COMPARISON THEOREMS FOR ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS

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1 COMPARISON THEOREMS FOR ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS BY C. A. SWANSONO) Compariso theorems of Sturm's type will be obtaied for the liear elliptic partial differetial equatios (1) lu = 2 Dy(aiiDju) bydyu + cu = 0, i.l=l (2) Lv = 2 Dy(AyjDjV) ByDyV + Cv = 0 i,l=l o ubouded domais i «-dimesioal Euclidea space. The boudary of is supposed to have a piecewise cotiuous uit ormal vector at each poit. Poits i are deoted by x=(x\ x2,..., x) ad differetiatio with respect to x' is deoted by Dt, i= 1, 2,..., «. The coefficiets aih by, c, Ayt, By, ad C i (1) ad (2) are assumed to be real ad cotiuous o u, ad the matrices (aw) ad (Ay,) symmetric ad positive defiite i. A "solutio" of (1) (or (2)) is supposed to be cotiuous i u ad have uiformly cotiuous first partial derivatives i, ad all derivatives ivolved i (1) (or (2)) are supposed to exist, be cotiuous, ad satisfy the differetial equatio at every poit i. Some recet results of Clark ad the author [2], [7] apply to bouded domais i \ I the self-adjoit case by = By = 0, i=l, 2,..., «, the variatio of lu is defied as (3) V[u] = \ T (.au-aii)dyudfu+(c-c)u2 dx. The followig result [2] is typical of those to be exteded to ubouded domais. Theorem A. Let R be a bouded domai i E whose boudary P has a piecewise cotiuous uit ormal. Suppose by = By=0 i (1), (2), /'= 1, 2,..., «. If there exists a otrivial solutio u of lu=0 i R such that u=0 o P ad K[w]=;0, the every solutio of Lv = 0 vaishes at some poit i R. I [7] the author exteded this to the geeral secod-order liear elliptic equatios (1) ad (2). Theorem A geeralizes a theorem of Hartma ad Witer [4] i which the coditio F[«] 0 is replaced by the poitwise coditios C^c ad (fly Ai}) is positive semidefiite i. I the case «=1, Theorem A reduces to Preseted to the Society, August 30, 1966; received by the editors July 13, ) This research was supported by the Uited States Air Force Office of Scietific Research uder grat AF-AFOSR i = l i = l

2 ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS 279 Leighto's geeralizatio [5] of the classical Sturm-Picoe theorem. I this case, because the solutios of secod order ordiary liear differetial equatios have oly simple zeros, it is easy to obtai the followig modificatio of Theorem A : If there exists a otrivial solutio of lu=0 i (a, ß) such that u{a) = u{ß) = 0 ad V[u] >0, the every solutio of Lv=0 has a zero i {a, ß). I the case «= 2, Protter [6] obtaied poitwise coditios o the coefficiets i (1) ad (2) to esure the coclusio of Theorem A. Our purpose here is to exted Theorem A to ubouded domais i E. Apparetly o geeral results are kow eve i the case «= 1. Our results will costitute a extesio of the Sturm-Picoe theorem i 4 directios : (i) to «-dimesios ; (ii) to oselfadjoit differetial equatios; (iii) to coefficiets satisfyig a geeral coditio of the type V[u]^0; ad (iv) to ubouded domais. However, differetial equatios of order higher tha 2 ad geeral boudary coditios will ot be cosidered here. Let Da deote the «-disk {xe E: \x x0 <a} ad let Sa deote the boudig («- l)-sphere, where x0 is a fixed poit i E. Defie Ra = RDa, Pa = PDa, Ca = R Sa. Clearly there exists a positive umber a0 such that 7?a is a bouded domai with boudary Pa u Ca for all a = a0. Let Q[z] be the quadratic form i «+1 variables zx, z2,..., z+1 defied by (4) ' Q[z] = 2 AtiztZj-2z+x 2 B^ + Gz2^, i.l = l (=1 where the cotiuous fuctio G is to be determied so that this form is positive semidefiite. The matrix Q associated with Q[z] has the block form ß = (-* ~g)' '-<** where BT is the «-vector {Bx, B2,..., B). Let B * deote the cofactor of 7? i Q. Sice A is positive defiite, a ecessary ad sufficiet coditio for Q to be positive semidefiite is det Q = 0, or (5) G det {Au) = - 2 BMi=i The proof is a slight modificatio of the well-kow proof for positive defiite matrices [3]. Let Ma be the quadratic fuctioal defied by (6) Ma[u] = f F[u] dx, where (7) F[u] = 2 AijDiUDjU-2u 2 BlDiu + {G-C)u2. i.l i

3 280 C. A. SWANSON [February Defie M[u] = lima^œ Ma[u] (wheever the limit exists). The domai 35 M of M is defied to be the set of all real-valued cotiuous fuctios u i u such that u has uiformly cotiuous first partial derivatives i a for all a^a0, M[u] exists, ad u vaishes o. Defie (8) [", V]a = «2 AUiDlV ds where («) is the uit ormal to Ca ; (9) [u, v] = lim [u, v]a a-* <x> JCa wheever the limit o the right side exists. Lemma 1. Suppose G satisfies (5) i R. If there exists u e 35 M ot idetically zero such that M [u] < 0, the every solutio v of Lv = 0 for which [u2 v, v] 2:0 vaishes at some poit ofrup. Proof. Suppose to the cotrary that there exists a solutio v ^0 i u P. For u e 35M defie X* = vdy(u v), Y^v-^A^Dp, i i.i i=l,2,...,. The followig idetity i will ow be established : (10) 2 AyjXiXi-2u^BiXl + Gu2+ 2 Dy(u2Y*) = /,[i/] + «a»-1l». i.i i i The left member of (10) is equal to -2 2 Ay (vdyu - udyv)(vd)u - udjv)-2 BiivDyU - UDyV) V i,l V i + Gu2 + T 2 AvDiuDiv + ^ 2 ivdy{ayidjv)-ayjdyvdjv). V i.i V.1 Sice {A) is symmetric, this reduces easily to the right member of (10). Sice Lv=0 i, (11) f F[u]dx = j \jiaijxix1-2ujibixi + Gu2\dx + j ^DfyPY^dx. The first itegrad o the right side is a positive semidefiite form by the hypothesis (5). Sice u=0 o Pa, it follows from Gree's formula that f 2 A("2 Y') dx = f JRa i JPavCa V 2 "2"*Y* ds = y AyMyDjV ds.

4 1967] ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS 281 Hece (8) ad (11) yield Sice [u2jv, v] = 0 by hypothesis, This cotradictio establishes Lemma 1. f F[u] dx = [u2lv, vl, JRa M[u] = lim f F[u] dx = 0. a-<= jra Lemma 2 (self-adjoit case). Suppose 7?t=0 i (2) ad (7), i= 1,2,...,«. 7/ «ere exists u e S)M o? idetically zero such that M [u] ^ 0, the every solutio v of Lv=0for which [u2jv, v]^0 vaishes at some poit of Ru P. Proof. I this case we ca take G=0, ad the first itegrad o the right side of (11) is a positive defiite form. Hece Í 2 «a 7 AvXW i 0, equality holdig iff X{ is idetically zero for each 1=1,2,...,«; i.e., «is a costat multiple of t>. The latter caot occur sice u = 0 o P ad v^oop, ad therefore JRa F[u] dx > [u2 v, v]a. It follows that M[u]>0, cotrary to the hypothesis Af[i/] = 0. I additio to (6) cosider the quadratic fuctioal defied by ma[«] = 2awAw7Lw 2u2_ibiDiu cu2\ dx, whose Euler-Jacobi operator is / ad let «i[w] = lima_.c0 ma[u] (wheever the limit exists). The domai m of m cosists of all real-valued cotiuous fuctios u i R u P such that u has uiformly cotiuous first partial derivatives i Ra for all a^a0, m[u] exists, ad u vaishes o P. The variatio of m[u] is defied as V[u] = m[u]-m[u], that is (12) V[u]= i \^{ai)-alj)dludju-2u^{bi-bi)diu + {C-c-G)u2\ dx, with domai % = %m M. The aalogues of (8), (9) for the operator / are {u, v}a = «2 aaidjv ds, lca J {u, v} = lim {«, v}a. a-* oo Theorem 1. Suppose G satisfies (5). If there exists a otrivial solutio we of lu = 0 such that {«, wj^o ad V[u]>0, the every solutio v of Lv = 0 for which

5 282 C. A. SWANSON [February [u2/v, v]ïzo vaishes at some poit of Ru P. The same coclusio holds if the coditios V[u]>0, [u2 v, v]^0 are replaced by V[u]^0, [u2 v, v]>0 respectively. Proof. Sice «=0o a, it follows from Gree's formula that ma[u] = - uludx+{u,u}a. Sice lu=0 ad {u, u} z 0, we obtai i the limit a -> oo that m[u] S 0. The hypothesis V[u]>0 is equivalet to M[u]<m[u]. Hece the coditio M[u]<0 of Lemma 1 is fulfilled ad v vaishes at some poit of u P. The secod statemet of Theorem 1 follows upo obvious modificatio of the iequalities. Theorem 2 (self-adjoit case). Suppose > = =0 i (1) a«c7 (2), i= 1,2,...,«. If there exists a otrivial solutio «e33 of lu = 0 such that {u, u}z%0 ad V[u]^0, the every solutio v of Lv = 0 for which [u2/v, v]^0 vaishes at some poit of u. This follows from Lemma 2 by a proof aalogous to that of Theorem 1. I the case that equality holds i (5), that is we defie G = - 2 2W/det {Ay,), (=i S = 2 Dy(by-Bi) + C-c-G. i = l It follows from (12) by partial itegratio that (13) V[u]=\ r2(%-^i;)a" y/ + 8M2j dx + a, where Q = lim f y(by-by)u2yds. a->» JCa y L is called a " strict Sturmia majorat " of / whe the followig coditios hold : (i) (ayj Ayj) is positive semidefiite ad 3 2:0 i ; (ii) 0 2:0; ad (iii) either S>0 at some poit i or (% Atj) is positive defiite ad c^o at some poit. A fuctio defied i is said to be of class C2,1( ) whe all of its secod partial derivatives exist ad are Lipschitzia i. Theorem 3. Suppose L is a strict Sturmia majorat of I ad all the coefficiets ayj ivolved i I are of class C2,\R). If there exists a otrivial solutio u e 35 oflu = 0 such that {u, u}zi0, the every solutio v of Lv = 0 for which [u2/v, v]^0 vaishes at some poit of u. Proof. V[u] exists sice m g 35, ad hece each term o the right side of (13)

6 1967] ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS 283 exists by the strict Sturmia hypothesis. Sice au e C2,1{R), i,j=l,2,...,, Aroszaj's uique cotiuatio theorem [1] guaratees that the otrivial solutio u caot vaish idetically i ay ope subset of R. I the case that S>0 at some poit i R it the follows from (13) that V[u]>0. I the case that S = 0 i 7? it follows from (13) ad the positive defiite hypothesis o {a A) that F[w] = 0 oly if 7) w=0 for each i'= 1, 2,..., «i some ope set S of R, that is, u is costat i S. Sice c= 0 at some poit x0 e S, the differetial equatio (1) would ot be satisfied at x0. Hece F[«]>0 also i the case that 8=0. The coclusio of Theorem 3 the follows from Theorem 1. Theorem 4 (self-adjoit case). Suppose bi = Bt=0 i (1) ad (2), :'= 1, 2,..., «, C^c, ad {a^ A^) is positive semidefiite i Ru P. If there exists a otrivial solutio ue'si of {I) such that {u, u}^0, the every solutio v of {2) for which [u2\v, v] = 0 vaishes at some poit of RU P. This is a immediate cosequece of Theorem 2. We assert that the same coclusio holds eve if {Ai}) is oly positive semidefiite, provided F is a strict Sturmia majorat of / ad all the coefficiets ay are of class C2,1{R). I fact, uder these assumptios K[«]>0 as i Theorem 3, i.e., M[u]<0 by the proof of Theorem 1, ad Lemma 2 remais valid for positive semidefiite {Aif) provided the hypothesis Af[w] = 0 is replaced by M[u]<0. With trivial modificatios the above theorems ad proofs remai valid i the case that F is a bouded domai, i.e., Ca is void for a = a0. I particular Theorem 2 implies Theorem A ad Theorem 1 implies the author's recet result [7] for the geeral elliptic equatios (1), (2) o bouded domais. I the case «=2 cosidered by Protter [6], the coditio 8^0 of Theorem 3 reduces to {AXXA22-A2X2)^ Dtf-Bd-rC-c\ = AXXB22-2AX2BXB2 + A22B\. If F is a bouded domai, Theorem 3 the reduces (with trivial modificatios) to the author's result i [7]. It is iterestig to ote the followig oe-dimesioal istaces of Theorem 2, i which R is a ope iterval {a, ß). Whe «=1 ad bx = Bx = 0, the differetial equatios (1), (2) have the form (14) {au')' + cu = 0, a > 0, (15) {Av')' + Cv = 0, A > 0. Theorem 5. If there exists a otrivial solutio u of (14) i {a, oo) such that u{a)=0, a{x)u{x)u'{x) -> 0 as x -> oo, ad (16) P [{a-a)u'2 + {C-c)u2]dx = 0, Ja

7 284 C. A. SWANSON [February the every solutio v of (15) for which A(x)u2(x)v'(x)/v(x) has a oegative limit as x -> oo has a zero o [a, oo). Uless v is a costat multiple of u, v has a zero i (a, o). Proof. The first statemet follows immediately from Theorem 2. To prove the secod statemet, recall from the proof of Lemma 1 that for all a 2t a0, Sice the solutios of secod order ordiary liear differetial equatios have oly simple zeros, a applicatio of L'Hospital's rule yields *-* f(x) Thus the limit of the first term o the right side of (17) as a -> oo is oegative. The secod term is oegative for all a ad zero iff «is a costat multiple of v. Hece M [u] >0 uless v is a costat multiple of u. This cotradicts the hypothesis (16). The ext result applies to the case that a, ß may be sigular poits of the differetial equatios (14), (15); the possibility that they are ±oo is ot excluded. The proof is similar to that of Theorem 5 ad will be omitted. Theorem 6. If there exists a otrivial solutio u of (14) i {a, ß) such that a(x)u(x)u'(x) -> 0 as x -> a ad as x >- 8, ad (18) f [(a-a)u'2 + (C-c)u2]dx > 0, Ja the every solutio v of (15) for which A(x)u2(x)v'(x)lv(x) has a oegative limit as x -> ß ad a opositive limit as x > a has a zero i (a, ß). If the left side o/(18) is oly oegative, the same coclusio holds uless v is a costat multiple of u. I the special case that a, ß are ordiary poits of (14) ad (15), this reduces to the followig geeralizatio of the classical Sturm-Picoe theorem ; our result is a slight extesio of Leighto's theorem [5]. Theorem 7. If there exists a otrivial solutio u of (14) i [a, ß] such that u(a) = u(ß) = 0 ad the left side o/(18) is oegative, the every solutio of (15) except a costat multiple of u has a zero i (a, ß). As a example of Theorem 5, cosider the differetial equatios (19) w" + (2«+l-x> = 0, (20) v" + [2«+1 - x2 +p(x)]v = 0, o a half-ope iterval [a, oo), where p(x) is a polyomial. Equatio (19) has the well-kow solutio u(x) = exp ( x2/2)77(x), where 77(x) deotes the Hermite

8 1967] ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS 285 polyomial of degree «. Clearly we. Sice every solutio v of (20) satisfies v'{a)lv{a)~q{a) as a -> oo, where q{a) is a polyomial, it follows that the hypothesis u2{a)v'{a)lv{a) -> 0 as a -> oo is fulfilled. Hece if a is a zero of 77(x), the every solutio of (20) has a zero i (a, oo) provided / oo Ja p{x)u2{x) dx > 0. Refereces 1. N. Aroszaj, A uique cotiuatio theorem for solutios of elliptic partial differetial equatios or iequalities of secod order, J. Math. Pures Appl. 36 (1957), Coli Clark ad C. A. Swaso, Compariso theorems for elliptic differetial equatios, Proc. Amer. Math. Soc. 16 (1965), F. R. Gatmacher, The theory of matrices, Vol. I, Chelsea, New York, Philip Hartma ad Aurel Witer, O a compariso theorem for self-adjoit partial differetial equatios of elliptic type, Proc. Amer. Math. Soc. 6 (1955), Walter Leighto, Compariso theorems for liear differetial equatios of secod order, Proc. Amer. Math. Soc. 13 (1962), M. H. Protter, A compariso theorem for elliptic equatios, Proc. Amer. Math. Soc. 10 (1959), C. A. Swaso, A compariso theorem for elliptic differetial equatios, Proc. Amer. Math. Soc. 17 (1966), Uiversity of British Columbia, Vacouver, B. C, Caada