Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
3.3 Elliptic and parabolic equations
We observe that in the situations covered by Theorems 3.2.3, 3.2.5 one can generally expect only stability estimates of logarithmic type, and indeed, for a solutionuto the Cauchy problem (3.2.5) the estimates
u2(0)≤C/|lnF|,
whereC=C(M) and a priori∂αu2()≤ M,|α:m| ≤1, have been obtained by Fritz John [Jo2] in the situation of Corollary 3.2.4. So Theorem 3.2.2 is very important due to the much better H¨older-type stability obtained.
Observe that Metivier [Me] found analytic nonlinear equations such that a non- characteristic Cauchy problem has several smooth solutions. A surface is nonchar- acteristic at a solution of a nonlinear partial differential equation if is is nonchar- acteristic with respect to a linearization of this equation at this solution.
COUNTEREXAMPLE3.2.6 (METIVIER[ME]). For the semilinear equation of a third order
(∂4+∂3)(∂42u+∂12u−∂22u+(∂4u)2+(∂1u)2−(∂2u)2)=0
inR4there are two differentC∞-solutions in the neighborhood of the origin which coincide whenx4<0. Obviously, the surface{x4=0}is noncharacteristic at any smooth solution to this equation.
It is instructive to mention also the simple 2×2 system found in [Me]
∂3u+v∂1u=0,
∂3v−∂2v=0
inR3 with a similar property. Indeed, according to [Me] there are two different C∞-solutions (u,v) to this system in a neighborhood of the origin inR3 which coincide when x3 <0. Again, the surface {x3=0} is noncharacteristic at any smooth solution to this system.
The Cauchy problem for some nonlinear (elliptic) partial differential equations is of applied importance. In particular, we mention the continuation of the wave field beyond caustics [MT].
3.3 Elliptic and parabolic equations
Now we derive from Theorem 3.2.2 and Theorem 3.5.2 (which is obtained in- dependently on section 3.3) particular and more precise results for elliptic and parabolic equations of second order.
We consider the elliptic operatorAu=
aj k∂j∂ku+
bj∂ju+cuwith real- valuedaj k∈C1() andbj,c∈ L∞(). Here the sums are over j,k=1, . . . ,n.
Theorem 3.3.1. For any domainεwithε⊂∪, a solution u to the Cauchy problem(3.2.5)satisfies the following estimate:
u(1)(ε)≤C(F+ u1(1)−κ()Fκ), (3.3.1)
where C andκ ∈(0,1) depend only onε, and F= f2()+ g0(1)()+ g1(0)().
PROOF. Letx ∈∪. Then there is a functionψ∈C2() with nonvanishing gradient insuch that 0< ψ(x) andψ <0 on∂\. This function is pseudo- convex inwith respect toA, hence by Theorem 3.5.2 we have the bound (3.3.1) whereεis replaced by a neighborhoodV(x) ofxin.V(x),x∈ε form an open covering of compactε, so there is a finite subcoveringV1, . . . ,VJ. By the choice,
u(1)(Vj)≤C(F+ u1−κ(1) j()Fκj)≤C(F+ u1−κ(1) ()Fκ)
whereκ=mi n(κ1, . . . , κJ). The last inequality can be easily obatined by con- sidering the casesF ≤ u(1)() andu(1)()≤ F. From the definition of the L2-norm we have
u2(ε)≤ u2(V1)+ ã ã ã + u2(VJ)
and using the above bounds inVjwe complete the proof of (3.3.1).
Corollary 3.3.2. For any domainεwithε⊂∪, a solution u to the Cauchy problem(3.2.5)satisfies the following estimate:
u(1)(ε)≤C(F+ u1−κ2 ()Fκ), (3.3.2)
where C andκ ∈(0,1) depend only onε, and F= f2()+ g0(1)()+ g1(0)().
This corollary follows from Theorem 3.3.1 and known interior Schauder-type estimates for elliptic boundary value problems. Indeed, let0be a subdomain of containingεwith0⊂∪. From known interior Schauder-type estimates (Theorem 4.1)u(1)(0)≤C(F+ u2()). Using this inequality in the bound (3.3.1) withreplaced by0we yield
u(1)(ε)≤C(F+(F+ u2())1−κFκ).
Now Corollary 3.3.2 follows by applying the inequality (a+b)1−κ ≤a1−κ + b1−κ, which is valid for all positivea,b.
Under additional regularity assumptions onone can similarly obtain (3.3.2) foru(2)(ε) by using higher order interior Schauder-type estimates and (3.3.1).
In particular, Theorem 3.3.1 implies uniqueness of a solution to the Cauchy problem (3.2.5) for such equations: when F=0, a solution u is zero as well.
Uniqueness is valid even under less restrictive assumptions that the coefficients aj k are Lipschitz (and not necessarily real-valued) (see the book of H¨ormander [H¨o2], section 17.2). An optimal assumption onc(∈Ln/2) is made in the paper of
3.3. Elliptic and parabolic equations 59 Jerison and Kenig [JK], where they considered the stationary Schr¨odinger operator u+cu=0.
These results show that an elliptic equation of second order with Lipschitz prin- cipal coefficients possesses the so-calleduniqueness of the continuationproperty for their solutions: if a solutionuis zero on a subdomain0of a domain, then u =0 on. To deduce it from Theorem 3.3.1, we introduce a ballBwith closure in0. Thenu satisfies the homogeneous equation in the domain \B and has zero Cauchy data on the part∂Bof its boundary, so by Theorem 3.3.1uis zero in
\B.
We mention some important and surprising counterexamples.
COUNTEREXAMPLE3.3.3 (Pli˘s [Pl2]). There is an elliptic equation a11(x3)∂12u+∂22u+∂32u+b1(x)∂1u+b2(x)∂2u+c(x)u =0
with real-valued coefficientsa11∈Cλ(R) for anyλ <1,b1,b2,c∈C(R3), that has aC∞(R3)-solutionu such thatu=0 whenx3≤0, but it is nonzero in any neighborhood of any point of{x3=0}.
There are nonzero solutions of similar equations with compact supports.
Another interesting question is about the minimal size of the setwhere on prescribes the Cauchy data. In the two-dimensional case uniqueness and stability hold for, which is closed and of positive measure on∂∈C1. We will show this by using the harmonic measure and some results of potential theory.
First we consider harmonic functionsu ∈C1(∪). We assume that|∇u|<F onand|∇u|<Mon. We recall that the harmonic measureà(x;) of a closed set⊂∂with respect to a pointx∈is a harmonic function ofxthat can be defined as follows. Let a sequence of functions gk+∈C(∂) be monotonically convergent to the characteristic function χ() of the set,g+k ≥g+k+1, andà+k
are harmonic functions with the boundary Dirichlet data gk+. Then theà+k are monotonically convergent toà. We observe that ifmeas1 >0, thenà(x;)>0 for anyx ∈. This follows from the standard representation of a solutionà+k to the Dirichlet problem via Green’s kernelG(x,y), which is positive and continuous whenx∈,y∈∂. We have
à+k(x)=
∂G(x,y)g+k(y)d(y)≥
G(x,y)d(y)>0 becausegk+≥1 on.
The function s(x)=ln|∇u(x)| is a subharmonic function in . When x∈
∂, we have lim inf(s(y)−(1−à(y)) lnM−à(y) lnF)≤0 as y→x∈∂.
Indeed, whenx∈, the left side is not greater than
lim inf(lnF−(1−à(y) lnM−à(y) lnF)
≤lim inf(1−à(y))(lnF−lnM))≤0,
because à(y)≤1 by the maximum principle, and F≤ M. When x∈∂\, the function à is continuous at x and equal to 0 there, so the left side under
consideration is equal to lim inf(s(y)−lnM)≤0. Since the functions(x)−(1− à(x)) lnM−à(x) lnF is subharmonic and its upper limit on∂is≤0 by the maximum principle, s(x)≤(1−à(x)) lnM−à(x) lnF for all x∈. Taking exponents of both parts, we will have
|∇u(x)| ≤M1−à(x)Fà(x).
Since 0< à(x)<1, we have a conditional stability estimate that implies unique- ness (whenF→0).
For recent results on minimal requirements on the uniqueness setwe refer to the review paper of Aleksandrov, Bourgain, Gieseke, Havin, and Vymenetz [AlBGHV], where they give a particular answer to the question, for whatdo both uniqueness and existence for the Cauchy problem hold. As far as we know, even inR2it is still an unresolved question.
In the three-dimensional case the assumptionmeas2 >0 is not sufficient. Re- cently, Wolff [Wo] (see also the paper of Bourgain and Wolff) [BoW] found a strong counterexample that disproves an old conjecture of Bers and M.A. Lavrentiev that this assumption is sufficient. We will formulate Wolff’s counterexample.
COUNTEREXAMPLE3.3.4. There is a nonzero C1(R3+)-function, u harmonic in the half-space R3+= {x3>0}, and a closed subset ⊂∂R3+ of positive two- dimensional measure such thatu= |∇u| =0 on.
Observe however, that a counterexample withu ∈C2(R3+) is not known.
Returning to equations of second-order in the plane, we observe that any elliptic equation
(a11∂12+2a12∂1∂2+a22∂22+b1∂1+b2∂2+c)u =0 (3.3.3)
with measurable and bounded coefficientsaj k,bj,cin a bounded plane domain can be reduced to the particular case of these equations with c=0 by the substitution u=u+v. Here u+∈ H2,p(),p>1, is a positive solution to the initial equation. The existence ofu+ can be shown for of small volume by using standard elliptic theory. The equation withc=0 can be reduced to an el- liptic system for the vector functionw1=∂1u,w2=∂2u. By the Bers-Nirenberg theory, any solution w=w1+i w2 of such a system admits the representation w(z)=es(z)f(χ(z)), where the functions s, χ are H¨older continuous on , χ is one-to-one there, and f is a complex-analytic function of z=x1+i x2 on χ(). Since the zeros of an analytic function f are isolated, so are the zeros of w = ∇u.
There is another useful concept for the uniqueness of the continuation. A point a∈is called a zero of infinite order of a functionu∈ L1() if for any natural numberNthere is a constantC(N) such that
B(a;r)|u| ≤C(N)rN. Ifu ∈C∞(), this definition is equivalent to the claim that all partial derivatives ofuare zero at a. From the above results it follows that ifais a zero of infinite order foru−u(a) for a solutionuto equation (3.3.3), thenuis constant in. Alessandrini observed
3.3. Elliptic and parabolic equations 61 that the same property holds for solutions to the important elliptic equation
∂1(a11∂1u+a12∂2u)+∂2(a21∂1u+a22∂2u)=0.
Indeed, this equation is the integrability condition for the overdetermined system
∂1v= −a21∂1u−a22∂2u,∂2v=a11∂1u+a12∂2uwith respect tov. So our equa- tion implies the existence of a solutionvto this system. This system with respect tou andv is again elliptic and satisfies all the conditions of the Bers-Nirenberg representation theorem, and we can repeat the above argument. In particular, all zeros of gradients of nonconstant solutions to the equation div(a∇u)=0 are of finite order ifais merely bounded and measurable.
For higher-order equations the situation is quite complicated. In fact, there are very convincing examples of nonuniqueness.
COUNTEREXAMPLE3.3.5 (Pli˘s [Pl1]). There is a fourth-order elliptic equation ((∂12+∂22+∂32)2+x3(∂12+∂22)2−1/2∂12+b1∂1+c)u=0
withC∞(R3) coefficientsb1,cand solutionuwith suppu= {0≤x3}. There are similar equations with compactly supported solutions. In the same paper there are examples of sixth-order elliptic equations with complex-valued smooth co- efficients in the plane that moreover do not have the property of uniqueness of continuation.
On the other hand, in some important cases from elasticity theory one has uniqueness and stability.
Exercise 3.3.6. Let A1,A2 be second-order elliptic operators with C2()- coefficients. Let be a bounded domain inRn, andaC4-smooth part of its boundary∂. Show that anH(4)()-solutionuto the Cauchy problem
A1A2u=0 in, ∂νju=gj on, j=0, . . . ,3, for any subdomainε⊂ε⊂∪satisfies the following estimate:
u(4)(ε)≤C(F+ u1−κ2 ()Fκ),
whereC, κdepend onε, 0< κ <1, andF= g0(4)()+ ã ã ã + g3(1)().
We observe that Theorem 3.2.1 cannot be applied to the operator A= because strong pseudo-convexity condition (3.2.2) is not satisfied for any func- tion φ. Indeed, the equality A(ζ)=0 implies that the left side of (3.2.2) is zero.
More general equations can be considered by using Carleman estimates with a second large parameterσ. In next result we considerψ(x)= |x−b|2and we let ε=∩ {ε < ψ},ϕ =eσψ.
Theorem 3.3.7. For a second order elliptic operator A and a bounded domain there are constants C1=C1(ε),C2=C2(ε, σ)such that
σ
(στϕ)3−2|α|e2τϕ|∂αu|2≤C1
e2τϕ|Au|2 for all u∈C02()and|α| ≤2provided C1< σ,C2< τ.
This result is obtained in [ElI] by careful examining how constants in the proof in [H¨o1] depend onσ. Using Theorem 3.3.6 one can handle arbitrary third order perturbations of the product of two elliptic operators of second order.
Exercise 3.3.8. Let A1,A2 be second-order elliptic operators with C2()- coefficients. LetA3be a third-order linear partial differential operator with bounded and measurable coefficients in. Letbe a bounded domain inRn, andaC4- smooth part of its boundary∂.
Show that anH(4)()-solutionuto the Cauchy problem
A1A2u+A3u =0 in, ∂νju =gjon,j =0, . . . ,3, for any subdomainε⊂ε⊂∪satisfies the following estimate:
u(4)(ε)≤C(F+ u12−κ()Fκ),
whereC, κdepend onε, 0< κ <1, and F = f(0)()+ g0(4)()+ ã ã ã + g3(1)().
To solve Exercise 3.3.8 we recommend first to coverε by a finite number ofC4-diffeomorphic images of the subset∗= {x:|x|<1,xn <−1/2}of the unit ball such that the same diffeomorphic image of∗= {x:|x| =1,xn <0}is contained in. Since the form of the fourth order equation from Corollary 3.3.8 does not change under these diffeomorphisms, it suffices to assume that ε= ∗ and=∗. Then chooseb=(0, . . . ,0,1), apply Theorem 3.3.7 twice (to A1(A2u)) to derive a Carleman estimate forA1A2with the second large parameter σ, and repeat the proof of Theorem 3.2.2 by using this new Carleman estimate.
COUNTEREXAMPLE3.3.9. For fourth order equations prescribing three boundary conditions is not sufficient for uniqueness. By the Cauchy-Kovalevsky theorem for analyticand coefficients there many solutions near. The following example gives global solutions inwhen=∂.
Indeed, letbe the unit ball inR3, letk12,k1 =πbe the first (smallest) Dirichlet eigenvalue andu1be a corresponding eigenfunction
(+k12)u1=0, in, u1=0 on∂.
Observe that we can assumeu1(x)=si n(πr)/r, wherer= |x|. Letk2=2πand u(x)=
K(x−y)u1(y)d y
3.3. Elliptic and parabolic equations 63 where K(x)= −ei k2|x|/(4π|x|) is the radiating fundamental solution to the Helmholtz operator+k22. We have
(+k22)u=χ()u1 inR3.
Then (+k12)(+k22)u =0 in,u=∂νu =∂ν2u=0 on∂, butu is not zero almost everywhere in.
To prove this statement it suffices to show that u=∂νu =∂ν2u=0 on ∂.
Sinceu1=0 on∂, the functionχ()u1∈ H2,p(R3) for anyp>1. From known regularity properties of potentials,u∈C2(R3). Hence to complete the proof it is sufficient to show thatu(x)=0 when|x|>1.
As known from the theory of the Helmholtz equation [CoKr], Theorem 2.10, K(x−y)=
cn,m(x)jn(k2|y|)Ynm(y/|y|)
where the sum is overn =0,1,2, . . . ,m= −n, . . . ,n, the jn is the spherical Bessel function andYnm are standard spherical harmonics, and the series is uni- formly convergent onwhen|x|>1. Combining this series representation with the integral formula foru, we complete the proof if we show that
u1(y)jn(2π|y|)Ynm(y/|y|)d y=0
for all n=0,1,2, . . . ,m= −n, . . . ,n. Since u1,jn(2π|y|) do not depend on spherical angles, by using polar coordinates this equality follows from basic or- thogonality property of spherical harmonics when n=1,2, . . .. When n=0, Y00 is constant and by using again polar coordinates and the formulas u1(y)= si n(πr)/r,j0(r)=si nr/r, the needed equality is reduced to
1
0
si n(πr)si n(2πr)dr =0 which follows by elementary integration.
In the remaining part of this section we consider the second-order parabolic op- eratorAu=∂tu+Au, whereAis the second-order elliptic operator considered above with coefficientsaj k∈C1();bj,c∈L∞(). We lett =xn+1and choose m=(2, . . . ,2,1).
Theorem 3.3.10. Let=G×I and=γ×I , where G is a domain inRn, γ is a C2-smooth part of its boundary, and I =(0,T).
Then for any domain with closure in ∪, a solution u to the Cauchy problem(3.2.5)satisfies the following estimate:
∂αu2(ε)≤C(F+ u1−κ2 ()Fκ) when|α:m| ≤1, (3.3.4)
where C andκ ∈(0,1)depend onε, and F is given in Theorem3.3.1.
PROOF. We first consider , which is the half-ball {|x|<1,xn <0} in Rn+1. We will make use of the weight function ϕ(x)=ex p(−σxn). Then
ζ =(ξ1, . . . , ξn−1, ξn−iτσexp(−σxn), ξn+1). The equality Am(x;ζ)=0 is equivalent to the equalities
aj kξjξk=annτ2σ2ex p(−2σxn), ξn+1−2
aj nξjτσexp(−σxn)=0.
(3.3.5)
The left side of (3.2.1) is σ2exp(−σxn)
4
aj nξj
2
+4(ann)2τ2σ2exp(−2σxn)
+2/τ
(∂kajl)akqζjζlζq.
Standard calculations show that the last sum consists of termsσbξjξlex p(−2σxn), σ3bτ2ex p(−3σxn) wherebare bounded functions. The first equality (3.3.5) and ellipticity of A imply that|ξ| ≤Cτσex p(−σxn). Observing powers ofσ and choosing σ large we achieve positivity of the left side of (3.2.2). We can as- sume thatis{|x|<1,xn <−ε}. Now from Theorem 3.2.2 we have the bound (3.3.4) withMinstead ofu2(). To obtain (3.3.4) from this bound we can use the known Schauder-type estimates for second-order parabolic boundary value prob- lems [LSU]∂αu2(ε/2)≤C(F+ u2(ε/4)) when|α:m| ≤1 and argue as in the proof of Corollary 3.3.2.
By using the substitution y1=x1, . . . ,yn−1=xn−1,yn=xn+ g(x1, . . . ,xn−1,xn+1), yn+1 =xn+1 making parallel to the t-axis, one re- duces any cylindrical domain B−×I (B is the unit half-ball in Rn) to the half-ball inRn+1. Observe that parabolicity of the equation is preserved under this substitution. To reduceGtoB−, one can argue now as in the elliptic case.
The proof is complete.
As in the elliptic case, a minor reduction of the regularity assumptions on the coefficients is possible. In particular, we mention the paper of Knabner and Ves- sella [KnV], in which they consider the one-dimensional case (n=1) and obtain stability estimates assuming only thata11, ∂1a11are continuous. Similar results are unknown in higher dimensions.
Also, as for elliptic equations, we can claim that under these regularity assump- tions on the coefficients we have the following lateral uniqueness continuation property: ifu=0 onG0×Ifor some nonempty open subsetG0ofG, thenu=0 on.
An interesting consequence of Theorem 3.3.10 and of t-analyticity of solu- tions of parabolic boundary value problems witht-independent coefficients is the following result generalizing the backward uniqueness property.
Corollary 3.3.11 (Local backward uniqueness). Let u be a solution to the evo- lution equation(3.1.1), where A is an elliptic partial differential operator of second order in G (with t-independent coefficients) satisfying the regularity assumptions of Theorem 3.3.10. The domain of A is H(1)0 (G)∩H(2)(G).