function ψ be pseudo-convex with respect to all scalar operators forming the principal part ofA. Letψ <0on∂\.
Then there are constants C, κdepending on, ,A, ϕ, εsuch that a solution uto the Cauchy problem(3.5.1)satisfies the bound
u(1)(ε)≤C(F+ u1−κ(1) (0)Fκ) (3.5.2)
where F = f(0)(0)+ u(0)()+ ∇u(0)()).
This result gives better stability estimate compared with Theorem 5.3.2 gaining 0.5 in indices of boundary norms.
Theorem 3.5.3. LetAbe a principally diagonal t-hyperbolic system with time independent coefficients. Let the surfacebe non-characteristic with respect to all scalar operators forming the principal part ofA.
Then there is an open subset(0)ofwhose closure containssuch that a solution to the Cauchy problem(3.5.1)is unique in(0).
Complete proofs of Theorems 3.5.2, 3.5.3 are given in the paper of Eller, Isakov, Nakamura, and Tataru [EINT]. It was also observed in this paper that Theorems 3.5.2, 3.5.3 imply uniqueness results in the lateral Cauchy problem for the classical isotropic Maxwell’s and elasticity systems.
First we consider the dynamical Maxwell’s system
∂t(εE)=curlH+j,
∂t(àH)=curlE, di v(εE)=4πρ, di v(àH)=0 (3.5.3)
for the electric and magnetic fields E=(E1,E2,E3),H=(H1,H2,H3) in the medium with electric permittivity and magnetic permeabilityε, à∈C2(), the density of electrical currentj(x,t), and the electrical charge densityρ(x,t). Here is a bounded domain inR4.
Differentiating the first equation (3.5.3) with respect tot, applying the curl to the second equation and using it to replace∂tcurlHin the first equation we obtain
∂t2(εE)+1/à(curlcurlE+∂tàcurlH+∂t(∇àìH))=∂tj.
Similarly,
∂t2(àH)+1/ε(curlcurlH−∂tεcurlE−∂t(∇εìE)+curlj)=0.
Using thatcurl curl= −+ ∇di vand utilizing the last two equations in (3.5.3) to substitute
di vE=1/ε(4πρ− ∇εãE), di vH= −1/à(∇àãH),
we conclude that the Maxwell’s system (3.5.3) implies the following 6×6 prin- cipally diagonal system
εà∂t2E−E+AE,1=à∂tj− ∇((4πρ)/ε), εà∂t2H−H+AH,1= −curlj, (3.5.4)
where we introduced the matrix operators of first order
AE,1(E,H)=2à∂tε∂tE+à∂t2εE+∂t(∇àìH)+∂tàcurlH− ∇(1/ε∇εãE), AH,1(E,H)=2ε∂tà∂tH+ε∂t2àH− ∇(à∇àãH)−∂tεcurlE−∂t(∇εìE).
Specifying Theorems 3.5.2, 3.5.3 to the system (3.5.4) we obtain two uniqueness of the continuation results for the Cauchy problem for the Maxwell’s system (3.5.3).
Corollary 3.5.4. Let a function ψ be pseudo-convex with respect to the wave operatorεà∂t2−in. Letψ <0on∂\whereis a C2- (hyper)surface inR4.
Then there are constants C, κ ∈(0,1)depending on, ε, à, ψ, δsuch that for any solution(E,H)to the Maxwell’s system (3.5.3) we have
E(1)(δ)+ H(1)(δ)≤C(F+M1−κFκ) (3.5.5)
whereF= E(1)()+ H(1)()+ j(1)()+ ρ(1)() and M= E(1)()+
H(1)().
Observe that to get the Cauchy data on, one has to useE,Hand the equations (3.5.3) to obtain normal components of∇E,∇Hon.
Corollary 3.5.5. Let the coefficientsε, àbe time independent. Let a surfacebe non-characteristic with respect to the wave operatorεà∂t2−,j=0, ρ=0in , andE=H=0on.
Then there is a neighborhood V ofsuch thatE=H=0in∩V .
Exercise 3.5.6. By using first two equations (3.5.3) to substitutecurlH, ∂tHonto the first equation (3.5.4) show that any solution to (3.5.3) satisfies
εà∂t2E−E+2à∂tε∂tE+à∂t2εE− ∇(1/ε∇εãE)+∂tà∂t(εE) +1/à∇àìcurlE+1/(2à)∂t(∇à2)ìH)
=∂t(àj)− ∇((4πρ)/ε)
3.5. Systems of partial differential equations 83 In particular, whenàdoes not depend on time the equations forEdo not involve H, i.e. the system is uncoupled.
A similar equation holds forH.
Now we will consider the dynamical isotropic elasticity system ρ∂t2u−àu−à∇di vu− ∇(λdi vu)−3
j=1
∇àã(∇uj+∂ju)ej=F (3.5.6)
for the displacement vectoru=(u1,u3,u3) of an elastic mediumof density ρ∈C1() with the Lame parametersλ, à∈C2(). For results on initial boundary value problems for the system (3.5.6) we refer to Ciarlet [Ci].
To principally diagonalize the system (3.5.6) so we introduce the functions di vu=v, curlu=w.
(3.5.7)
Dividing the equations (3.5.5) byρand applying the operatordi vto the both parts of the resulting equality we obtain
∂t2v−(λ+2à)/ρv− ∇(à/ρ)ãu− ∇(à/ρ)ã ∇v−2/ρ∇λã ∇v−λ/ρv−
∇1/ρã ∇(λv)− 3
j=1
∂j(∇à)/ρã(∇uj+∂ju)− ∇à/ρã(∇v+u)=di v(F/ρ) and using the known identity= ∇di v−curlcurlwe obtain the equation
∂t2v−(λ+2à)/ρv+A1;4U=di v(F/ρ) (3.5.8)
whereU=(u,v,w) and
A4;1U= −2(∇(à/ρ)+(∇(λ+à))/ρ)ã ∇v− ∇(1/ρ)ã(λv) +(∇(à/ρ)+ ∇à/ρ)ãcurlw
−3
j=1
∂j(∇à/ρ)ã(∇uj+∂ju)−λ/ρv.
Similarly, applying thecurl and using thatcurl(fu)= f curlu+ ∇f ×uafter relatively lenghty but standard computations we obtain
∂t2w−à/ρw+A5;1U=curl(F/ρ), (3.5.9)
where
A5;1= −2∇(à/ρ)ì ∇v+ ∇(à/ρ)ìcurlw− ∇(1/ρ)ì ∇(λv)
−(∂2(∇à/ρ)ã(∇u3+∂3u)−∂3(∇à/ρ)ã(∇u2+∂2u)− ∇à/ρã ∇w1)e1
−(∂3(∇à/ρ)ã(∇u1+∂1u)−∂1(∇à/ρ)ã(∇u3+∂3u)− ∇à/ρã ∇w2)e2
−(∂1(∇à/ρ)ã(∇u2+∂2u)−∂2(∇à/ρ)ã(∇u1+∂1u)− ∇à/ρã ∇w3)e3.
So we have for U=(u1,u2,u3,v,w1,w2,w3) the principally diagonal system (3.5.5), (3.5.8), (3.5.9) where the diagonal entries of the diagonal of the principal part are the isotropic wave operators∂t2−à/ρexcept for the equation (3.5.7) where the principal part is the wave operator∂t2−(λ+2à)/ρ.
The principal diagonalization (3.5.6), (3.5.8), (3.5.9) of the isotropic elasticity system and Theorem 3.5.2 imply
Corollary 3.5.7. Let a function ψ be pseudo-convex with respect to the wave operatorsρ/à∂t2−, ρ/(λ+2à)∂t2−in. Letψ <0on∂\.
Then there are constants C, κ∈(0,1), depending onδsuch that for any solution uto the elasicity system (3.5.6)
u(1)(δ)≤C(F+ u(2)(0)1−κFκ) (3.5.10)
where F= f(0)(0)+ u(2)()+ ∂νu(1)() Similarly, Theorem 3.5.3 implies
Corollary 3.5.8. Let the coefficientsρ, à, λbe time independent. Let a surface be non-characteristic with respect to the wave operatorsρ/à∂t2−, ρ/(λ+ 2à)∂t2−.
Then a solution to the Cauchy problem for the elasticity system with the Cauchy data onis unique innear.
This sharp uniqueness of the continuation result was used by McLaughlin and Yoon [McLY] to get first uniqueness result forà, λin the so-called elastic sonog- raphy, when one recovers elastic parameters from a complete knowledge of the displacementuinsideQ.
Recently, Imanuvilov, Isakov and Yamamoto [IIY] obtained a most natural Car- leman estimate for the elasticity system on compactly supported functions.
Theorem 3.5.9. Let the functionψ∈C3()be pseudo-convex with respect to the wave operatorsρ/à∂t2−, ρ/(λ+2à)∂t2−in.
Then there are constants C1(σ),C2such that for C2< σand C1< τ
e2τϕ(τ2|u|2+τ(|divu|2+ |curlu|2))≤C
e2τϕ|Aeu|2 (3.5.11)
for all functionsu∈C20(). HereAeuis the left side in (3.5.6).
By using the same new unknown functionsv,w(3.5.7) and Carleman estimates for the Laplace operator given in section 3.3 one can obtain uniqueness of the continuation results for time independent solutions to the elasticity system (3.5.6) under the same regularity assumptions on its coefficients [AITY].
There are important systems which can not be principally diagonalized but have a special “upper triangular” principal part. As an example we consider the system
3.5. Systems of partial differential equations 85 of thermoelasticity
Aeu+A1;1(u,v)=0,
∂tv− 3 j,k=1
aj k∂j∂kv+A2;1(di vu,curlu,u,v)=0 (3.5.12)
for the displacement vectoruand the temperaturev. HereA1;1is a linear matrix partial differential operator of first order with respect tov an of zero order with respect to u withC1()-coefficients, the symmetric matrix (aj k)∈C1() and is positive in, and A2;1 is a first order linear partial differential operator with L∞()-coefficients. Hereis a bounded domain inR4. Other examples include von Karman system and systems for elastic plates [Is13], [Is14 ].
Theorem 3.5.10. Let a functionψbe pseudo-convex with respect to the wave op- eratorsρ/à∂t2−, ρ/(λ+2à)∂t2−in. Let⊂∂be a C2-(hyper)surface andψ <0on∂\.
Then there are constants C, κ, depending onδsuch that for any solution(u,v) to the thermoelasicity system (3.5.12)
u(1)(δ)+ v(δ)≤C(F+ u(2)(0)1−κFκ)
where F= f(0)(0)+ u(2.5)()+ ∂νu(1.5)()+ v(1.5)()+
∇v(0.5)().
A proof is given in [ElI]. We will outline basic ideas of this proof.
Using (3.5.12), from (3.5.6),(3.5.8), (3.5.9) withF= −A1;1(u,v) we have A(d)U=A1U+A2vin
(3.5.13)
whereA(d) is a diagonal 7×7 matrix linear partial differential operator with diagonal operatorsρ/à∂t2−orρ/(λ+2à)∂t2−,U=(u,di vu,curlu), and A1,A2are linear partial differential operators of first and second order withL∞()- coefficients.
Let us introduce a cut-off functionχ ∈C∞(R4),χ=1 on2ε,χ=0 on\ ε. DefiningU0 =χU,v0=χvand using the Leibniz’ formula we derive from (3.5.12), (3.5.13) that
A(d)U0=χA2v0+A2;1(U,v)
∂tv0−
Aj k∂j∂kv0=A3;1(U,v) (3.5.14)
where A2;1,A3;1 are (matrix) linear partial differential operators with L∞()- coefficients. Moreover, A2;1 is of second order but it does not involve ∂t2v and second order derivatives ofU, andA3;1is of first order and does not involve∂tv.
Using the Carleman estimate of Theorem 3.4.3 for each of the seven first equa- tions of (3.5.14) and summing the results over components we yield
(σ τϕ)3−2|α|e2τϕ|∂αU0|2
≤C
e2τϕ(
|α|≤1
|∂αU|2+
|β|≤2,βn+1≤1
|∂βv0|2+
|α|≤1,αn+1=0
|∂αv|2). Similarly, from Theorem 3.3.12 (applied to (στϕ)1/2v0) and from the last equation (3.5.14) we have
σ
(στϕ)4−2|α|e2τϕ|∂αv0|2
≤C
(στϕ)e2τϕ(
|α|≤1
|∂αU|2+
|β|≤1,βn+1=0
|∂βv|2).
We can chooseτlarge and use the choice ofχto eliminate the integrals over2ε in the right sides. Denoting by. . .terms bounded bye4τεand bounding the terms withUin the right side of the previous inequality by preceding inequality we yield
σ
2ε
|α|≤2,αn+1≤1
(στϕ)4−2|α|e2τϕ|∂αv|2≤C
2ε
e2τϕ
|α|≤2,αn+1≤1
|∂αv|2+ ã ã ã Dividing the both parts bye4τεand choosingσandτ large we conclude thatv=0 in2ε. Similarly,U=0 in2εfor any positive ε. This completes the proof of uniqueness of (u,v) in0. Stability estimate can be obtained as in the proof of Theorem 3.2.2.
Uniqueness of the continuation results for the thermoelasticity system are used in control theory. For some generalisations and applications we refer to the paper of Eller, Lasiecka, and Triggiani [ElLT].