Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
3.2 General Carleman estimates and the Cauchy problem
One-dimensional case was considered in some detail by Watanabe [W].
3.2 General Carleman estimates and the Cauchy problem
Let m be a multi-index with positive integer components mj satisfying the following condition: m1= ã ã ã =mq >mq+1 ≥. . . . We define ∇q as (∂1, . . . , ∂q,0, . . . ,0).
We consider the differential operator A(x;∂)=
aα∂α (the sum is over|α:m| ≤1),
where|α:m| =α1/m1+ ã ã ã +αn/mn. We define itsm-principal partAmas the sum of the same terms with|α:m| =1. We introduce them-principal symbol of this operator
A(x;ζ)=
aαi|α|ζα, |α:m| =1.
We will assume that theaαare inL∞() and the coefficients of them-principal part are inC1(). In sections 3.2–3.5,is a bounded domain inRn.
Let a functionϕbe inC2() and∇qϕ =0 on. We introduce the exponential weight functionw(x)=exp(τϕ(x)).
We remind that by C we denote generic constants depending only on A,A, , , ϕ. Any additional dependence will be specified in parentheses.
Theorem 3.2.1. Suppose that either (a) Am(x;ξ)=0for allξ ∈Rn\{0}or (b) the coefficients of Amare real-valued.
If the conditions
Am(x;ζ)=0, ζ =ξ+iτ∇qϕ, τ =0 (3.2.1)
imply that
j,k≤q
(∂j∂kϕ(∂Am/∂ζj)(∂Am/∂ζk)+τ−1I∂kAm(∂Am/∂ζk))> δ (3.2.2)
infor some positive numberδ, then there is a constant C such that τ
|∂αu|2w2d x≤C
|Au|2w2d x, |α:m|<1 (3.2.3)
when C < τ for all functions u in C∞0 ().
The conditions of this theorem on the function ϕ are called strong pseudo- convexity conditions. When A is the Laplace operator (A(ζ)=ζ12+ ã ã ã +ζn2), then these conditions are certainly satisfied for strictly convex functionsϕ, but not only for them. When Ais the wave operator (A(ζ)=ζ12+ ã ã ã +ζn2−ζn2+1), this
convexity concept is adjusted to space-time geometry, which is discussed in more detail in Section 3.4.
A proof of Theorem 3.2.1 is quite long and technical. General ideas are to use differential quadratic forms and to utilize a energy integrals methods of Friedrichs, Leray, and Lewy from the theory of hyperbolic partial differential equations of higher order. In the isotropic case (m1=. . .mn), a proof is given by H¨ormander in his book [H¨o1], section 8.4, and in the general case by Isakov in the paper [Is12].
Further development of Carleman estimates in the isotropic case can be found in the paper of Nirenberg [Ni]. More recently, Tataru [Tat3] included boundary terms in the isotropic case. There are several applications of Carleman estimates, for example to solvability of the equationAu= f for any (regular)f, to uniqueness of propagation of singularities, and to exact and approximate boundary controllability for partial differential equations. In this section we will derive from them a theorem about uniqueness and stability in the general Cauchy problem. These estimates were introduced in 1938 by Carleman [Ca] exactly for this purpose (in a particular case of first order systems in the plane with simple characteristics).
In case of second order operators the conditions of Theorem 3.2.1 can be relaxed.
Theorem 3.2.1. Let us assume that A is a partial differential operator of second order with real-valued principal coefficients (m=(2, . . . ,2)).
Let a functionψ∈C2()and the conditions (3.2.1) Am(x;ξ)=0,
(∂Am/∂ξj)∂jψ=0, ξ =0, imply that
(∂j∂kψ)(∂Am/∂ξj)(∂Am/∂ξk)
(3.2.2) +
((∂k∂Am/∂ξj)∂Am/∂ξk−∂kAm∂2Am/∂ξj∂ξk)∂jψ >0 in. Moreover, let us assume that
A(x;∇ψ(x))=0,x ∈, and let us introduce
ϕ=eσψ. (3.2.4)
Then there are constants C1(σ),C2such that (3.2.3) τ3−2|α|
|∂αu|2w2≤C
|Au|2w2+
∂(τ|∇u|2+τ3|u|2)w2
when C2< σ,C1< τ,|α| ≤1,for all functions u∈ H(2)().
The conditions of Theorem 3.2.1on functionψare called the pseudo-convexity conditions.
We will show that for second order operatorsA=
aj k∂j∂kpseudo-convexity of ψ implies strong pseudo-convexity ofϕ given by (3.2.4) for largeσ. Then Theorem 3.2.1follows from results of Tataru [Tat3].
3.2. General Carleman estimates and the Cauchy problem 53 Indeed, from (3.2.4) we have
∂jϕ=σ ϕ∂jψ, ∂j∂kϕ=σϕ(∂j∂kψ+σ ∂jψ∂kψ).
From these formulae by standard calculations the left side in (3.2.2), m= (2, . . . ,2),q =n, is
4
σ ϕ(∂j∂kψ+σ∂jψ∂kψ)ajlζlakpζp+2/τI
∂kajlζjζlakpζp
=4σ ϕ
(∂j∂kψ+σ∂jψ∂kψ)aj kakp(ξlξp+τ2σ2ϕ2∂lψ∂pψ) +2/τI
∂kajlakp(ξj+iτσ ϕ∂jψ)(ξl+iτσ ϕ∂lψ)(ξp−iτσ ϕ∂pψ)
=2σ ϕ
2
(∂j∂kψ+σ ∂jψ∂kψ)ajlakp(ξlξp+τ2σ2ϕ2∂lψ∂pψ)
+
∂kajlakp(−∂pψξjξk+2∂lψξjξp+τ2σ2ϕ2∂jψ∂lψ∂pψ)
. We will denote the last expression byH.
To achieve positivity ofHwe can use homogeneity with respect (ξ, τσ ϕ) and assume that|ξ|2+τ2σ2ϕ2=1. First we considerτ =0. Passing to the limit in (3.2.1) as τ goes to zero we obtain the equalities (3.2.1). Moreover, standard calculations show thatHbecomes the sum of the left side in (3.2.2) and of
2σ2ϕ
∂jψ∂kψajlakpξlξp=2σ2ϕ
ajl∂jψξl
2
.
Hence His positive for any x∈andξ =0. By compactness and continuity argumentsC−1σ ϕ <Hwhenστϕ <C−1. Now we considerC−1< στϕ. The sum of the terms ofHcontaining the highest power ofσ is
σ
∂jψ∂kψajlakp(ξlξp+σ2τ2ϕ2∂lψ∂pψ)
=σ
⎛
⎝
jl
∂jψξl
2
+σ2τ2ϕ2
ajl∂jψ∂lψ2
⎞
⎠≥C−1σ3τ2ϕ2,
because by the conditions of Theorem 3.2.1 A(x,∇ψ(x))=0 and because of compactness and continuity reasons. The modulus of the remaining terms inHis bounded byCσ ϕ(|ξ|2+σ2τ2ϕ2). Summing up the above inequalities and using that|ξ|2+σ2τ2ϕ2 =1 we conclude that
H> σϕ(C−1σ3τ2ϕ2−C(|ξ|2+σ2τ2ϕ2))> σϕ(C−3σ−C)>0, providedσ >C4.
We mention that for elliptic, parabolic, and hyperbolic operators of second order there are Carleman estimates in Sobolev spaces of negative orderH(−1)which are useful when handling equations in the divergent form with reduced regularity of solutions [Im], [IIY], [IY2]. In [IIY] these estimates are derived from the Carleman
⌫
⍀⑀
⍀0 ⍀
FIGURE3.1.
estimate (3.2.1) replacinguby a cut-off function multiplied by a negative power of the Laplacian with the parameterτ and using appropriate commutators bounds for pseudo-differential operators.
As a first application of Carleman estimates we obtain uniqueness and stability results for the following Cauchy problem:
Au= f on, ∂νju=gj,j ≤m1−1 on,
∂αu∈L2() when|α:m|<1. (3.2.5)
Here is a part of ∂of the class Cm1 that is open in∂. We define as ∩ {ϕ > ε}. We illustrate our problem in Figure 3.1
Theorem 3.2.2 (Uniqueness and Stability). Let ϕ be a function satisfying the conditions of Theorem3.2.1. Let us assume thatϕ <0on∂\.
Then there are constants C, κ∈(0,1)depending on, , ϕandεsuch that for a solution u to the Cauchy problem(3.2.5)we have
∂αu2(ε)≤C(F+M1−κFκ)when|α:m|<1, (3.2.6)
where F isf2()+
gj(m1−j−1/2)()(the sum is over j ≤m1−1), and M is the sum of∂αu2()overαwith|α:m|<1.
This theorem guarantees uniqueness ofu on0, provided that we are able to find a strongly pseudo-convex functionϕ that agrees withand:ϕ <0 on
∂\.
PROOF. By using extension theorems we can find a functionu∗with the Cauchy data (3.2.5) such that the norms from the left side of (3.2.6) are bounded byC F.
By subtracting this function fromu, we may assume that its Cauchy data onare zero.
Letχbe aC∞function that is 1 onε/2and 0 near∂\, 0≤χ ≤1. Thenv defined asχuis contained in ˚H(m1)(0).
3.2. General Carleman estimates and the Cauchy problem 55 Using Leibniz’s formula for the differentiation of the product we conclude that A(χu)=χAu+Am−1u, where the last operator involves only ∂αu with
|α:m|<1. Observe that the bound (3.2.3) stated for infinitely smooth compactly supported functions by approximation can be extended onto u∈ H˚(m1)(). By applying tovthis estimate, we get
τ
w∂αv22(ε/2)≤C
w f22()+
w∂αu22() .
Sincev=uonε/2, we can replacevbyu, provided that we replacebyε/2
on the left side. By choosingτ >2Cand subtracting from both sides the integrals overε/2multiplied byC, we can shrinkto\ε/2 on the right side. Since ε⊂ε/2, we obtain
w∂αu22(ε)≤C(w f22()+
w∂αu22(\ε/2)).
We have exp(τε)<w on ε,w <exp(τ) where is supϕ over , and 1≤w ≤exp(τε/2) on\ε/2. Replacingwby its minimum on the left side and by its maximum over closures of the integration domains on the right side and dividing both sides by exp(2τε), we get
∂αu2(ε)≤Cexp(τ(−ε))F+Cexp(−τε/2)M. Let us choose
τ =max{(−ε/2)−1ln(M/F),C(ε)},
whereC(ε)>0 is needed to ensure thatτ >2C. Due to this choice, the second term on the right side does not exceed the first one. After substituting the above expressionτ, we obtain (3.2.6) withλ=ε/(2−ε).
The proof is complete.
In many interesting situations the pseudo-convexity condition is not satisfied, but one can still prove uniqueness in the Cauchy problem. A classical example is Holmgren’s theorem.
We recall that a surfacegiven by the equation{γ(x)=0,x∈},γ ∈C1(), is called noncharacteristic with respect to the operatorAif for the principal symbol Am(x;ζ)(m1= ã ã ã =mn) of this operator we have Am(x;∇γ(x))=0 whenx∈ .
Theorem 3.2.3. Let us assume that the coefficients of the operator A are (real) analytic in a neighborhood of.
If a surfaceis noncharacteristic with respect to A, then there is a neighborhood V ofsuch that a solution u to the Cauchy problem(3.2.5) is unique in0= ∩V .
For a proof of this classical result we refer to the book of Fritz John [Jo4]. It is based on solvability of the noncharacteristic Cauchy problem for the adjoint operator in analytic classes of functions, on the density of such functions inCk
andHp,kspaces, and on Green’s formula. An immediate corollary is the following global version of this result.
Letτbe a family of noncharacteristic surfaces in a neighborhood ofgiven by the equations{γ(x,t)=0}, 0≤τ ≤1, whereγ ∈C1(Q) for some open setQin Rn+1containingτ× {τ}. We assume that the boundary ofτdoes not intersect . We introduce1−τ= ∪(σ∩) over 0< σ < τ.
Corollary 3.2.4. Let us assume that A has real-analytic coefficients inand the surfacesτ,0≤τ ≤1, are noncharacteristic with respect to A. Let us assume that theτdo not intersect∂\and that0∩⊂.
Then a solution u to the Cauchy problem(3.2.5)is unique in0.
This result follows from Theorem 3.2.3 by a standard compactness argument.
Indeed, by this theoremu =0 on1−ε(0), and ifu =0 onτ, then it is zero on τ−ε(τ) for some (small) positive ε(τ). The intervals (τ −ε(τ), τ) form an open covering of the compact interval [δ,1−δ] for anyδ < ε(0). There is a finite subcovering (τ1−ε1, τ1), . . . ,(τk−εk, τk) of this interval, and one can assume thatτj−εj < τj+1< τj. Moving from jto j+1, we conclude thatu=0 onδ
for anyδ, and therefore on0.
For a discussion of these classical results and for an explicit construction of the uniqueness domains0we refer to the books of Courant and Hilbert [CouH], pp. 238, and John [Jo4]. We observe that uniqueness domain in Corollary 3.2.4 is sharp, while uniqueness results of Theorem 3.2.2 are typically not sharp.
The assumption of analyticity of the coefficients is too restrictive for many applications, so the following result of Tataru [Tat2] is quite important.
Letx=(x,x) wherex∈Rk,x∈Rn−k. We say that a functionϕ∈C2() is stronglypseudo-convex with respect to the operator A(m=(m1, . . . ,m1)) if
∇ϕ(x)=0 and conditions (3.2.1) are satisfied for anyξ =(ξ,0) at any point x ∈.
Theorem 3.2.5. Let us assume that A is a differential operator with x- independent coefficients. Let ϕ be a strongly pseudo-convex function in , =∩ {ε < ϕ}, and0⊂∪.
Then a solution u to the Cauchy problem (3.2.5) is unique in0.
In fact, Tataru proved a stronger result assuming analytic dependence of the coefficients onx. A crucial idea of his proof is to apply touthe pseudodifferential operator
e(∂)2/τ, (3.2.7)
which is the convolution with the Gaussian kernel inx-variables (τ/(2π))(n−k)/2e−τ|x−y|2/2,
while using Carleman estimates inx. This idea brought weaker results, which appeared earlier in the paper of Robbiano [Ro]. In Section 3.4 we will show that