SOME ANM063 OF THE PICONE IDENTITY APPLIED T0 FOURTH , ORDER DIFFERENIIAL EQUATIONS ............ Thesis for the Degree of Ph. D, ‘ MICHIGAN STATE UNIVERSITY COREEN L. METT ’ ‘ 1973 ‘ ........... Iv’fl‘1‘m... y w J 7 ‘36”,- LI L R A R Y I Michigan Stan: fiwUmvcrsity This is to certify that the thesis entitled "Some Analogs of the Picone Identity Applied to Fourth Order Differential Equations" presented by ‘ Coreen L. Mett has been accepted towards fulfillment of the requirements for I_, 1’ , (V 1’ " " I: 5.. degree in 3“” //3' x: /. (.3 6 mid/woo. M Major profesér u...” \“ - ' " ' 7 Date \J (A I.) L” .3 L / (7 /_j r 0-7639 9‘ L mane av “" 0‘ i HOAB & SONS‘ I 800! BINDERY INC. ; : LIBRARY BIND! Rs i ABSTRACT SOME ANALOGS OF THE PICONE IDENTITY APPLIED TO FOURTH ORDER DIFFERENTIAL EQUATIONS By Coreen L. Mett In this paper we obtain Sturmian-type theorems and lower bounds for eigenvalues of some fourth order problems. Our results are attained by using some generalizations of the classical Picone identity, which are themselves derived from a generalized technique of Picard. SOME ANALOGS OF THE PICONE IDENTITY APPLIED TO FOURTH ORDER DIFFERENTIAL EQUATIONS By Coreen L. Mett A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1973 To Mom and Dad, who gave me life and the love of it. 11 ACKNOWLEDGMENTS I wish to acknowledge my indebtedness to the wisdom and patience of Professor Dunninger, and to thank him for his guidance in the development of this paper. iii TABLE OF CONTENTS INTRODUCTION 0.1 Sturmian Theorems 0.2 Lower Bounds for Eigenvalues Chapter I. {MAXIMUM PRINCIPIES AND BASIC IDENTITIES I . 1 Maximum Principles 1.2 Identities A. Cimmino's identity B. Kreith's identity C. Dunninger's identity D. A new identity Chapter II. COMPARISON AND OSCILLATION II.1 Sturm-type Comparison Theorems II.2 Oscillation Results Chapter III. EIGENVALUE PROBLEMS III.1 Barta-type Lower Bounds III.2 Comparison Theorems for Eigenvalues Chapter Iv. REMARKS 0N PARTIAL DIFFERENTIAL EQUATIONS BIBLIOGRAPHY iv Page 15 15 26 31 31 40 50 55 INTRODUCTION Sturm-type comparison theorems and lower bounds for eigen- values will be considered for various fourth order linear ordinary differential equations. The results are derived from certain extensions of Picone's [26] classical identity. In each case our studies are motivated by well known results for similar problems in connection with second order linear ordinary differential equa- tions. 0.1 Sturmian Theorems The classical Sturm-Picone comparison theorem (see [15], pp. 225-226) asserts that if u is a nontrivial solution to the boundary value problem (a(x)u')' + c(x)u = 0 in (O,L) U(0) = U(L) = 0 and if v is a solution of the equation (A(x)v')' + C(x)v = O in (O,L) where the coefficients satisfy the relationship a2A>O,Czc in (0.!) then v must have a zero in (0,L) unless u and v are linearly dependent. This fact follows readily from the Picone identity ([15], p. 226) u2v' L L 2 SAv'z' uv' 2 (0-1) (auu' ' A_—;-)IO = £[u(au')' - u v + A(u' - f;-) , 2 + (a - A)(u) ‘de. For the fourth order equations (0.2) (a(x)u")" - c(x)u = 0 (O 3) (A(x)v")" - C(x)v = O the situation is somewhat more difficult as is evidenced by the following theorem of Leighton and Nehari ([22], p. 327). Theorem. If v is a solution of (0.3) (A(x)v")" - C(x)v = 0 with A > O and C > 0, and if the values Of v, v', v", and (Av")' are nonnegative but not all zero at x = 0, then the func- tions v(x), v'(x), v"(x), and (A(x)v"(x))' are all positive for x > 0. Hence every equation of the type (0.3) has at least one solution without zeros. Therefore, if u is a nontrivial solution of (au")" - cu = O in (O,L) U(0) = U'(0) = U(L) = U'(L). then it cannot be asserted that every solution v of (0.3) has a zero in [0,L]. As a consequence, fourth order equations are studied either in the context of oscillatory behavior, as was done by Leighton and Nehari [22], or by considering classes of solutions which exclude those of the type occurring in the theorem, as was done by Diaz and Dunninger [5, 6], Dunninger [10], Kreith [17, 18, 20], Swanson [31], and Wong [35]. We shall follow the latter course. In Chapter II we obtain Sturmian-type results for the fourth order equations (0.2) and (0.3) under the assumption that solutions v of (0.3) are positive at a point and satisfy v" s 0 in the interval (0,L). In fact, our results will be established for certain systems of second order equations which contain (0.2) and (0.3) as special cases. Some oscillation results are also presented which compare the oscillatory behavior of (0.2) and a related second order equation. Aside from resolving the above complication, (namely, the formulation of a fourth order Sturmian result) there has been dif- ficulty in obtaining a natural analog of the second order Picone identity (0.1) which can be used to treat fourth order equations. One such identity was presented by Cimmino in 1930 [3], and later by Leighton [21] and Kreith [20], another by Kreith in 1969 [18], and a third by Dunninger in 1971 [10]. It is of interest that each of these identities as well as some new identities, can be derived from a more general identity which results from a technique due to Picard [25]. We shall carry this out in Chapter I. 0.2 Lower Bounds for Eigenvalues If u is an eigenfunction correSponding to the lowest eigenvalue A1 of the problem Lu 2 (a(x)u'>' + cu = -xp(x)u a > 0, o > 0 i“ (M) U(0) = U(L) = 0. then it is an immediate consequence of Picone's identity (0.1) that Al 2 inf %% (0,L) where the infimum is taken over functions v > O in [0,L]. Lower bounds, of this nature, for eigenvalues were first established by Barta [2] and later considered by Duffin [8], Dunninger [10], Hersch [12, 13], Hersch and Payne [14], Ogawa and Protter [23], Protter [27], and Protter and Weinberger [30]. In Chapter III we establish analogous results for some fourth order eigenvalue problems. Further lower bounds are established by comparing the lowest eigenvalue of fourth order problems with the lowest eigenvalues of related second order problems. These results are motivated by some recent works of Protter [28] and Hersch [13]. CHAPTER I MAXIMUM PRINCIPLES AND BASIC IDENTITIES I . 1 Maximum Principles In this paper we shall make use of some well known maximum principles (see e.g. [29], pp. 6-9) which for our purposes may be stated as follows. Let u be a classical solution of the differential in- equality (1.1) I u" + c(x)u 2 0 in a bounded interval (0,L). Let c(x) E C[O,L]. Principle I. Assume c(x) S 0 in (O,L). If u attains a nonnegative maximum value M fit an interior point of (0,L), then u E M. Principle II. Assume c(x) s 0 in (0,L). If u (non- constant) has one-sided derivatives at x = 0 and x = L and if u $.M. in (0,L) where M 2 0, then u'(0) < 0 if u(0) =‘M, whereas u'(L) > 0 if u(L) = M. Principle III. Assume there exists a classical solution g(x) of the differential inequality g" + c(x)g s 0 in (O,L) such that g(x) > 0 in [0,L]. Then the function u/g satisfies Principles I and II. Remark. Analogous results hold for solutions of u" + c(x)u S O in (0,L), yielding an associated minimum principle. These principles are obtained by applying the above results to the func- tion (-u). 1.2 Identities Various integral identities have been employed by several authors ([3], [10], [18], [20], [21]) to obtain Sturmian-type comparison theorems for fourth order differential equations. We shall now indicate how these identities, as well as a new identity, may be derived from a single integral identity. The method we use is a generalization of a technique used by Picard ([25], p. 151) in connection with certain problems for second order equations. For completeness we show that Picard's method readily yields Picone's identity .12 V") + (u' ~ u {-Vde. V u L (1.2) (uu' - u2 '37-)‘3 = £[ (uu" where for simplicity we have let a A E 1 in (0.1). To Green's identity L I 2 uu'I0 = €[uu" +-(u') ]dx we add the following identity for an arbitrary sufficiently smooth function P L Pu2]g = 3(2Puu' +'P'u2)dx and complete squares to obtain ,24,‘,, , 2 ,22 (uu + Pu )IO = £[uu + (u + Pu) + (P - P )u ]dx . Upon setting which incidentally is a solution to the Riccati-type equation we obtain the Picone identity (1.2). In order to derive the fourth order identities, we begin with the Green's identity x, L (4) 2 (1.3) (uu - u'u")\O = g[uu - (11") ]dx and add the obvious identity 2 0 2 2 0 {I _ L 2 0 0 2 0 (1.4) (Plu + P2(u) + P3uu ){0 — t[;[(P1u) + (P2(u ) ) + (2P3uu')']dx for arbitrary sufficiently smooth functions P1, P2, and P3. We thus obtain 11/ 2 2 L [uu - u'u" + Plu + P2(u') + 2P3uu']\0 (1.5) t = £[uu(4) - (u")2 +Piu2 + 2P1uu' + P2'(u')2 + ZPZU'U" 2 + 2P5uu' + 2P3(u') + 2P3uu"]dx . Guided by the form of the Picone identity (1.2) for second order equations we attempt to choose the functions P1, P2, and P3 so that the integrand in (1.5) will be analogous in form to that of (1.2). A. Cimmino's identity. Recalling that the Wronskian W(a1,az,...,an) of the n functions a1,az,...,an is defined as 0’1 0’2 an I I ! a1 02 CYr) W(al,az,--..On) = : E , “(01) = a1, (n-1) (n-1) (n-1) (1’1 (1'2 . . . an we observe that the integral in identity (1.2) can be written in the form L n u2 " WSv,u) 2 g {(uu - Gf'v ) + [ W(v) ] ]dx . It is then natural to seek a fourth order identity with an integral of the form I 2 (1.6) I) {(uu(4) - ”— vm) - [wax v where u is a nontrivial solution of (1.7) u“) - c(x)u = O in (0,1,), and y and v are linearly independent solutions of the boundary value problem (1.8) v(4) - C(x)v = 0 in (O,L) (L% \MD=V@)=WW)=WQJ=0- Observing that w(UTY9V) = U y V = u 10 we find, upon defining y V ylvl O: ytvt 3 I”: yuvn that W(u,y,v) = uw - u'a"+ u"o and W(y,v) = a . The integral (1.6) can thus be written (4) . Zvv )-(u"-cg,—U'+%u)2]dx. L (1.10) £[(uu(4) - u By rearranging terms, the right hand side of (1.5) can be written as . _ 2 , 2 . 2 u P3u) + (u ) (2P3 + P2 + P2) 2 2 I I I + 2uu (P2P3 + P1 + P3) + u (P1 + P3)]dx, and by comparing (1.11) with (1.10), we are led to choose I P -.Q_ U and P 4’1. O 3 It follows (see e.g. [21]) from (1.7), (1-3). (1-9), and the definitions of o and m that (1.12) a" - 2w 0 in (0 .4.) 11 2 2 II II . (1.13) (1) +00 +Ou> 'wo =0 m (0.0- In view of (1.12) we find that 2P +P'+P2-0 3 2 2“ ' Furthermore, the choice implies that '= P2P3+P1+P3 O , and then (1.13) implies 2 (4) 2 ow" - w'o"+ w v I _ r j- = = - P1 +P3 - 2 -C v 0 With the above choices of P1, P2, and P3 identity (1.5) reduces to [2(011/1/ _ LOU.) _ fl (out! _ O'U') _ B (LOU. _ uwl)]\{1 o o o 0 (1.14) 4. (4) . = £[(UU(4) _ U2 L.) _ (UH _ L U. + Q u)2]dx V o O which is the desired identity. This identity was first introduced by Cimmino in 1930 [3], and was recently found by Leighton [21] and Kreith [20]. B. Kreith's identity. Upon setting P3 = O, the right hand side of (1.5) reduces to 12 L (1.15) [[uu(4) - (u" - qu')2 + (u')2(P2' + 12:) + 29 15 Choosing 1uu' + Piu2]dx. v P2 - v' the expression (1.15) becomes L n III P V' (4) n I L2 L I I 2 (1.16) £[uu - (u -u v') +v' (u +—;nr-U) 2 v' 2 I _P __/_I; o 1+ (P1 1 v )u ]dx Hence, by the choice _ - v_”' Pl _ v (1.16) becomes L In m u 2 (4) (A) uv 2 v uv 2 u v £[uu - (u" - v' ) +-;T'(u' - v ) - v ]dx . With the above choices of P1 and P2 the identity (1.5) now yields Kreith's identity [18] e w - m -5; W - Wm: (1.17) {I (4) " III p - (4) 2 V ‘x_ 2 2__ uv 2 - £[(uu - u v ) - (u" - u' v') +Iv' (u' - -;-) ]dx. C. Dunninger's identity. Upon setting P2 = 0, the right hand side of (1.5) reduces to L (1.18) £[uu(4) - (u" - P3u)2 +2P3(u')2 +2(P1 + P$)uu' 2 2 13 Choosing the expression (1.18) becomes I I III L II I II gpiuc‘) - (u" - i)2 + 2 1,—-[u' + l- (P + :— - v ‘2, )u]2 v v 2v" 1 v (1.19) 2. 111?- ..v_ E’.’ ___V'V"2 +U[PI+(V) 'ZVII(P1+V ' v2)]}dx. Hence, by the choice P = _ (v'v" + vvé) l 2 v (1.19) becomes {1 II I (4) (4) u __uv2 .Y— ._uv2_2V ]Euu -( v>+2 ( v) ,]d With the above choices of P and P3, the identity (1.5) now yields 1 Dunninger's identity [10] ['3 (vum " LIV/fl) 'I' 3:: [3' (VU' - uv')] - $- (VUH _ UV")]\3 (1.20) (I 2 (4) II II I = g[(uu(4) - u Z ) - (u" - 2%")2 + 2 %—’(u' - u %—)2]dx. D. A new identity. If we now set P1 = P3 = 0 the identity (1.5) reduces to L (1.21) [uu,” - [1.11" + P2(U')2]‘3 = £[UU(4) _ (u" _ PZU')2 + (P: + PZ') (u')2]dx . Upon choosing we arrive at the following new identity 11/ V'SU'ZZ L L (4) V" 2 (1.22) [uu - u'u" +' v ]\0 = £[uu + ;—'(u') " IIZ -(u -3%)]dx. Remark: Although the identities, as they have been pre- sented, can only be applied to equations of the form u(a) ll 0 - c(x)u v(4) - C(x)v ll 0 they may easily be extended to cover the more general equations (1.23) (a(x)u")" + 2(b(x)u')' - c(x)u ll 0 (1.24) (A(X)V")" + 2(B(X)V')' - C(X)V II C We shall illustrate this point later on. In Chapter II we will show how Dunninger's identity (1.20) can be extended to an identity which is valid for systems of second order equations which contain the equations (1.23) and (1.24) as a special case. Moreover, we shall obtain Sturmian-type comparison theorems for such systems. The identity (1.22) (actually a slight generalization of it) will be used subsequently in connection with various oscillation problems (Chapter II) and eigenvalue problems (Chapter III). CHAPTER II COMPARISON AND OSCILLAIION II.1 Sturm-type Comparison Theorems In this section and the succeeding sections all solutions are assumed to be classical, and any coefficients appearing are assumed sufficiently differentiable. No further mention of these facts will be made. Consider the fourth order equations (2.1) (a(x)u")" + 2(b(x)u')' - E(x)u = o in (0,2) (2.2) (A(x)v")" + 2(B(x)v')' - C(x)v = o in (0,2). Upon setting e = i f =‘% b = g- d = % c=-(E+%'2'+b") h=-(C+%2-+B"), Whyburn showed in 1930 [33] that (2.1) and (2.2) were equivalent to the following systems u" +'bu = ew (2.3) in (0.L) w" + bw = -cu 15 16 v" + dv fz (2.4) in (0.4.) . -hv z" + dz respectively, when a and A, and hence e and f, never vanish in (O,L). In what follows we shall consider the systems (2.3) and (2.4) under somewhat less restrictive hypotheses. Namely, we do not require e and f to have fixed signs. Consequently, the systems (2.3) and (2.4) will be more general than the equations (2.1) and (2.2) To compare solutions of (2.3) and (2.4) we derive an identity which is closely related to Dunninger's identity (1.20). We could again use Picard's technique. However, proceeding more directly, and being motivated by the form of the left hand side of (1.20) we consider the following expression {5-(vw' - uz') + f [$‘(vu' - uv')] - §—-(vw - uz)]\3 (2.5) = 2_ u - I E, g_ t _ c _ 2;, _ u £{v (vw uz ) + v [v (vu uv )3 v (vw uz)} dx. Upon performing the differentiation under the integral, rearranging terms, and making use of (2.3) and (2.4) the right hand side of (2.5) becomes L I II II t£[uw" - u"w + ‘2-5 (u' - u 2")2 - u2 E—+ 2uu" _z_ - u2 u]dx v v v v v2 =L[(h-)2+2-z-(u'-u!;)2+2(d-b)2-z-- w2+2 B- S c u v v u v e an v u222 - f 2 ]dx. V 17 Then completing squares on the last three terms and substituting back into (2.5) we obtain the following formal identity for solu- tion pairs (u,w) and (v,z) of (2.3) and (2.4), respectively [3 (vw' - uz') + f; [5 (vu' - W'H ' 5" (W ' uznfi; (2.6) L = g{(h-c)u2 - e(w - $592 +22 V |N (u' - ”—jl’if + 2(d-b)u2 + (e-f) (:—§)2]dx. Theorem 2.1. Suppose there exists a function w such that u is a nontrivial solution of (2.3), and suppose there exists a function 2 such that v satisfies system (2.4) and v > 0 at some point in (0,L). If (1) f 2 e 2 0, d 2 b in (O,L), and either L 2 (2) £(C-h)u dx 2 0 (3) Z‘< 0 in (0:L) (4) u é kv, k a constant or 4' 2 (5) J; (c-h)u dx > o (6) 250 in (0,2,) then, under any boundary conditions on u, v, w, and 2 such that the left hand side of identity (2.6) is nonnegative, v must have a zero at some point in [0,L]. 18 Egggf. Suppose to the contrary that v has no zero in [0,L]. Since v > 0 at some point in (0,L), we have v > O in [0,L], and the identity (2.6) is valid. In view of the hypotheses (l), (2), and (3) and the fact that the left side of (2.6) is non- negative, we readily find that L . _z_ ._uv 2 (2.7) osgvm --v)dx. On the other hand hypothesis (3) and the fact that v > 0 in [0.1.] imply L . (2.8) gfim' -u—::-)2dxs0. Inequalities (2.7) and (2.8) together imply that 52-...1. v-2"_'= (v) V(u v) 0 which contradicts hypothesis (4). In a shmilar manner hypotheses (5) and (6) lead to the contradiction " 2 o s £(h-C)u dx < o . Hence, in either case, v must vanish at some point in [0,L]. Remark. Theorem 2.1 holds for a variety of general boundary conditions. 0f importance, from a physical point of view, we cite the following boundary conditions (I) U(0) = U'(0) = ML) " U'(L) 0 (II) u(0) 3 "(0) = 0(0 = WU.) " 0 19 which correspond to clamped ends and supported ends respectively in the boundary value problem for the vibrating rod (see [4], pp. 295-296). Remark. Under boundary conditions (I) or (II) the con- clusion of Theorem 2.1 is valid without hypothesis (4). Indeed, if v has no zero in [0,;3, then v > 0 in [0,L]. But u(0) = u(L) = 0, and hence u and v must be linearly independent. Our next results show that under some additional hypotheses, namely that the coefficient d is nonpositive, when the boundary conditions (I) or (II) hold, the conclusion of Theorem 2.1 can be sharpened to assert that v must have a zero in the open interval (0.L)o Theorem 2.2. Suppose there exists a function w such that u is a nontrivial solution of (2.3), and suppose there exists a function 2 such that v satisfies system (2.4) and v > 0 at some point in (0,L). If (1) f2e20,02d2b in (0,1,) and either L 2 (2) £(c-h)u dx 2 o (3) 2<0 in (0.0 01' L 2 (4) £(c-h)u dx > o (5) z 5 0 in (09‘): and if u satisfies boundary condition (I) U(0) = U'(0) = U(L) = U'(L) = 0. then v must vanish in (0,L). Proof. Suppose v does not vanish in (0,L). Then v > 0 in (0,L), and by continuity v 2 O in [0,L]. Consequently the identity (2.6) is valid on the interval (0.8) where 0 < a < B < L. [u :l_ (W. _ uz!) + 3 [% (VU' - uv')] " :17" (W ' 112)]‘2 (2.9) B 2 22 Z a UV'Z TENN)" -e +23“ 'T’ + 2(d-b)u2 3+ (e-f)(:—z—)2}dx. To establish that (2.6) is valid in [0,L] we consider the various possible cases depending on the behavior of v at the boundary points. Case I. v(0) # 0 and v(L) # 0. In this case identity (2.6) is obviously valid in [0,L], and moreover the boundary term is zero. Hence the conclusion follows from Theorem 2.1. Case II. v(0) = 0 or v(L) = 0. Suppose v(0) = 0 and v(L) # 0. From the first equation of system (2.4) and the hypotheses (l) and (3), or (1) and (5), it follows that v" + dv = fz S 0 . 21 Since d s 0, Principle II implies v'(0) > 0. Hence an applica- tion of L'Hopital's rule yields I F 11m+3=11m+37=0 x~0 v x~0 v and consequently lim L1-(W' -u2')=0 (2.10)$ x—.o+" z u u u lfin '— C— (vu' - uv')] = lim -" lim (zu' -'- zv') = 0 X-°0+v V X“O+v 3040+ V u' __ l U n lim+--(vw-uz)—1im+(uw--uz)=0. \X-‘O V x-oo V Another application of L'Hopital's rule (if necessary) readily verifies that the integrand in (2.9) may be extended to a continuous function in the interval [0,L]. Therefore, it now follows that (2.9) is valid in [0,L], and moreover the boundary term is zero. Since u(L) = 0 but v(L) * 0, u and v are linearly independent. Proceeding as in Theorem 2.1 we infer that v must have a zero in (0,L). A similar argument is valid in the case v(0) # 0 and v(L) = 0. Case III. v(0) = v(L) = 0. By Principle II, v'(0) # 0 and v'(L) f 0. Since u'(0) = u'(L) = 0, u and v are linearly independent. The rest of the proof follows as in Case II. Corollary. Suppose there exists a function w such that u is a nontrivial solution of system (2.3), and suppose there exists a function 2 such that v satisfies system (2.4) and v > 0 at some point in (0,4). If 22 (1) f2e20,02d2b in (0,2,) L 2 (2) £(c-h)u dx 2 o (3') z < 0 at some point in (0,4,), and if u satisfies boundary condition (I) U(0) = u'(0) = U(L) = u'(L) = 0. then either v or 2 must vanish at some point in (0,L). Proof. If 2 does not vanish in (0,L), then by (3') z < 0 in (0,L), and the result now follows from Theorem 2.2. Remark. In Theorem 2.2 the hypothesis d s 0 was only used, in conjunction with Principle II,to obtain that v' is non- zero at an endpoint where v vanishes. However, this condition on the coefficient d can be removed if we assume the existence of a positive function Q satisfying the hypotheses of Principle III. That is, we assume there exists a function g > 0 in [0,L] such that g"'+ dg s O in (0,2). The following analysis shows that under this new hypothesis we can again conclude that v' is nonzero at endpoints where v is zero. Since 2 s 0, we have from the first equation in system (2.4) that v" + dv = £2 5 0. Then if, for example, v(0) = 0, by Principle III 23 (E)'\O > 0. Hence 0 < L(0)V'(0) - V(0)C.'(0) = V'LO) 2 (0) g (0) Q Consequently v'(0) > 0, and we can proceed to prove Theorem 2.2 as before. Theorem 2.3. Suppose there exists a function w such that u is a nontrivial solution of (2.3), and suppose there exists a function 2 such that v satisfies system (2.4) and v > 0 at some point in (O,L). If (1) f2e20,02d2b in (0,1,) (2) 2(0) = 2(4.) = 0 and either (3) hs0,haé0 in (0.4,) L 2 (4) £(c-h)u dx 2 0 01' (5) 1130 in (0,4,) ’v 2 (6) £(c-h)u dx > 0, and if u and w satisfy the boundary condition (11) u(0) = w(0) = ML) = WU.) = 0 then v must have a zero in (0,L). 24 Proof. Suppose v does not vanish in (0,L). Then v > O in (0,L), and v 2 0 in [0,L]. From the second equation in system (2.4) together with hypothesis (3) z" +'dz = -hv 2 0, i 0 in (0,L). Then hypothesis (2) and Principle I imply z < 0 in (0,L). Similarly under hypothesis (5) z" + dz = -hv 2 0 in (0,L), and hypothesis (2) and Principle I imply z S 0 in (0,L). The remainder of the proof consists of showing that (2.9) is valid in [0,4] from which the conclusion follows. The details are similar to those shown in (2.10), and hence are omitted. Note that here we only know that lim 2 " lim 2';- + ' + an v x~0 v exists and is finite. Hypothesis (2) is used to show that the boundary term vanishes. Remark. In the special case when the coefficients c,b, and e in (2.3) are identically equal to h,d, and f, respectively, in (2.4), the above theorems are separation theorems for the system (2.3). That is, under the given hypotheses, the zeros of linearly independent solutions of (2.3) separate each other. Remark. In the case f,e > 0 the systems (2.3) and (2.4) can be transformed back to the fourth order equations (2.1) and (2.2) by the substitutions indicated on page 15. Consequently, our results contain some recent results of Dunninger [10]. 25 Remark. Theorem 2.1 is also valid for the systems of dif- ferential inequalities u" + bu = ew in (0.!) uw" + buw 5 -Cu and II 2 zv + dvz 2 £2 in (0%.) vz" +-dvz 2 -hv Furthermore, if z < O in Theorem 2.2, and if h i 0 in Theorem 2.3, these theorems also hold for the above systems of inequalities. Remark. We can treat in the same manner nonlinear dif- ferential inequalities by allowing coefficients c(x,u,w), b(x,u,w), e(x,u,w), d(x,v,z), f(x,v,z), and h(x,v,z) as long as the co- efficients are continuous in x for all values of the other vari- ables. 26 11.2 Oscillation Results Under consideration now is the boundary value problem (2.11) (a(x)u")" - c(x)u = 0 in (O,L) (2-12) U(0) '-' U'(0) = ML) - U'OL) = 0 and the second order equation (2.13) (A(x)v') ' + C(x)v '3 O in (0,1,) . The identity (1.22) L II l”_ln _Y__'_ IZ‘L: (4) V__ l2 [uu u u + v (u ) ]‘0 £[uu + v (u ) I - (u" - u' 5—)21dx is readily generalized to the following identity. In I _ t u _A_‘_’_'_ 02 L [u(au ) au u +' v (u ) 1‘0 (2.14) vI [u(au")" + Sé%;1l(u')2 - A(u" - u' ;-)2 + (A-a)(u")2]dx. 05—16 In fact, in Picard's technique, identity (2.14) is derived merely by adding the Green's identity I." 2 [u(au ) - au'u"]‘0 = £[u(au")" - a(u") ]dx to the identity 'ZL L '2' L 1'2 2P 'nd P201) ‘08£[P2(U ) ] dx-£[P2(u) + zoux to obtain 27 [u(au")' - au'u" + P2(u')2]\g (2.15) 2 L 1P2 2 2 P2 2 = &[u(au")" - A(u" - K-'u') + (A-a)(u") + (Pi +-2r)(u') ]dx . Identity (2.14) follows upon setting Provided identity (2.14) is valid, the following observa- tion can be made. Remark. If u is a solution of (2.11), (2.12) and v is a solution of (2.13) with A.) 0, and if L 2 t[;(€=i'-t‘\)(tl") dx 2 0. then identity (2.14) reduces to the inequality 5 2 2 (2.16) 0 5 gm. - C(u') ]dx . Moreover, equality in (2.16) implies, from (2.14), that {a II 0=£A(u"-uv 2 v ) dx Then since A > 0, we conclude that I u v u" V '=v(‘"7'-> a o, ' kv for some constant k. and hence u We shall use inequality (2.16) to relate conjugate points of solutions to the second order equation (2.13) and solutions to the fourth order equation (2.11). 28 Definition. (cf. [34], p. 17). For the second order equa- tion (2.13), the first coniugate point n1(x0) of x = x0 is the smallest x1 > x0 such that there exists a nontrivial solution v of (2.13) with v(x(9 = v(x1) = 0 . Definition. (cf. [34], p. 82). For the fourth order equa- tion (2.11), the first conjugate point fi1(xo) of x = x0 is the smallest 21 >xO such that there exists a nontrivial solution u of (2.11) with u(x0) = u'(x0) = u(§ Theorem 2.5. Let u be a nontrivial solution of (2.11) such that h1(0) exists. Let v be a nontrivial solution of (2.13) such that “1(0) exists. If (1) A>0 in (0, ‘1‘)1(0)) 01(0) (2) (a-A)(u")2dx 2 0 01(0) 01(0) 2 2 (3) I cu dx 5 g C(u') dx 0 then 01(0) 2 01(0)- Eroof. Suppose h1(0) < no. Then v does not vanish in (0, h1(0)]. Although v(0) - 0, v'(0) # 0 since nontrivial solu- tions of second order linear equations have simple zeros. Hence, 29 + as before, an application of L'HBpital's rule as x a 0 establishes the validity of identity (2.14)- Consequently hypotheses (l) and (2) together with W) = u'(o> = umlw» = u'fixlmn = 0 imply inequality (2.16) holds for L = fi1(0). In view of hypothesis I (3) we have equality in (2.16), and hence u a kv. Therefore v(0) = V(Ih(0)) = 0, which contradicts the assumption that 01(0) < n1(0). Hence fi1(0) 2 “1(0). Inequality (2.16) can also be used to obtain a disconjugary result. Definition. (cf. [34], p. 17). The second order equation (2.13) is said to be disconjugate in [0,L), 0 < L s m, if n1(0) does not exist in [0,L). Definition-(cf. [1]). The fourth order equation (2.11) is said to be disconjugate in [0,L), 0 <.L s m, if fi1(0) does not exist in [0,4). Theorem 2.6. Suppose that (l) A > 0 in (0,4,) 5 2 (2) £(a-A)(u") dx 2 0 2 5 2 (3) 3 cu dx 5 & C(u') dx for all g E (0,L), where u is any solution of (2.11) in (0,L). If (2.13) is disconjugate in [0,L), then (2.11) is also discon- jugate in [0,L). 30 Proof. Suppose, to the contrary, that (2.11) is not dis- conjugate in [0,L). Then there exists a solution u of (2.11) such that for some fi1(0) E (0,L) (2.17) u = u'<0> = u<01<0>> = u'<fil<0>) = 0. Since any v which satisfies (2.13) is disconjugate in [0,L), v does not vanish in (0, h1(0)]. Again, by an application of L'Hfipital's rule at x = 0, identity (2.14) is valid in [0, fi1(0)]. The boundary condition (2.17) and hypotheses (l) and (2) imply inequality (2.16) holds in (0, fi1(0)). Moreover, in view of hypothesis (3) for g = fi1(0) we have equality in (2.16), and hence u' E kv. But then by (2.17) vw)=wmwn=o which contradicts the fact that v is disconjugate in [0,L). Therefore (2.11) must be disconjugate in [0,L). CHAPTER III EIGENVALUE PROBLEMS III.1 Barta-type Lower Bounds In this section we shall be concerned with obtaining Barta- type [2] lower bounds for the smallest eigenvalues of the follow- ing prob lems £1 - nu E (a(x)u")" - c(x)u - flu ' 0 in (0.1.) (I) U(0) = U'(0) ' U(L) = U'(t) = 0 in - Au 5 (a(x)u")" - C(X)u - Au - 0 in (0.L) (II) u(0) - n"(0) - U(L) ' U"(t) ' 0 (a(x)U")" + VU" ' 0 in (0.4) (III) MD) .. u'(0) um - u'((.) = 0. Problem (I): Let u be an eigenfunction which corresponds to the lowest eigenvalue “1 of problem (I). If in addition u is positive in (0,.L) , then the following Darts-type inequality is valid [9] (3.1) a 2 inf ‘1. 1 (0.4.) " 31 32 Here v is any positive function which satisfies the boundary condition in (I). The proof of (3.1) is quite simple. Indeed, from (I) L £1100: - 01106:: = 0. and hence an integration by parts yields L l[u(£v - 01v)dx = 0. But u > 0 in (0,L), and so (3.1) follows. It is of interest to note that in general it is not known whether the eigenfunction corresponding to “1 has a fixed sign in (O,L) (see [9]). Hence, using a suitable modification of Dunninger's identity (1.20), we will construct similar bounds with- out this assumption. If the Operator L is defined as Lv E (A(x)v")" - C(x)v, then we obtain the following formal identity which is a generaliza- tion of (1.20) (see [10]). {'3 [v(au") ' - u(Av") '] 4- 552 [3- (vu' - uv')] 1 L (3.2) + 3— (Auv" - avu")}‘: - {[(A-a) (u")2 + (c-C)u2]dx L II I II {a + t([2 51,!- (u' - 9%”)z - A(u" - 33-)23dx + g $- (vnm - uLv)dx . Theorem 3.1. Suppose there exists a function v satisfying (1) V > 0 I.“ (0 rt) 33 (2) v"_<.0 in (0,4,). If u is an eigenfunction corresponding to the lowest eigenvalue “1 of (I), and if A 2 0 in (0,4,) and L 2 2 (3) v[u] .=. t(Luv-mm") + (C-c)u ]dx 2 0, then Lv (3.3) 0 2 inf - . 1 (0.1.) " Proof. We first note that from hypotheses (l) and (2) and Principle II, it follows that if v(0) - 0, then v'(0) * 0. Similarly if v({,) 3 0, then v'(L) * 0. Consequently the validity of identity (3.2) can be established following the procedure in Theorem 2.2. In view of the above hypotheses (3.2) reduces to L L II I II 1.2 2 . AL ._ 1.2 .. 2.2 gm, v)urdx V[u] [EU v (u “v) -A(u -uv)]dx20. Hence, Lv sup (0 - —) 2 0 (0.1.) 1 V from which inequality (3.3) follows. Remark. The above theorem is a slight improvement of a recent result of Dunninger [10]. Namely, we have eliminated the hypothesis that v is positive at the boundary points. Remark. In comparing inequalities (3.3) and (3.1) we note that we have not only succeeded in removing the fixed sign condi- tion, but we have also removed the condition that v must satisfy 34 the boundary conditions in (I). However, we must pay for this by adding the requirement that v" s 0, and consequently, it is easily seen that the eigenfunction u is not an admissible function in the inequality (3.3). Remark. Although we are not able to obtain a complementary upper bound, it should be pointed out that upper bounds are usually much more readily found. For example, the Rayleigh quotient char- acterization (see [11], p. 393) of the eigenvalue L {[Md'fi - «92de fl = min , 1 L g (pde where the mininunlis taken over all functions m satisfying the boundary conditions in (I), readily yields good upper bounds for 01- Problem (II): Proceeding exactly as in Theorem 3.1 we can readily establish an analogous result for the eigenvalue problem (II). Theorem 3.2. Suppose there exists a function v satisfying (1) v > 0 in (0,L) (2) v"$0 in (0,4,), and suppose A 2 0 in (0,L). If u is an eigenfunction correspond- ing to the lowest eigenvalue A1 of problem (II), and if (3) vtu] 2 o, 35 then (3.4) A1 2 inf 1:1. (0.0 In the special case that c E 0 we can obtain the following complementary upper bound. IDQQEEE 3.3. Suppose there exists a function v satisfying (1) v > 0 in (O,L) (2) V(0) = V"(0) - m.) = V"(L) - 0 (3) V[v] S 0 . If u is an eigenfunction corresponding to the lowest eigenvalue A of problem (II) with c a 0 and a > 0, then 1 (3.5) A1 s sup :43. (0.4.) Proof. Interchanging the roles of u and v, a and A, and c and C in identity (3.2) we obtain the identity {‘3' [u(Av") ' - v(au") '] + a a: [if (uv' - vu')] 1'. (3.6) “ 4. + ‘3' (uLv - mock. O—wfi We first must establish that (3.6) is valid. To this end it suffices to show that u >'0 in (0,L). For then it readily follows from 36 (II), Principle I, and the fact that A is positive that 1 u" s 0 in (0,L). Hence the validity of (3.6) is established in the same way that (3.2) was established. To obtain the positivity of u we first note that problem (II) can be expressed as the composition of the two problems u" '- w/a (3.7) . “(0) ‘ U(L) ' 0 w" .3 A10 (3.8) w(0) - w(.{,) = 0. Let G(x,§) denote the Green's function associated with the boundary value problem y" " f(X) (3.9) v(O) - v(t) = 0. Then from (3.7) we have L . - 1%). (3.10) u(x> c 0, it follows from Jentzsch's Theorem [16] that u 2 0 in (0,L). By applying Principle I to (3.8) and then to (3.7), it is easily seen that u >’0 in (0,L). Returning to identity (3.6), in view of the hypotheses, we obtain L Lv 2 0 S £(v - A1)V dx from which (3.5) follows. Remark. Since u > 0 in (0,L), it follows that u is an admissible function in (3.5). Hence upon setting A a > 0, C E c - 0 in (3.5), equality holds and therefore A1 = inf { sup €23 (0.L) where the infimum is taken over the class of functions v satisfy- ing v > 0 in (0,L) and the boundary conditions V(0) = V"(0) = V(L) = V"(L) = 0- Moreover, since u" s 0 in (0,L), it follows that u is also an admissible function in (3.4). Hence upon setting A a a > 0, C E c ' 0 in (3.4), equality holds, and therefore A1 = 3“? { inf fl} (0.L) where the supremum is taken over the class of functions v satisfy- ing v > 0, v" s 0 in (0,L). Problem (III): For eigenvalue problem (III) we obtain the following result from the identity (2.14) which is repeated here 38 I c 2 L I I [u(au")' _ au'u" + A v-i—Lvu 1“: a. £[U(au")" + $13.)... (u')2 (3.13) I I 2 - A(u" - 9+)?! + (A-a) (u") ]dx. Theoremg3.4. Suppose there exists a function v satisfy- ing (1) v>0 in (0,4,) (2) v has at most simple zeros at 0 and L, and suppose A > 0 in (0,L). If u is an eigenfunction corre- Sponding to the lowest eigenvalue Y1 of problem (III), and if L 2 (3) £(a-A)(u") dx 2 0, then I I (3.14) v1 2 inf - 9%1— (0 .4.) Proof. From identity (3.13), whose validity is established in the usual fashion, it follows that L (M) 2 4 (Av')' 2 0 s £[U(au")" 4' v (u') ]dx - {EL-yluu'flx + v (u') ]dx. Applying Green's first identity (integration by parts) to the first term on the right we have L (Av')' , 2 0 S &[y1‘+ v ](u ) dx. Hence 2 inf [- $51,794] Y 1 (o .1.) 39 with equality only if u' 2 RV, as was shown in the remark on page 27. But since v > 0 in (0,L), u' a kv implies u' has a fixed sign in (0,L) which contradicts the boundary conditions u(0) ' u(L) = 0. Consequently the inequality is strict and (3.14) is established. 40 111.2 Comparison Theorems for Eigenvalues It is the purpose of this section to establish comparison theorems for the lowest eigenvalue “1 of the problem (a(x)0")" - fl p(x)u ' 0. :30!) > 0 in (0.1,) (3.15) M0) = U'(0) ' nu.) ' U'Q.) = 0 and the lowest eigenvalues )‘1 and “,1, which are known to be positive (see [4], pp. 292-295) , of the second order problems (A(x)v')' + )‘v - 0, A > 0 in (0,1,) (3.16) V(0) ' v0.) ' 0 and z" + “.2 - 0 in (0,4,) (3.17) 2(0) ' 2(L) - 0 . In the case a =-.: 1 problem (3.15) governs the vibration of a non- homogeneous rod with linear density p(x), and (3.17) governs the vibration of a homogeneous string. Theorem 3.5. Let 1.1 be the lowest eigenvalue of problem (3.16), and let v be a corresponding eigenfunction. Let “.1 be the lowest eigenvalue of the problem (3.17). If u is an eigen- function corresponding to the lowest eigenvalue “1 of problem (3.15) and " 2 £(a-A) (u") dx 2 0, 41 then where p0 = [:a:] p. Proof. It is well known (see [4], p. 452) that v > 0 in (0,4,), and moreover (cf. Theorem 2.5) v'(0) 2‘ 0 and v'(4,) I‘ 0. Hence identity (3.13), whose validity is established in the usual fashion, yields 4 2 f 2 (3.18) o s t[(119 u dx - plow) dx . From Rayleigh's principle (see [11], p. 393) it follows that (3.19) for sufficiently smooth functions (9 vanishing on the boundary. In particular, since u is an admissible function in (3.19) we have L 2 L 2 £(u') dx 2 “13 u dx which combined with (3.18) yields 4 2 0 ‘ 3 U (“19 " 41151)“: and thus 2. u (3.20) “1 2 .14. p0 42 If equality holds in (3.20), then equality must hold in (3.18). Consequently, by the remark on page 27, u' a kv. But then the positivity of v in (0,4) again contradicts the boundary con- ditions u(0) - u(4) - 0. Therefore we have u n >_1_1, 1 p0 Remark. If 0 < p s 1, then (3.20) reduces to and if A a 1 in (3.16), then ‘1 - pl, and we have the well known result (see [24]) 2 01 > “,1 . We wish to obtain similar lower bounds for {11 for various classes of density functions p. To this end, we return to the derivation of identity (2.14) on page 26. Instead of making the choice in identity (2.15), we leave the function P2 unspecified and consider the identity [0(w") I - a“In" + P2(u I)2] ‘3 (3.21) 2 P P I. . gnaw)" - A(u" - :3- u')2 + (:2- + P2')(u')2 + (A-a)(u")2]dx. 43 If u is an eigenfunction corresponding to the lowest eigenvalue “1 of problem (3.15), if L 2 gun-Ann") dx 2 o, and if identity (3.21) is valid and the boundary term vanishes, then we have 4. 2 4. 2 (3.22) o sgnlp u dx - £Q(x)(u') dx where Pi ‘Q-r+% If (Q >>0 and if a1 is the lowest (positive) eigenvalue for the problem (QZ')' +on '0 in (0.4.) (3.23) 2(0) = 2(L) ' 0. then by Rayleigh's principle 4 2 (')dx S£Q¢ “1 4 2 (M where m is any function vanishing on the boundary. In particular, since u is an admissible function, we have g-Qm') «smingudx , and inequality (3.22) becomes ’* 2 055a (019 'aIQNX . 44 Hence we obtain the formal inequality alQ (3.24) {11 2 inf —- . (0.2,) " Motivated by some recent works of Protter [28] and Hersch [13] in which a similar problem for second order equations was considered, we let T AV. 1 -_0__ .41. (3.25) P2 1' v + 2T , where v is an eigenfunction corresponding to the lowest eigenvalue )1 in problem (3.16), T is an arbitrary smooth positive function in (0,4,), and 1' a min 1'. o [0.4,] A simple computation yields Pi ' Q ' r + 1’5 1'" AT T I 12 1. (3.26) - - —9--1=+—-Q (;9 - lug—)2 - ALB—+33%)- 41' T ‘1' 2 101.1 _ 91152.3 1 M); T 4 2r ’ T which implies Q >10 if we impose the further condition that (AT')' 5 0 in (0,4,). Therefore, by the choice for P2 given in (3.25), and for 'r > 0, (A1")' 5 0 in (0,4,) inequality (3.24) becomes the 45 formal bound 0 T x A I 2 A I I (3.27) ()1 2 inf —1-[-9-T—1+—(1-%— - Lil-1. (02L) p 4T For choices of T motivated by Protter [28], we obtain the following lower bounds for n1. Theorem 3.6. Let 01 be the lowest eigenvalue and let u be the corresponding eigenfunction for problem (3.15), and let v be an eigenfunction corresponding to the lowest eigenvalue II > 0 of problem (3.16) with L 2 £(a-A)(u") dx 2 o . If T = l'> 0 p and if (3.28) [Afim'so in (0.4.), then we have the inequality or). (3.29) n >-1—1-+a 1 p0 1N1 + O'1N2 where “I > 0 is the lowest eigenvalue of problem (3.23) po 5 max p > 0 [0.1] l 1 2 N1 5 4. min Ap[(;)'] 2 0 [0.1,] N 2- -% max [A(-1-)']' 20. [0.1.] p 46 Proof. Since v > 0 in (0,4), it follows (cf. Theorem 2.5) that v'(0) # 0 and v'(4) # 0. Therefore, with the choice of P as given by (3.25), the validity of (3.21) follows as before, 2 and moreover, the boundary term vanishes. Hence the formal bound (3.27) is now valid. For T = l- we note that To = min T =.__ p [0.4] po and (3.27) becomes “141 (3.30) 01 2 p0 + alNl 'I' aZNZ . Equality holds in (3.30) only if we have equality in (3.26) and in (3.22). Equality in (3.26) implies T = To, a constant, while equality in (3.22) taken together with (3.21) yields P2 (3.31) u" - X"“' 2 o in (0.2)- Since T a To, from (3.25) we have simply that P = 5%- , and 2 hence (3.31) yields u' a kv for some constant k. But then the positivity of v in (0,4) contradicts the boundary conditions on u. Therefore the inequality in (3.30) is strict. Remark. For p a l, (3.26) shows that Q =‘11: and thus problem (3.23) is equivalent to problem (3.17). As a result, 01 ' HI, and (3.29) simplifies to the previous bound of Theorem 3.5 01 > A1411. Remark. 1n the case A a 1, condition (3.28) becomes (%)" s 0 which implies p" 2 0 in (0,4). In other words, the vibrating rod with governing equation (3.15) is assumed less dense near the center, but may be of large density near the ends. 47 By choosing T = log (i) we are led in a similar manner to the following result. Theorem 3.6. Let n be the lowest eigenvalue and let 1 u be a corresponding eigenfunction for problem (3.15), and let v be an eigenfunction corresponding to the lowest eigenvalue x1 > 0 of problem (3.16) with L 2 IE0141) (u") dx 2 o . If 1 T = log (-) > 0 where 0 < p < 1 p and if [A(1°g p) '1 2 O in (0 94;): then 0’1"1"0 “1N4 N5 01 > N + 2 ' + “1 1?- 3 N3 3 where a1 > 0 is the lowest eigenvalue of problem (3.23) T = min (-1og p) > 0 [0.4.] N3 5 max (-p 103 p) > 0 [0.4.] 2 N4 5'}; min 559.1. 2 0 [0.4.] p N a-él‘ min [A(log p)']' 2 o. [0.1.] 48 In a similar manner the lowest eigenvalue 0 of problem 1 (3.15) can be compared to the lowest eigenvalue XI of the more general second order problem (A(x)v')'-+ xa(x)v = o, A > o, o > o in (0,4) (3.32) 0 . II V(0) = V“) Choosing P2 as given by (3.25), and thus replacing Ll by Ala, (3.26) becomes -Q,_blfi-eggfi,ew_£ T 4T2 2T ’ and the formal inequality (3.27) becomes 0' TKO 12 11 012 inf _1.[_0..1_'_+_ALT_2)_.-.LA.'L)._] . T 4T 2T The validity of identity (3.21) is established in the usual fashion, and we obtain the following bound. Theorem 3.7. Let ()1 be the lowest eigenvalue and let u be a corresponding eigenfunction of prdblem (3.15), and let v be an eigenfunction corresponding to the lowest eigenvalue )1 > 0 of problem (3.32) with L 2 g (a-A)(u") dx 2 o . If T=§>0 in (0.4.). and [Mgrj'so in (0.1.). then 49 01 > C"1"0’11 + 0’1N6 + “1N7 where 011 > 0 is the lowest eigenvalue of (3.23) To ‘5 min g>0 [0.1.]9 2 pAUQ)‘ N6 521; min —2-2-2- 2 0 [0.4] o [A(Q)']' N7 2 - - max 2 0 . [0.4] ° CHAPTER IV REMARKS ON PARTIAL DIFFERENTIAL EQUATIONS In this chapter we shall comment on the extension of the preceeding results to the case of elliptic partial differential equations. We consider the following analogs of equations (2.1) through (2 .4) , namely , (4.1) A(a(x)Au) + 2 div(b(x) grad u) - c(x)u = 0 in G ax1+ax2 axn (4.2) A(A(x)Av) + 2 div(B(x) grad v) - C(x)v - 0 in G and the corresponding systems All + b(x)u = e(x)w (4.3) in G Aw + b(x)w = -c(x)u and AV 4' d(x)v f(x)z (4.4) in G A2 + d(x)z - -h(x)v. We assume all solutions are classical and all coefficients are suf- ficiently differentiable in a bounded domain G with piecewise smooth boundary I“ in n-dimens ional Euclidean space E“. 50. 51 Although the analogs of the identities used in Chapters 11 and 111 can be derived by Picard's technique, this method is rather cumbersome. In fact, in only two dimensions, Picard's method would involve the introduction of twelve arbitrary (sufficiently smooth) functions via the obvious identity 2 2 2 a); l[[(Pllu + P2111x + Pzzuy + P23uxuy + 2P31uux + 2P32uuy) 3“ 353cm 2 u +P uu +2P 3l‘uuy)a 2 2 + (Plzu + P ux 4- P25 y 26 x y 33qu + 2P 24 2 2 2 a ({IHPIIU + P21ux + P22uy + P23uxuy + 2(P31uux + P32uuy) ]x 2 2 2 +[P12u +P24ux+P u +P 25y 26uxuy + 2033qu + P34uuy)]y]dxdy . A more natural and enlightening derivation of the identities for partial differential equations involves making use of the known one dimensional identities. For example, if we write equa- tion (3.2) in differential form rather than in integral form and rearrange terms in the boundary, we obtain N 11 1 1 11 [U(aU")' - au'u" + 2Auu'-:-’,— - (3(9):?)— ... 52%.”. V '- u(au")" - u2 Liz-'3: + 2 5%".- (u' - 9-3;)2 - A(u" - 23:)2 + (A-a) (u")2. 2 . a_ . L__ .2 Letting D1 5x1 and D1.1 331331, (i,j 1,...,n), we readily find that D V D (AD V) AD VD v .11.. 2 11 .._1 11 Dj[uDj(aDiiu) - aDjuDuu + 2AuDju v -u (-L-‘-'-—+ v2 )] 2 D v D v . - 9... .11. - _.1._2 uDjj(aDnu) v D 1(ADuv) 4- 2A v (Dju u v) .1 Dv Dva u-zun u%'+u2 -i—:——-u—)+(A-a)D - uD “”11 11 11 v iiuDjju' 52 Summing over i and j and using the divergence theorem, we easily obtain Dunninger's identity [10] C {igliva'LzLL nam1+A$3[;(va:- 3%)] (4.5) 0 at some point in G. If (1) f2e20,d2b in G, and either (2) £(c-h)u2dx 2 o (3) z<0 in G (4) u i kv, k a constant or (5) £(c-h)u2dx > o (6) 2‘0 in G then, under any boundary conditions on u,v,w, and 2 such that the left hand side of identity (4.6) is nonnegative, v must have a zero at some point in E - c u 1‘. Proof. Suppose v has no zero in El Then v > 0 in G, and identity (4.6) is valid. In view of hypotheses (l), (2), and (3) and the fact that the left hand side of (4.6) is nonnegative, we readily find that (4.7) 1 0 s g '3 \grad u - u 8rTar-EI—‘ledx . 54 On the other hand hypothesis (3) and the fact that v > O in G imply (4.8) éi-‘grad u - u 3231\2dx S 0. Inequalities (4.7) and (4.8) together imply that