On radial limit functions for entire solutions of second order elliptic equations in

Publicacions Matem`atiques, Vol 42 (1998), 509–519. ON RADIAL LIMIT FUNCTIONS FOR ENTIRE SOLUTIONS OF SECOND ORDER ELLIPTIC EQUATIONS IN R2 Andr´ e Boivin∗and Peter V. Paramonov† Abstract Given a homogeneous elliptic partial differential operator Lof order two with constant complex coefficients in R2, we consider entire solutions of the equation Lu = 0 for which lim r→∞ u(reiϕ)=:U(e iϕ) exists for all ϕ∈[0,2π) as a finite limit in C. We characterize the possible “radial limit functions” U. This is an analog of the work of A. Roth for entire holomorphic functions. The results seem new even for harmonic functions. 1. Introduction and Main Results Let Lv =c11vx1x1+2c 12vx1x2+c22vx2x2 be an homogeneous partial differential operator of order two with constant complex coefficients in R2satisfying the ellipticity condition c11ξ2 1+2c 12ξ1ξ2+c22ξ2 26=0 for all (ξ1,ξ 2)6=(0,0), ξ1,ξ2∈R. Keywords. Elliptic operator, L-entire functions, radial limit functions. ∗The first author was partially supported by NSERC (Canada). †The second author was supported by RFBR (grants No 96-01-01240 & 96-15-96846). 510 A. Boivin, P. V. Paramonov Let λ1,λ2be the (complex) roots of the characteristic equation c11λ2+ 2c12λ+c22 = 0. It follows from the ellipticity condition that λ1,λ2/∈R. We define ∂1=∂ ∂x1 −λ1 ∂ ∂x2 ,∂ 2 =∂ ∂x1 −λ2 ∂ ∂x2 if λ16=λ2, or ∂1=∂ ∂x1 −λ1 ∂ ∂x2 ,∂ 2 =∂ ∂x1 +λ1 ∂ ∂x2 if λ1=λ2. We then have the following decomposition of L: Lv =(c11∂1(∂2(v)),if λ16=λ2; c11∂2 1(v),if λ1=λ2. We also introduce the following new coordinates: z1=λ2 λ2−λ1µx1+1 λ2 x2¶,z 2 =λ 1 λ 1 −λ 2 µ x 1 + 1 λ 1 x 2 ¶if λ16=λ2; or z1=1 2µx1−1 λ1 x2¶,z 2 = 1 2 µ x 1 + 1 λ 1 x 2 ¶if λ1=λ2. The following “orthogonality” relations then are easily obtained: (1) ∂1z1=1 ∂ 1 z 2=0 ∂ 2 z 1=0 ∂ 2 z 2=1. Finally, we identify z=x1+ix2in Cand x=(x 1 ,x 2)inR 2and, for s= 1 and 2, we define Ts(z)=z s(which are linear nondegenerate transformations of R2). For any set Ein R2, denote by L(E) the family of all functions v, each defined on its own neighbourhood Ωvof E, such that Lv = 0 in Ωv in the classical sense. We note that for Eopen, one can take Ωv=E for all v. Functions in L(E) and L(R2) are called L-analytic on Eand L-entire respectively. It is well known that (for Eopen) each function v∈L(E) is realanalytic on E, and that each continuous function vsatisfying Lv =0on Ein the distributional sense is in L(E). From these facts, using (1), one can prove the following well known result [1, Chapter IV, §6, (4.77)] (see also [5] for a simple direct proof). On radial limit functions 511 Proposition 1. Let Dbe any domain in Cand Lbe as above. 1. If Dis simply connected and if λ16=λ2, then 1a) v∈L(D)if and only if there exist f1holomorphic in T1(D) and f2holomorphic in T2(D)such that v(z)=f 1 (T 1 (z))+f2(T2(z)) = f1(z1)+f 2 (z 2 ) for all z∈D. In particular, L-entire functions uare of the form u(z)=f 1 (z 1 )+f 2(z 2)where f1,f2are entire holomorphic functions. 1b) There exist in C\{0}a fixed analytic branch log(z1zν 2)of the multivalued function Log(z1zν 2)and a nonzero complex constant CLdepending only on Lsuch that ΦL(z)=C Llog(z1zν 2) is a fundamental solution of L, where ν=1if sgn(Im λ1)6= sgn(Im λ2), and ν=−1otherwise. 2. If λ1=λ2, then 2a) v∈L(D)if and only if there exist g1and g2holomorphic in T2(D)such that v(z)=T 1 (z)g 1 (T 2 (z))+g2(T2(z)) = z1g1(z2)+g 2 (z 2 ) for all z∈D. In particular, L-entire functions uare of the form u(z)=z 1 g 1 (z 2 )+g 2 (z 2 )where g1,g2are entire holomorphic functions. 2b) ΦL(z)=C Lz 1 z 2is a fundamental solution of L, where CLis a nonzero complex constant depending only on L. 3. If {vn}⊂L(D)and {vn}converges uniformly to von compact subsets of Das n−→ ∞ , then v∈L(D). We just note that 1b) and 2b) follow from 1a) and 2a) respectively, and from the definition of fundamental solution. It is not difficult to check that if sgn(Im λ1)6= sgn(Im λ2) (respectively sgn(Im λ1) = sgn(Im λ2)), then the increment of the polar argument of (z1z2) (respectively (z1/z2)) around the origin is zero, and thus some analytic branch of the function log(z1z2) (respectively log(z1/z2)) exists in R2\{(0,0)}. 512 A. Boivin, P. V. Paramonov Example 1. For the Laplacian L= ∆, one has λ1=i,λ2=−i, z1=z/2, z2=¯z/2 and ∂1=∂ ∂x1 −i∂ ∂x2 =: 2 ∂ ∂z, ∂2=∂ ∂x1 +i∂ ∂x2 =: 2 ∂ ∂¯z, Φ∆(z)= 1 4πlog ³z¯z 4´. For the Bitsadze operator L=∂2 ∂¯z2=1 4µ∂2 ∂x2 1 +2i∂ 2 ∂x1∂x2 −∂2 ∂x2 2¶, one gets λ1=λ2=−i,z1=¯z/2, z2=z/2 and ∂1=2∂ ∂¯z,∂ 2 =2∂ ∂z,ΦL(z)= 1 π ¯z z. In order to formulate our main results (Theorems 1 and 2), we need the following characterization of radially constant solutions of the equation Lv =0. Proposition 2. Let J={z∈C:ϕ1<arg z<ϕ 2 },ϕ 1<ϕ 2≤ ϕ 1+2πdenote an (infinite) open sector with vertex at 0.Letv∈L(J) and assume that v(z)=v(reiϕ)=v(e iϕ)does not depend on r. 1. If λ16=λ2, then there exist α,β∈Cand a fixed analytic branch log(z1/z2)of Log(z1/z2)in Jsuch that, for z∈J, (2) v(z)=αlog z1 z2 +β =αlog Ãcos ϕ+1 λ2sin ϕ cos ϕ+1 λ1sin ϕ!+β=: v∗ 12(eiϕ). 2. If λ1=λ2, then there exist α,β∈Csuch that, for z∈J, (3) v(z)=α z 1 z 2 +β =αÃcos ϕ−1 λ1sin ϕ cos ϕ+1 λ1sin ϕ!+β=: v∗ 1(eiϕ). (For this case, J=C\{0}is also allowed.) On radial limit functions 513 Example 2. For L= ∆, one has v∗ 12(eiϕ)=αϕ+β,ϕ1<ϕ<ϕ 2 , and for L=∂2/∂¯z2,v1(eiϕ)=αe−2iϕ +β, where αand βare any complex constants. Theorem 1. Let ube an entire solution of the equation Lu =0such that (4) lim r→+∞u(reiϕ)=:U(e iϕ) exists and is finite for all ϕ∈[0,2π). Then A) Uis of Baire class 1on S={eiϕ :ϕ∈[0,2π)}; that is, Uis a pointwise limit on Sof a sequence of continuous functions on S. B) There is an open set I=∪∞ j=1Ij, where the Ijare disjoint open arcs on S(and Ij=∅is possible for some j, but Ij6=S) with the following properties: B1) Iis everywhere dense on S; B2) On each Ij,U(eiϕ)is of the form v∗ 12(eiϕ)if λ16=λ2(respectively of the form v∗ 1(eiϕ),ifλ 1=λ 2 ), (see (2) and (3)); B3) The limit (4) is uniform on each compact subset of each Ij. Conversely, let Ube a function defined on Sand Ibe an open subset of Swith I=∪∞ j=1Ij, where the Ijare disjoint open arcs. If (A), (B1) and (B2) above are satisfied, then there exists an L-entire function u with the properties: a) lim r→∞ u(reiϕ)=U(e iϕ)for each ϕ; b) The limit in (a) holds uniformly on each compact subset of Ijfor each j. Moreover, if U1is of Baire class 1on Sand U1(eiϕ)=∂U(eiϕ)/∂ϕ on I, then the function ucan be chosen such that (a) and (b) are satisfied and lim r→+∞ ∂u(reiϕ) ∂r =0,lim r→+∞ ∂u(reiϕ) ∂ϕ =U1(eiϕ) for all ϕ∈[0,2π). Let Kbe a compact set in S. Let RP(K) (respectively RU(K)) denote the set of all functions gon Kfor which there exists u=ug∈L(R2) such that u(reiϕ)−→ g( e iϕ) for each ϕ∈K(respectively u(reiϕ)−→ g( e iϕ) uniformly on K)asr→∞. 514 A. Boivin, P. V. Paramonov Theorem 2. a) For each compact set Kin S,g∈RP(K)if and only if gis of Baire class 1on Kand there exists a countable family of disjoint open arcs {Ij}∞ j=1 in Ksuch that K\∪ ∞ j=1Ijis nowhere dense in Sand on each Ij,gis of the form v∗ 12(eiϕ)(when λ16=λ2) or v∗ 1(eiϕ)(when λ1=λ2) (see Proposition 2). In particular, RP(K)consists of all Baire class 1functions on Kif and only if Khas an empty interior on S. b) Let Kbe a compact set in S,K6=S. Then g∈RU(K)if and only if g∈C(K)and gis of the form v∗ 12(eiϕ)(when λ16=λ2) or v∗ 1(eiϕ)(when λ1=λ2) in each connected component of the interior of Kin S. In particular, RU(K)=C(K)if and only if Kis nowhere dense in S.IfK=S, then RU(K)contains only constant functions. 2. Proofs We first establish the following uniqueness theorem for L-analytic functions. Lemma 1. Let Dbe any domain in Cand v∈L(D). If the set Gv={z=x1+ix2∈D|∇v(z):=(∂v(z)/∂x1,∂v(z)/∂x2)=(0,0)} has at least one accumulation point inside D, then vis constant in D. Proof: From Proposition 1 and equations (1), one has ∂1v=f0 1(z1) for λ16=λ2and ∂1v=g1(z2) for λ1=λ2, where f0 1and g1are holomorphic on T1(D) and T2(D) respectively. By assumption, f0 1=0onT 1 (G v ) (respectively g1=0onT 2 (G v )). It thus follows from the uniqueness theorem for holomorphic functions that f1≡const in T1(D) (respectively g1≡0inT 2 (D)). An analogous study of ∂2vcompletes the proof of Lemma 1. Proof of Proposition 2: We shall consider only the case λ16=λ2, the proof for the case λ1=λ2being similar. Let v∈L(J), v=v(eiϕ). Let v0(z) = log(z1/z2) be some fixed analytic branch of Log(z1/z2)in J. Simple calculations show that ∂v0(z)/∂ϕ 6= 0 and ∂v0/∂r ≡0inJ. Fixing some ϕ0∈(ϕ1,ϕ 2), we can thus find αand βin Csuch that v−αv0−β= 0 and ∂(v−αv0−β)/∂ϕ = 0 on the ray {arg z=ϕ0}.It thus follows that ∇(v−αv0−β) = 0 on the ray {arg z=ϕ0}. Lemma 1 now gives the desired result. On radial limit functions 515 Proof of Theorem 1: The scheme of the proof is analogous to that of A. Roth [7] (see also [3, Chapter IV, §5A]). The main new tools are some recent results in approximation theory ([6] and [2]). Let u∈L(R2) satisfy (4), then A) is a consequence of limn→∞u(neiϕ)= U(eiϕ). Using a decreasing sequence of nested intervals and condition (4), one can prove that for each nonempty sector J00 with vertex at the origin, there exists a nonempty sector J0={ϕ0 1<arg z<ϕ 0 2 }⊂J 00 with ϕ0 1<ϕ 0 2≤ϕ 0 1 +2πsuch that uis bounded on J0(see [3, p. 164]). Fix any ϕ1and ϕ2with ϕ1<ϕ 2and [ϕ1,ϕ 2]⊂(ϕ 0 1,ϕ 0 2). Let un(z)=u(2nz). We claim that the sequence {un(z)}∞ n=1 converges uniformly on compact subsets of the “closed” sector J={ϕ1≤arg z≤ϕ2}. From (4), it will follow that the limit function vdoes not depend on r. Since v∈L(J) (see 3 of Proposition 1), Proposition 2 will give us B) in our theorem (see [3, p. 166] for more details). To prove the claim, it suffices to establish that {un}converges uniformly on the compact set K={ϕ1≤ arg z≤ϕ2,1≤|z|≤2}. In order to prove this last assertion, it is enough to check that |∇un|is uniformly bounded on Kand to use AscoliArzela’s theorem. Notice that sup{|un(z)||z∈J 0 ,n≥1}<+∞, and d:= dist(K, ∂J0)>0 (here and in the sequel, ∂E is the boundary of a set E). Denote by Φ the fundamental solution of L, which is found in Proposition 1, and set B(a, δ)={z∈C||z−a|<δ}, where a∈Cand δ>0. Fix ψ∈C∞ 0(B(0,d)) such that ψ=1inB(0,d/2). Now fix z0∈Kand put ψ0(z)=ψ(z−z 0 ). Then ψ0= 0 outside the ball B(z0,d)⊂J0and ψ=1onB(z 0 ,d/2). One has ([6, p. 255]) unψ=Φ∗L(u n ψ), so that in B(z0,d/2), we can write (in the case λ16=λ2) un(z)=Φ∗(Lunψ+a11∂1un∂2ψ+a11∂2un∂1ψ+unLψ)(z). Since ψLun≡0 and a11∂sun∂3−sψ=a11∂s(un∂3−sψ)−unLψ (s=1 and 2), we obtain that, in B(z0,d/2), un=Φ∗(a 11∂1(un∂2ψ)+a 11∂2(un∂1ψ)−unLψ) =a11(∂1Φ) ∗(un∂2ψ)+a 11(∂2Φ) ∗(un∂1ψ)−Φ∗(unLψ). Now the desired uniform estimate for |∇un(z0)|can be obtained by making trivial estimates in the formula ∇un(z0)=a 11£(∇∂1Φ) ∗(un∂2ψ)+(∇∂ 2 ψ)∗(u n ∂ 1 ψ) ¤ −(∇Φ) ∗(unLψ)) ¯¯¯z=z0 . The proof for the case λ1=λ2is similar. 516 A. Boivin, P. V. Paramonov Let us now prove the second part of Theorem 1. Let I=∪∞ j=1Ij,U, U1be as in (the second part of) Theorem 1. Put I0=S\I, and for j=0,1,... let Jj={z∈C\{0}|e iarg(z)∈Ij}. Finally set F0= {z∈J0||z|≥1},F j={z∈J j|dist(z,∂Jj)≥1},j=1,2,..., and F=∪∞ j=0Fj. Notice that each Fjand Fare closed subsets of Cand that the Fj(j≥0) are pairwise disjoint. We note that if they are infinitely many Fj, they are pushed to ∞(i.e. they are eventually outside any fixed compact set). It follows that there exist pairwise disjoint neighbourhoods Ωjof Fj,j=0,1,..., with Ωj⊂Jjfor j≥1. We first want to show that there exists a neighbourhood Ω0 0of F0, Ω0 0⊂Ω0, and a function f∈C1 loc(Ω0 0) such that (5) lim r→∞ f(reiϕ)=U(e iϕ), lim r→∞ ∂f(reiϕ) ∂ϕ =U1(eiϕ), lim r→∞ ∂f(reiϕ) ∂r =0, for each eiϕ ∈I0. The proof of this elementary fact is included for completeness. Let A0={|z|<2},As={2s−1<|z|<2s+1}), s=1,2,..., and let {χs}∞ s=0 be a partition of unity on Csubordinate to {As}∞ s=0 such that χs(z)=χ s (|z|) and |∇χs|≤c/2s, where cis a constant independent of s. Since Uand U1are of Baire class 1 on S, there exist sequences of continuous functions {Vs},{Ws}on Ssuch that Vs(eiϕ)−→ U( e iϕ) and Ws(eiϕ)−→ U 1( e iϕ), for all eiϕ ∈S(and thus in particular for all eiϕ ∈I0). In addition we can choose the continuous functions Vsand Wsso that they are bounded by 2s/2. Since Vsand Wsare uniformly continuous on S, there exists δs,0< δ s<2 −s , such that |eiϕ −eiϕ0|<δ simplies |Vs(eiϕ)−Vs(eiϕ0)|<1/2s and |Ws(eiϕ)−Ws(eiϕ0)|<1/2s. Since by assumption I0is nowhere dense in S, there exist open neighbourhoods Nsof I0,s=0,1,..., such that Ns=∪k≥1Isk is the union of finitely many open arcs Isk whose closures are disjoint and each Isk is of length less than δs. Now for each s≥0, define Ωs 0=N(ϕ) s×(2s−1,2s+1)(r)and Ωsk 0= I(ϕ) sk ×(2s−1,2s+1)(r)in the (ϕ, r)-plane. We further require that the Ns (s≥0) be chosen such that Ωs 0⊂Ω0. We note that, by construction, Vsand Wsare almost constant on each of the sets Isk. Fix ϕsk ∈I0∩Isk.Forz=reiϕ ∈Ωsk 0, let fsk(z):= On radial limit functions 517 αskϕ+βsk, where αsk,βsk ∈C, are chosen such that fsk(eiϕsk )= V s (e iϕsk ) and ∂fsk/∂ϕ =αsk =Ws(eiϕsk ), so that |αsk|≤2 s/2. Let fsbe the function defined on Ωs 0which is equal to fsk on Ωsk 0. And let f=P∞ s=0 fsχs. Then fis well-defined on some neighbourhood Ω0 0of F0. It is not too difficult to see that fsatisfies (5). In the sequel, we identify Ω0and Ω0 0. Using the localization scheme of Vitushkin (similarly to [4, Lemma 2.2(8), Corollary 6.3]), one can prove that for each R>0, there exists {fR n}⊂L(F R 0), where FR 0=F0∩{|z|≤R}, such that fR n−→ f in C1 jet(FR 0)asn→+∞(see [4] and [2, section 2.1]; in our particular case, since the interior of F0is empty and the union of all the lines in C\F0is everywhere dense, we only need a very simple part of the localization scheme). Let us now consider the Banach space V=ng∈C1(R2)¯¯kgk:= sup z∈R2©max{|g(z)|,|∇g(z)|}(1 + |z|2)ª<∞o with norm k·k. This space satisfies the conditions (1)-(4) of [2]. From the fact that Vis locally equivalent to the space C1(R2) and from the approximation properties of fon FR 0mentioned above, it follows also that there exists a locally finite family of balls covering F0such that for each ball Bin this family and for each ε>0, there exists gsuch that Lg = 0 on some neighbourhood of F0∩Band kf−gkF0∩B<ε i.e. fis approximable locally on F0in the norm of Vby (local) Lanalytic functions. Theorem 2 in [2] now states that this is equivalent to global approximation, that is, for each ε>0, there exists an L-analytic function gon (all of) F0such that kf−gkF0<ε. Denote by R2 ∞=R2∪ {∞} the one-point compactification of R2. Since R2 ∞\F0is connected and locally connected (that is, F0is a RKL-set in the terminology of [2] (the letters stand for Roth-KeldyshLavrentieff)), we can use an analog of Runge’s theorem obtained in [2, Theorem 1] to approximate in the norm of VL-analytic functions on F0 by L-entire functions. We thus conclude that we can find an L-entire function hsuch that kf−hkF0≤1. Using the estimate (6) |∂ψ(z)/∂ϕ|<|∇ψ(z)||z|, this gives that (5) is satisfied when his substituted for f. Now define v(z)=h(z)inΩ 0and v(z)=U(e iarg(z))in∪ ∞ j=1Ωj. Then v∈L(Ω), where Ω = ∪∞ j=0Ωjis a neighbourhood of F, and Fis a RKL-set. Thus again by [2, Theorem 1], we can find u∈L(R2) with kv−ukF≤1. It suffices to notice, using (6) with ψ=u−v, that uis the desired L-entire function. Theorem 1 is proved.