リーマン面で定義された方程式 △u=quのエバンス型の解について（1）
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(2) Vol. 22, No. 2 Journal of Hokkaido University of Education (Section II A) January 1972. Evans' Solution of the Equation ^u=qu on Riemann Surfaces (I). Takeyoshi SATO The Department of Mathematics, Iwamizawa Branch, Hokkaido University of Education. fe^X^: P --7^®^%^^^^^'®^; J%=^ 0 ^-.^y^UJ 0^^-01. -C. 0. Introduction Throughout this paper, let R be an open siemann surface. We consider the differential equation. (0,1) Au=qu, on R, where A is the Laplacian and q (z) is a non-negative continuouosly differentiable function of local parameters z=x-{-iy such that the expression q{z)dxdy is invariant under the change of local parameters z. We always assume that the function q{z) does not vanish constantly. We shall call a function ?/(z) a sohltion of (0,1) on an open subset D of R, if it is a twice continuously differentiable function satisfying the equation (0,1) on D. An Evans' sohltion. e{z) of the equation (0,1) on R is a solution of (0,1) on R satisfying e{z} — > oo, as z — >A, where A is the Alexandroff's ideal boundary point of R. In this paper, we shall point out a property of the ideal boundary of R on which there exists an Evans' solution satisfying. (0,2) \ e(z)g(z)dxdy<+w B. and no bounded solution of (0, 1) except the constant zero (i. e., R^Oq;,), and prove these Riemann surfaces belong to the class Og. And, it is well known that from the only existence of an Evans' solution which does not satisfy the condition (0,2) we can not say R^Oy. In 1 we shall make remarks on the properties of energy-finite functions f{z) (i. e., so that. Ds{f)+\IBf2(z)q(z)dxdy<^, where Ds^f) is the Dirichlet integral of / over R). These properties are similar to those of Dirichlet functions in the sense of Constantlnescu and Cornea's book [2], but the space of Dirichlet functions is a Banach algebra and our space is only a Hilbert space, and the space of these functions was considered by F. IVIaeda ([3]). In § 2 we define one compactification of Riemann surfaces. At last, in § 3 we shall prove the main theorem. For the properties on the solutions of the equation (0,1), we refer to Myrberg's works. [4] and [5].. (.72).
(3) ; 22^ ^2^- WMaiWNag (^2^ A) fl^^ 47^1^ 1. A space of functions with finite energy The space E^R} associated with a Riemann surface R and the equation Alt=qu is the totality of real functions ,(2) on R satisfying the following four conditions;. (1,1) /(z) is bounded on R, (1,2) ,(2) is a Tonelli function on R ([2] and [8]), (1,3) the Dirichlet integral Ds^f) of / over R is finite, (1,4) the integral \ f2{z~)q^z)dxdy is finite. 'R. For /i, ,2, / in E(R), let. (1,5) (/i, f^s=Ds(.fi, ,2) + \J^z)f^z)(l^z)dxdy, 'R a>6) 11/11^= (/,/)^. In the following we shall identify two functions in E(R) which are equal q. p. on R ( q. p. means " except for a polar set").. Lemma 1,1. (F. Maeda [3]) ||.|| is a norm in E{R) and E{R) is a Pre-Hilbert space with respect to the inner product (1,5), Let C be the set of all infinitely differentiable functions with the compact support on R.. Obviously, CC^(^?). We define the subspace Eo{R) of E{R) to be the closure of C in E(,R) with respect to the norm(l,6). Thus if fo^Eo(R'), then there exist functions /„ in C (n=l, 2, ...) such that ||/n-/o|| —^ 0, as n-)>oo. Next we state a lemma corresponding to. Theorem IG in III § 1 of [8]. Lemma 1,2. Let (/„) be a sequence of fimctions in Eo{R) and f a bounded function on R such that (/n) converges to f on R uniformly on each compact subset of R and \\fn\\R<K<<^ for every n. Then f belongs to Eo(R~) and (1,7) lim{g,f^s=(g,f)s n-foo. for every g in E{R~). Proof. At first we prove that / belongs to the space E<iR). Since Z)^(/)<; ||/n||2< Kz and (/n) converges to / uniformly on every compact set of R, f belongs to the Royden's. Algebra by Theorem 1C of III in [8]. And, from. /2 (z) ^ (z) ^^ = lim \ f\ (z) q (^) rf^ <^ ^m [| /„ ]|2 < K\. 'S,n. .. Ti-^oo. ^R,n. we have. Ifi. f2^g(z)dxdy <^K2. by q{z)^0 on R, and so f{z) belongs to E(R). Next we prove (1,7). From. (73). n-»°o.
(4) Vol. 22, No. 2 Journal of Hokkaido University of Education (Section II A) January 1972. _(fn-fVQ^dxdy<\\fn-f\[2<afn\\+\\f\\V<(K+\\f\\V,. i-R. •. we see, by Schwarz's inequality, for any g in E(^R~). fngqdxdy- (' fgqdxdy]~<.\_ _ (A-/)2^^^ „ g2qdxdy. ^R-Rm " ' JB--"m'~' '/ •3R-Rm ~ t'S-Bm. <. (K+1| /1|)2 \ g2gdxdy -^ Q, as m -> oo, fS-Bm. and so. (1,8) Urn \ gfnqdxdy=\ gfqdxdy n-».co ^M-Srn, 'IR-R^. for every m. Since (/n) converges to / uniformly on every compact set,. (1,9) Urn \ gfnqdxdy= \ gfqdxdy. n-^=o ^R^ .).»,„. Hence, by (1,8) and (1,9), we have. Urn \ gfnqdxdy-= \ gfqdxdy. n-*oo. *)ii-. -. '. •JB'. for every g contained in E{R), and so, from. UmD(g,fn)=D(g,f),. n->oo. which was proved in Theorem 1C of III in [8], we obtain li^n (fn, g~)s=<if, g')R n->-oo. for every g in E{R~). At last, we remark that / is contained in Eo(,R). Let 5£(J?) be the space which consists of all solutions on R of the equation (0,1) belonging to E{,R'). Then F. Maeda ([3]) remarked that EoW is orthogonal to SE^R), i.e., {u,f)s=Q for any u^SE^R) and f^EoW. Hence,. in our case we see that by (1,7) for any u in SE (^?) (/, u)R-=lim (/„, u')n=0, n->.oo. for fn is contained in Eo(R'). And so / belongs to Eo(R~). Q.E.D. It is L. Myrberg's work ([5]) that there exist always Green's functions G(z, w) of the equation (0,1) with respect to an arbitrary Riemann surface R unless g^O on R. Let p. be a positive regular measure on R and S(/-() be the support of p.. The potential G^z, p.) is. defined by G{z, fi) == \G(z, w)rf^(w). If G(z, fi) < oo for a point 2 in R, then G(z, <u)>0 and G(z, /^) is a solution of the equation (0,1) on R—SW. The energy of a measure p. is defined by. W=^G{z,zu)d^^dfJt{w). For such potentials we can apply the general potential theory with a positive symmetric kerner (see, for example, Ninomiya [7J). For the next lemma we may use the method in the proof of Hilfssatz 7,5 and Satz 7,2 in [2]. Hence we have. (74).
(5) ''22^. ^2^-. (^2^. A). ?47^1^. Lemma 1,3. (Constantinescu and Cornea) A potential G{z, fJ.) with a measttre p. on R. belongs to E{R) if and only if p. has finite energy. And we have (G(z, ^, G{z, v})H=2Tt ^G<iz, /J.')dv^ =27r ^G{z, v~)dp.^z), for p. and v ivith finite energy, respectively. For a relative compact domain D with the relative boundary consisting of a finite number of analytic closed Jordan curves (in the following such a domain of R is said to be analytic), and a continous function / on a set A containing -D, let fo be the continuous function on A such that /D=f on A -D and /j is a solution of the equation (0,1) on D. A continuous function. / defined on an open subset U of R is said to be a super solution of the equation (0,1) if for any point ZQ in U there exists a relative compact analytic domain Do of ^? such that ZQ^DQ^DQ (Z.U and fo ^f on £/for any analytic subdomain D of Dy with Zo£JDC5C-Do- A nonnegative constant and Green's function of the equation (0,1) are supersolutions of the equation (0,1) on R. From Perron's method (Proposition 1,13 in [3]), any non-negative supersolution ll{z') of (0,1) on R has the greatest solution among the class of smaller solutions of (0,1) than U. A non-negative supersolution of (0,1) is called a potential type supersolution, if the above minorant of this supersolution is zero. Green's functions of (0,1) are potential type supersolutions of (0,1).. Lemma 1,4. (F. Maeda [3]) Any potential type supersolulion of (0,1) which belongs to E{R) belongs to Ey^R). Lemma 1,5. For any point a in R^Ogb and any positive number /t, the function min{G (z,a), ^} ?'s <3 potential G{z, IJL) with a measure p. on R whose energy is finite.. Proof. Since min{G{z,d), ^} is a potential type supersolution of (0,1), from the Riesz decomposition theorem (Theorem 2,2 in [3]), there exists a measure on R, such that min{G{z,a), A}=G'0?,,u).. From ([5]) G(z, w) q (z) dxdy = 2Tt,. 's. we have. 2;r \d^.=\{ \ G(z, tu')q^z)dxdy\dfj,{w}= \ G{z, p^q^dxdy ^\ G(z,a')q(,z~)dxdy==2n, '5. and so, ^ ^ 1 and ||^||2= ^G(z, fJt)dfJt <. L Q.E.D.. (75).
(6) Vol. 22, No. 2 Journal of Hokkaido University of Education (Section II A) January 1972. § 2. On Royden,s compactification of Riemann surfaces By the method of Constantinenscu and Cornea's Q-compactification ([2]) we can construct a topological space R* with the following conditions (2,1) R* is a compact Hausdorff space, (2,2) R* contains R as an open and dense subset, (2,3) every function / in E{R~) can be continuously extended to R*, (we also denote this extended function by the same notation /). (2,4) E{R) separates points in R* (i.e,, if p and q are two distinct points in R*,. then there exists a function / in E(R) such that f(F)^ f \C[))And this topological space R* is uniquely determined up to a homeomorphism fixing R element wise. The set F=R*—R is a compact subset of R* and we call F the ideal boundary. of R*. We distinguish the following subset of the ideal boundary of R; A={p^F: /(^)=0 for every / in E^R)}. This is a compact subset of F. Since the function min{G(z, a), ^} for every fixed point z on R and any positive number ^ belongs to E{R) from Lemma 1,3, and 1,5, G{z, a) is continuously extended to R*. We shall denote this extended function by G(z, p) for p in R* in the following.. Lemma Lemma 2,1. For the function G(z,p'), there exists a sequence {a.n) of points in R converging converging to the Alexander ff's ideal bounday point such that the sequence {G(z,a.n)} of Green's functions converges to G{z, p~) uniformly on every compact subset of R. Then G{z, p~) is a. solution of (0,1) on R. Proof. We take a countable dence subset (Zn) of R. We can find a sequence (Um.n) (n, m=l, 2, .,.) of neighbourhoods of p in R* such that m,n —' ^m,n+l ci.nu Um+l.m. H7?n u,,n=^,. n -=1. and lim sup{ | G (zm, w) - G (2,,,, ^) I: we Um, n} = 0.. n-*°o. We take a sequence (a») such that a,, is contained in Un,n[~\R- Then. UrnG^z, an) =G <iz, p). 7t-»-oo. for z=Zm {m=l, 2, ...). On the other hand, by the Harnack type inequality, G(z, fin) is a bounded sequence of solutions on each compact subset of R except for a finite number of terms. Hence, by choosing a subsequence, we may assume G{z, a.n) converges to G{z, p~) uniformly on each compact subset of R. Q.E.D.. Lemma 2,2. For every point p on R* and any positive number ^, the function. min {G {z,p), ^.} belongs to EoW. And. (76).
(7) : 22^ ^2^ (^2^ A) TO 47^1^ (2,5) \\min{G{z, p), /{}||2 < 2^. Proof. For any point a^R, it is evident from Lemma 1,3 and 1,4. For any point p on F, we can take a sequence (fin) of points on R from Lemma 2,1 such that. HmG{z, an)=G{z,p). n~->oo. uniformly on every compact set on R. And from. the case that p^-R,. \\min{G{z, an), /{}||<27r^. Since min{G(z, a.n), ^^EQ^R}, we have. min[G{z,p), ^}^E^R), from Lemma 1,2.. Furthermore by (1,7) and Lemma 1,3. \\min{G{z, p), ^}\\2=lih Un {min[G{z, an), ^}, min{G{z, a,n),^}~) nl-».oo 7t->.03. = lim lim l-st \ G {z, /<„,,) dp.n ^ 2'K^ lim \ dp.n <. 2Tt^, m->co. n->.co. »'. n-».co. where min{G{z, On), ^}=G{z, /<»). Q.E.D. 3. The main theorem. Lemma 3,1. If e{z) is a solution of the equation (0,1) satisfying the condition e(z)q{z)dxdy< oo, then there exists a positive measure p. on F, such that. IJB. e^=\_G(z,p)d/^p) 'r. on RsOgb,. Proof. Let (Rn~) be a normal exhaustion of R. Then there exists a measure /J.n on the boundary 9Rn of Rn, such that e{z)=G{z, p.n) on Rn, e{z) ;> G(z, /^n,) on R. From. 27T \ dp.n =\\\ G {z, w) q (2') dxdy \dp.n {w)=\G {z, p.n) q (z) dxdy <,\ e(^z)q{z)dxdy< oo, fB. we can choose a subsequence of (/^n) and a measure /Jt, such that, denoting this subsequence by the same notation, lim \fdiJ.n-=\fdiJ. n->.oo. for every finitely continuous function / on 7?*. Hence we have \G{z,p~)dfJt{p')=lim \G{z, zu)dfin{w~)=e{z), on R. Q.E.D. n-»oo. Lemma 3,2. Let p be a fixed point on F and K be a compact subset of R. Then for every positive number e, there exists a neighbourhood U(p~) of p in R* such that. (77).
(8) Vol. 22, No. 2 Journal of Hokkaido University of Education (Section II A) January 1972. \G{z,g)-G(z,p)\<£ for q^U{p') and z^K. Proof. At first we remark that for fixed points Zy^R and p^F, there exist neighbourhoods U^Zo) of the point ZQ and V( ^), ^o) °f the point ^?, such that. \G{z,q)-G{z,p)\<& for z^.U{Zo) and q^V{p, Zy). Since G{z, p) is a continuous function of p on R*, we can choose a neighbourhood V{p, Zo) of the point p so that \G{z,,q~}-G{z,,p~)\<s for q^V{p, Zo). We can also find a neighbourhood U' of ZQ with the compact closure in R and a positive constant M such that G(,ZQ, 2) ^ M if z^R— U'. There exists also a neighbourhood. U(Zo) of ZQ such that U{zo)dU' and C-IG(ZQ, q)^G{z, q)<.cG{zo, q) for any z^U^Zo) and q^R—U' where c=l+s/M. Hence for every z e ?7(zo) and ^EV^o,^),. \G{z, g) -G{z, p~)\ < \G{z, q) -G(z,, q~)\+\G^, q) -G^ p)\ +\G(zo,p~)-G(z,p~)\<3s. Then, our assertion is evident from the compactness of K. Q.E.D.. Lemma 3,3. Any function ivhich is represented by an integral. ^G{z,p)d^p) ivith a finite measure p. on T, can be uniformly approximated on every compact set of R by the function with the form I. I. S tiG (2, ^,), where ti > 0 <2%rf S ^ = \ ^z=l. ~. J^i. Proof. Making a suitable Riemann sum we can prove this lemma from the preceding lemma. Q.E.D.. Lemma 3,4. For ^>0, ti>0 and pi^F (?"=1, 2, ..., /), the function. min{J]tiG(z,pi), ^} (=1. is a potential type super solution of (0,1). ;. Proof. Since S ti inin{G{z, pi), ^ti} belongs to Ey^R) from Lemma 2,2 there exists a (=1. ;. potential type supersolution which is larger than ^ tz min{G{z, pi), ^ti}, from Lemma 4,5 in i=l. [33. Then from min{T,tiG{z,pt), ^}<^ S ti min[G{z, pi), -^-}, (=i. ~. ^i. >-. '. ". I. t(J. min{'^tiG(,z, pi), ^} is also a potential type supersolution of (0,1). Q.E.D. (=1. (7S).
(9) : 22^ ^2-^- »»1WNS^ (^-2^ A) W^^IH Theorem 3,5. The existence of an Evans' solution e{z) of the equation (0,1) satisfying the condition e (2) q (2) (/A'rfy < oo. IB. implies that the subset A of the ideal boundary F tuith respect to the compactification R* of R is empty, zvhere R belongs to the class Oqg, Proof. From Lemma 3,3 we can take a sequence of functions fn such that (/n) converges to e{z) uniformly on every compact set of R and. fn{z)=T,tiG{Z,pi), f^l. ~. where pi is a point contained in F and ^ is a positive number (%=!, 2, ..., /). By Lemma 3,4 min{fn, ^) is a potential type supersolution for every n, and so, by the Riesz decomposition theorem (F. Maeda [3]), there exists a positive measure Vn such that. min{fn{z), ^=G{z, Vn) for every positive number ^. Then,. 27T \ dvn =\\\ G{z,zv)q (z) dxdy \ dvn = \ G{z, Vn) q (.z~) dxdy <\ fn{z)q{z)dxdy=^ti\ G^z, pi)q{z)dxdy <.2n^ ti=2Tt\d/ji, <R. 1=1. •JB. ~. ~. ~. l-i. that is,. (3,1) ^dvn<.^, where p. is a measure associated to e{z) in Lemma 3,1, since for every point tv in R,. G{z, w)q {z) dxdy = 27T,. 'B. and for every point p in 7"',. G (.z, p) q {z) dxdy <. Urn inf \ G (2, w) q (z) dxdy = 2Tt.. 'K. W-rp^. R. And furthermore, from (3,1) we have \\^n\\2=\G(,Z, Vn')dVn<.^\dVn ^ ^ \d/J.. Then Lemma 3,4, 1,3 and 1,4 show that min(fn, ^) belongs to Eo(R~), for min^fn, /i) is a potential of a measure fn with finite energy. We have. \\min{fn, ^= {G{z, Vn), G(z, Vn))s = 27T \ G (2, Vn) dVn {.Z) = 1-K \ mln (/n, -Q fi^u < 27T^ ^ fi?/^, from (3,1) and Lemma 1,3, that is,. \\min{fn,^\\H^[2^\dp.}. 1/2. for every integer n.. Since the sequence {min^fn, ^)) converges to the function min{e{z), /i) uniformly on every compact set of R, it follows that min{e{z), ^) belongs to Eo(R') from Lemma 1,2. Hence e{z) takes the value zoro on A. From our assumption (e(z)—)- oo, as z tends to the ideal. (79).
(10) Vol. 22, No. 2 Journal of Hokkaido University of Education (Section II A) January 1972 boundary of R) we can say that J==^. Q.E.D.. Let M, Rx, FS[ and As be the Royden's algebra associated with a Riemann surface R, Royden's compactification, Royden's boundary and Royden's harmonic boundary of R, respectively. ([2], [8]). Lemma 3,6. // the Royden's harmonic boundary Ajs of R is not empty, then the subset J of the boundary T is not empty either. Proof. By the method of Q-compactification ([2]), we can embed R to the topological. product ZT{/(/) : /e=MUC} and 77{7(/) : /€=£LJCL respectively, where /(/) is the real line in which the values of f{z) fall. Then we identify R*x and R* to the closure of the image of R in each product space. Let Ttf be the projection of each product space to the component I(.f). Then Ttj plays the role of the extended function of / on R to RM and J?* respectively. We consider a mapping 0 of RM to -R*, such that for any point p on R^i,. 7r/(0(^))=7r/(^), for every / in E(R). If p belongs to Jy, then 7T/(0(^)) =0 for every / in Ey^R), since EQ^R) is contained in the space of Dirichlet potentials. That is, the image ^{p) of p belongs to J. Hence we have ^(Jjy)CJ. Q.E.D.. Theorem 3,7. // there exists an Evans' solution e{z) of the equation (0,1) on -ReOgs. such that IjR. e (z) q (z) rf.rfi?y < c>o,. then R belongs to the class Og.. Proof. Since R^Og is equivalent to the condition Jx=0 ([8]), from. Theorem 3,5 we can understand this theorem easily. Q.E.D.. Remark. It is well known that from the existence of only Evans' solution e{z) on R, we can not say R^Og. For example ([6]), let R={z: r<l} and. (?(z)==4(l+r2)/(l-r2)2, where r=\z\. Then the equation (0,1) possesses an Evans' solution e{z) =l/(l—r2) on R. Hence e(^z)q{z)dxdy=oo,. R. and .R does not belong to the class Og. References [1] Brelot, M. (1965), Elements de la theorie classique du potentiel. Center de documentation universitaire, Paris, p. 209.. (.80).
(11) : 22^ ^2^ (^ 2 ^ A) R^p47^1^ [2] Constantinescu, C. and Cornea, A. (1963), Ideale Rander Riemannscher Flachen. Springer, Berlin. p. 244. [3] Maeda, F. (1968), Boundary value problems for the equation Au—qu=Q with respect to an ideal boundary. J. Sci. Hiroshima Univ. Ser-A-I. 32, p. 85-146.. [4] Myrberg, L. (1954), Ober die Integration der Differential Gleichung ^u=c{P)'u auf offenen Riemannscher Flachen. Math. Scand. 2, p. 142-152. [5] Myrberg, L. (1954), Uber die Existenz der Greencher Function des Gleichung Jw,==c(P)-M auf Riemannscher Flachen. Ann. Acad. Sci. Fenn. A.I. 170. [6] Nakai, M. (1963), On Evans' solution of the equation Au=P'u on Riemann surfaces. Kodai Math. Sem. Rep. 15, p. 79-83. [7] Ninomiya, N. (1969), Potential theory. Kyoritsu, Tokyo, p. 190. [8] Sario, L. and Nakai, M. (1970), Classification theory of Riemann surfaces. Springer, Berlin, p. 446.. (S2).
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