Unique continuation with weak type lower order terms : the variable coefficient case

Publicacions Matem`atiques, Vol 39 (1995), 187–200. UNIQUE CONTINUATION WITH WEAK TYPE LOWER ORDER TERMS: THE VARIABLE COEFFICIENT CASE Guozhen Lu1 Abstract This paper deals with the unique continuation problems for variable coefficient elliptic differential equations of second order. We will prove that the unique continuation property holds when the variable coefficients of the leading term are Lipschitz continuous and the coefficients of the lower order terms have small weak type Lorentz norms. This will improve an earlier result of T. Wolff in this direction. 1. Introduction Unique continuation problems for variable coefficient elliptic differential equations of second order have been studied by many authors. We refer the reader to [7], [8] and [9] for the most recent results and many references therein. The following result we prove here is an improvement of a theorem in [9]. Theorem 1. If d≥3, then there is a constant =d,λ >0making the following true. Assume that Ω⊂Rdis a domain and L= d i,j=1 aij(x)∂2 ∂xi∂xjis an elliptic operator with Lipschitz coefficients on Ω,A:Ω→Rand B:Ω→Rare functions such that (1.1)-(1.3) below hold: lim r→0||A||Lp∞(D(a,r)) ≤d (1.1) lim r→0||B||Ld∞(D(a,r)) ≤d (1.2) 1The author is supported in part by NSF Grant #DMS93-15963. AMS classification 35, 42. 188 G. Lu for each aΩ, where p=d 2if d≥5, and p>2if d=3and d=4. Assume also that u∈W22 loc(Ω) and satisfies (1.3) |Lu|≤A|u|+B|u|. Then if uvanishes on an open set it vanishes identically. In the above, “elliptic” means that the matrix aij(x) is real and positive definite for each x∈Ω, i.e., there exists a positive constant λsuch that λ−1|ξ|2≤ d  i,j=1 aij(x)ξiξj≤λ|ξ|2 for each x∈Ω and ξ=(ξj)d j=1 ∈Rd;W22 is the Sobolev space, i.e., functions whose second derivatives are in L2;||A||Lq∞(D(a,r)) is the weak type norm defined as follows: ||A||Lq∞(D(a,r)) = supλ>0λq|{x∈D(a, r):|A(x)|>λ}|)1/q. This theorem is a refinement of the previous result of [9], in which the theorem was shown to be true for A∈Lp loc and B∈Ld loc, and is an extension to the variable coefficient case of the result in [3]. Such type of extension to weak type zero order case was considered earlier in [6], which sharpens the result of [2]. It is known that the Lipschitz condition can not be replaced by any weaker Holder condition (see [4]). For the previous known results for unique continuation in this direction, we refer the reader to [7] and [8], where the unique continuation and strong unique continuation when B= 0 and A∈L d 2 loc for aij(x)∈C∞ were proved. For an alternate way of sharpening the result with Lp potential by using Campanato-Morrey type condition, we refer the reader to [1], [5], and [10]. We will use the covering lemma (Lemma 1’) of [9] and the method of freezing coefficients on appropriate convex and compact sets to prove our theorem. We will adapt the idea of bounding the weak Ldnorm of Bfrom [3]. However, unlike the constant coefficient case (see [3]), we will not be able to control the weak Ldnorm of Balone when we deal with the variable coefficient case because we can not drop the terms containing the weak Lpnorm of Adue to the presence of other quantities (see Section 3 for more details). Thus, in contrast to [3], in order to prove Theorem 1 we shall show either the weak Lp(pis as in Theorem 1) norm of the zero order term Aor the weak Ldnorm of the lower order term Bover the union of some (many enough) disjoint sets would not be too Unique continuation for elliptic equations 189 small if the unique continuation property failed. In any case, it would lead to a contradiction and the theorem follows. Section 2 of this paper is the Carleman inequalities needed for Theorem 1, and Section 3 is the proof of the theorem. One word about the notation: we always assume d≥3 in this paper; constants depend on the dimension dand the elliptic constant λonly unless otherwise specified and may differ from lines to lines; we write xyto mean x≤Cy and x≈yfor xyand yx. Acknowledgement. I am greatly indebted to T. Wolff for his encouragement and helpful discussions. 2. Carleman type of inequalities Hereafter, f∧and fvdenote the Fourier and inverse Fourier transforms respectively. We now define the multiplier Nkby (ek·xu)∧=Nk(ek·x u)∧, where Nk(ξ)= 1 |ξ|2−ik ·ξ−|k|2. Take φ∈C∞ 0(D(0,2)), and φ=1onD(0,1), where D(0,a)={x:|x|< a}. Set φ(k)(x)=φ(|k|x). Then we can define three multiplier operators, T,T1, and T2by  Tf =Nkˆ f,  T1f=Nk(1 −φk)ˆ f,  T2f=(Nk·φk)ˆ f. Thus we have the following lemma Lemma 2.1. Suppose k∈Rd,E ⊂Rdcompact and convex, |E|≥ |k|−d,u∈W22 has compact support, then if i) Suppose either a) d≥5and θ> d−4 d(d−1) ,1 2−1 q=1 t≥2 d;orb) d=3or d=4,0<θ< 1 d+1 and 1 2−1 q=1 t>1 2−θ(d−1) 2; The following holds: ||T2(ek·xu)||q≤C−1 θ|k|d t−2(|k|d|E|)θ||ek·xu||2,E. ii) Suppose either a) d≥5,1 2−1 q=2 dand θ> d−4 d(d−1) ,orb)d=4, 2 d>1 2−1 q>1 2−θ(d−1) 2and 0<θ< 1 d+1 ;orc)d=3,1 2−1 q> 1 2−θ(d−1) 2and 0<θ< 1 d+1 .Ifd=3or 4, assume |k|≥1. Then ||ek·xu||q2≤C−1 θ(|k|d|E|)θ||ek·xu||2,E. 190 G. Lu We note in the above that || · ||qp is the Lorentz norm defined by ||f||qp =(q∞ 0sp−1|{x:|f(x)|>s}|p/qds)1 pand ||f||2,E =||fgE||2 where gEis defined for the compact and convex set Eby gE(x)= min{T≥1:x∈TE}where TE is the expansion of Earound the barycenter of E. Remark. 1) We note the values of θ>θ d= max d−4 d(d−1) ,0in Lemma (2.1) is sharper than those in the similar type of inequalities in [9] (see Lemma 6.4 in [9]). However, it is of no essential use for the present purpose. As long as we can take θ<1 d, it will be good enough in the later argument in Section 3. 2) In the above lemma (ii) if we set 1 2−1 q=1 p, then we can certainly take p=d 2if d≥5; p>2ifd=4(θcan be as small as possible provided that p>2 is very close to 2); and p>2 and any θ>0ifd=3 (actually p= 2 is allowed). In (i) of the lemma, we can take t<pfor the corresponding pwe just mentioned. 3) We note we may assume k=e1in proving lemma (2.1), since inequalities (i) and (ii) a) are scale invariant and in (ii) b) and c), the scaling works for |k|large. Proof of Lemma (2.1): Let N(ξ)= 1 |ξ|2−iξ1−1=Ne1(ξ). Let also N1(ξ)=N(ξ)·(1 −φ)(ξ) and N2(ξ)=N(ξ)φ(ξ), where φ(ξ)=φ(e1)(ξ) as before. Then (2.2) N1(ξ)≤(1 + |ξ|2)−1. It is shown in Lemma 6.3 of [9] that ||(N2ˆ f)v||q1≤C||f||p1 provided (2.3) 1 ≤p1≤s, 1 q1 <1 p 1 +1 2−s 2p 1 , where s=2d+2 d+3 is the Stein-Tomas exponent in the restriction theorem for the Fourier transform, and pdenotes the conjugate of p. By duality, (2.4) ||(N2ˆ f)v||p 1≤C||f||q 1. Unique continuation for elliptic equations 191 We note (2.3) is equivalent to (2.5) 1 q 1 >1 2+1 p 1 2 d−1and 1 ≤p1≤s. We now let p1=2d d+4 (which is less than s) and θ> d−4 d(d−1) when d≥5. Then by (2.5), once we select q1such that 1 p 1 2 d−1<1 q 1 −1 2<θ, for such q1and p1(2.4) holds. Since N2(ξ) has compact support, then for any q≥p 1we have ||(N2ˆ f)v||q≤C||(N2ˆ f)v||p 1≤C||f||q 1 ≤C|E| 1 q 1 −1 2||f||2,E ≤C|E|θ||f||2,E. But 1 2−1 q≥2 dis equivalent to q≥p 1, we are then done for the case d≥5 by scaling. When d=3ord= 4, we let 0 <θ< 1 d+1 and qsatisfy 1 2−1 q= 1 t>1 2−θ(d−1) 2.Thus 1 q<θ(d−1) 2. We can then pick p1such that 1 q≤1 p 1 <θ(d−1) 2because 1 p 1 is increasing as p1does and is at most 1 s=d−1 2d+2 . Consequently we can pick q1such that 1 p 1 2 d−1<1 q 1 −1 2<θ. Since q≥p1and N2has compact support, we will get ||(N2ˆ f)v||q≤C||(N2ˆ f)v||p 1≤C||f||q 1≤C|E|θ||f||2,E. This proves (i) of Lemma (2.1). We now give the proof of (ii) of lemma (2.1). By (2.2), N1(ξ)isa Bessel potential of order 2. Thus for any qsatisfying the assumption of Lemma (2.1) (ii) we have ||(N1ˆ f)v||q≤C||f||2, which of course leads to the desired conclusion by combining with the estimate for ||(N2ˆ f)v||qobtained in part (i). We now fix p0=2d d+2 and use p 0=2d d−2to express its conjugate. We assume mk(ξ)= iξ−k |ξ|2−iξk−|k|2for k∈Rd, i.e., a multiplier such that (ek·xu)∧(ξ)=mk(ξ)(ek·xu)∧(ξ) for u∈C∞ 0(Rd), where f∧is the Fourier transform of f. Set φ(k)(x)= φ(|k|x) as before. We now introduce the notations of three multiplier operators (2.6)  Sf =mkˆ f,  S1f=mk(1 −φk)ˆ f,  S2f=mkφkˆ f. Thus ek·xu=S(ek·xu)=S1(ek·xu)+S2(ek·xu). We have the following lemma: 192 G. Lu Lemma 2.7. If k∈Rd,E ⊂Rdcompact and convex set, |E|≥ |k|−d,u ∈W22 has compact support. Then for any θ> d−2 d(d−1) , the following two inequalities hold: i) ||S2(ek·xu)||q≤C|k|d s−1(|k|d|E|)θ||ek·xu||2,E provided 1 d≤ 1 2−1 q=1 s ii) ||ek·xu||p 02≤C(|k|d|E|)θ||ek·xu||2,E. We will prove Lemma (2.7) in the case k=e1and then by the scaling property the general case will follow. We need a closer look of the proof of Lemma 6.4 in [9]. Proof: Recall, when k=e1,m e1(ξ)=m(ξ)= iξ−e1 |ξ|2−iξ1−1. Let m1(ξ)= 1−φ(ξ)m(ξ),m 2(ξ)=φ(ξ)m(ξ), where φ(ξ)=φe1(ξ). Then clearly, (2.8) m1(ξ)≤C(1 + |ξ|2)−1/2. It is shown in Lemma 6.4 of [9] that (2.9) ||(m2ˆ f)v||p 0≤C||f||r provided 1 r>1 2+d−2 d(d−1) , where p 0=2d d−2. Since m2has compact support, then for any q≥p 0, ||(m2ˆ f)v||q≤C||(m2ˆ f)v||p 0≤C||f||r. Note 1 2−1 p 0 =1 d,thus1 2−1 q≥1 dis equivalent to q≥p 0. Given any θ> (d−2) d(d−1) , we can select rsuch that 1 2+d−2 d(d−1) <1 r<1 2+θ. Thus, for so selected r, (2.9) holds. So for any qsatisfying 1 2−1 q≥1 d ||(m2ˆ f)v||q≤C||f||r≤C|E|θ||f||2,E. Thus part (i) of lemma (2.7) will follow by scaling. We now prove (ii) of Lemma (2.7). We also assume k=e1first. Then ||(m1ˆ f)v||p 02≤C||f||2 since |m1(ξ)|≤(1 + |ξ|2)−1/2. Since the proof in [9] will apply to the weak type version, thus ||(m2f∧)v||p 02≤C||f||r2 if 1 2+θ>1 r>1 2+d−2 d(d−1) . So, ||(m2ˆ f)v||p 02≤C|E|1 r−1 2||f||2,E ≤C|E|θ||f||2,E, then ||Sf||p 02≤C|E|θ||f||2,E. By scaling, this shows (ii) of Lemma (2.7). Unique continuation for elliptic equations 193 3. Proof of Theorem 1 We first make a reduction. Set ed=(0,··· ,0,1). Lemma 3.1. To show Theorem 1, it is sufficient to show that for any given λ>0there is >0such that if (3.2)-(3.6) below holds then u=0, where (3.2) Ω = Rd\D(−ed,1 2),u:Ω→R (3.3) supp u⊂D(−ed,1),0∈supp u (3.4) u∈W22 loc(Ω) (3.5) |Lu|≤A|u|+B|u|, where (3.6) L= d  i,j=1 aij(x)d2 dxidxj , λ|ξ|2≤ d  i,j=1 aij(x)ξiξj≤λ−1|ξ|2,||aij||Ω≤ where ||aij||Ω= sup x,y∈Ω |aij(x)−aij(y)| |x−y| (3.7) ||A||Lp∞(Ω) ≤, ||B||Ld∞(Ω) ≤ where pis the same as in Theorem 1. Proof of Lemma 3.1 follows from a modification of the reductions in [9] and [3]. We now let Kbe the convex hull of Supp u{xd≥−1/4}. Select φ:Rd→Rsuch that φ(x) = 0 when xd≤−1/3 and φ= 1 on a neighborhood of ∂K. Set v=φu, then (3.8) |Lv|≤A|v|+B|v|+χ 194 G. Lu where χ∈L2has the property: supp χ⊂A1A2 where A1=D(−ed,1{x:−1/3≤xd≤−1/4},A2= a compact subset of IntK. Let Γ be the cone {k∈Rd:kd≥4|k|2−k2 d}. We now let >0 in Lemma (3.1) be small, Mlarge and define µ=(A|v|+B|v|+M−1/2|Hv|)2dx. By Lemma 1’ (instead of Lemma 1) in [9], taking p=2,C=D 2Med,2M 100 , then we can select {kj}and {Ej}with {Ej}disjoint, compact and convex sets such that (3.9) M 2≤|kj|≤2M,kj∈Γ (3.10) ||ekj·x(A|v|+B|v|+M−1/2|Hv|)||L2(Rd\(1+T)Ej) ≤2−1/2e−1/2C−1T||ekj·x(A|v|+B|v|+M−1/2|Hv|)||2, provided T≥0, and (3.11) |Ej|−1≥C−1Md (3.12) diam Ej≤CM−1/2 (3.13) Ejcontains a disc of radius (CM)−1. Because the family of weights {ek·x}is invariant by linear coordinate changes, we note the Carleman inequalities in Lemmas (2.1) and (2.7) remain valid if the Laplacian is replaced by any other constant coefficient second order elliptic operator, with bounds depending on ellipticity. We now let Ljbe the constant coefficient operator obtained by freezing the coefficients of Lat the barycenter of Ej. Besides (3.0)-(3.13), we furthermore have the following Unique continuation for elliptic equations 195 Lemma 3.14. Under the assumptions (3.2)-(3.7), we can select {kj} and disjoint, compact and convex sets {Ej}satisfying (3.11)-(3.15) and also the following inequalities (3.15)-(3.18) for any θ1>θ d= max d−4 d(d−1) ,0, and θ2>d−2 d(d−1) . (3.15) ||A||Lp∞(Ej)(Md|Ej|)θ1+||B||Ld∞(Ej)(Md|Ej|)θ2≥C1 (3.16) ||A||Lt(Ej)Md t−2(Md|Ej|)θ1+||B||Ls(Ej)Md s−1(Md|Ej|)θ2≥C2 (3.17) ||A||Lt(Ej)Md t−2(Md|Ej|)θ1+||B||Ld∞(Ej)(Md|Ej|)θ2≥C3 (3.18) ||A||Lp∞(Ej)(Md|Ej|)θ1+||B||Ls(Ej)Md s−1(Md|Ej|)θ2≥C4 provided that t<pis very close to pand s<d. Proof: To show (3.15), we will apply Lemmas (2.1) (ii) and (2.7) (ii) and Lemmas (8.1) and (8.2) in [9] together with the properties (3.9)- (3.13) of the sets {Ej}and kj. We also note below that 1 2−1 q=1 p,pis as in Theorem 1, and p 0=2d d−2. ||ekj·x(A|v|+B|v|+M−1/2|Hv|)||L2(Ej) ≤||A||Lp∞(Ej)||ekj·xv||Lq2(Ej)+||B||Ld∞(Ej)||ekj·xv||Lp 02(Ej) +M−1/2||ekj·xHv||L2(Ej) ||A||Lp∞(Ej)(Md|Ej|)θ1||ekj·xLjv||2,Ej +||B||Ld∞(Ej)(Md|Ej|)θ2||ekj·xLjv||2,Ej+||ekj·xLjv||2,Ej. The last inequality follows from Lemmas (2.1) and (2.7) for Ljand by using (3.12) after using Lemma (8.1) in [9]. On the other hand, by (3.6) and (3.12), we have |Ljv|≤|Lv|+CM−1 2gEj|Hv|. Hence |Ljv|≤A|v|+B|v|+CM−1 2gEj|Hv|+χ. Therefore, by dropping χin view of Lemma (8.2) in [9] and using the exponential decay property (3.10), (3.19) is bounded by ||A||Lp∞(Ej)(Md|Ej|)θ1+||B||Ld∞(Ej)(Md|Ej|)θ2+ ·||ekj·x(A|v|+B|v|+CM−1 2gEj|Hv|)||L2(Ej)