Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment of Mthemtics Pennsylvni Stte University University Prk, PA 1680 Emil: ryhm@mth.psu.edu Abstrct As prt of 00 VIGRE mini-course t University of Uth, we present properties of eigenvlues nd eigenfunctions for the second-order Sturm- Liouville boundry vlue problem. Most of our proofs re dpted from [1] nd re given using vritionl methods. Exmples re lso discussed. Introduction We define the Sturm-Liouville opertor L on bounded intervl [, b] vi Lu := (pu ) + qu, where p C 1 ([, b]), q C([, b]), p ν > 0, ν R, nd q 0. We re interested in the Sturm-Liouville Eigenvlue Problem (SLEP) with Dirichlet boundry conditions: Lu = λu, λ R (1) u() = u(b) = 0 () Definition 1. Vlues of λ for which (1),() hs nontrivil solution re clled eigenvlues nd nontrivil solution u corresponding to λ is clled n eigenfunction. The pir (λ, u) is clled n eigenpir for the SLEP (1), (). First, we present n exmple of how PDE cn led to SLEP. This exmple cn be found in []. 1
Exmple 1. In this exmple we use seprtion of vribles to show how we cn get solutions of the two dimensionl Lplce s eqution We look for solutions of the form From Lplce s eqution we get u xx + u yy = 0. u(x, y) = X(x)Y (y). X Y + XY = 0. Seprting vribles nd ssuming X(x) 0, Y (y) 0 we get X X = Y Y. Since the left hnd side of this eqution depends only on x nd the right hnd side depends only on y we get tht X (x) X(x) = Y (y) Y (y) = λ, where λ is constnt. This leds to the Sturm-Liouville differentil equtions X = λx, Y = λy. (3) It follows tht if X is solution of the first differentil eqution in (3) nd Y is solution of the second eqution in (3), then u(x, y) = X(x)Y (y) is solution of Lplce s prtil differentil eqution. Existence of minimizers Using stndrd vritionl methods of Lgrnge multiplier type, we look for minimizers of the functionl F [u] = b p(u ) + qu dx, whose corresponding Euler-Lgrnge eqution is given by the Sturm-Liouville eqution, over A = {u H 1 0 ([, b]) : so tht minimizer will yield the eqution b F [u] = λu, u dx = 1}
b pu φ + quφ dx = λ for ll φ in H0 1 ([, b]). The minimiztion will yield some suprising nd useful results bout the eigenvectors of the opertor. For our purposes we consider the minimiztion problem in closed subspce V of H0 1 ([, b]). First, to gurntee minimum, we need F [u] to be bounded below nd thus require the ssumption tht p(x) θ > 0 nd q(x) > 0. Then F [u] hs n infimum over A, which we cll α, nd minimizing sequence u k. To use our compctness results we would like the u k to be bounded in H0 1 ([, b]), but this follows from the Poincire inequlity nd coercivity of the functionl F, which we see in the chin of inequlities M F [u k ] = b b uφ p(u k) + qu k dx θ u k L + 0 C u k H 1 0 ([,b]). As H 1 0 ([, b]) is Hilbert spce (i.e. reflexive Bnch spce) u k H 1 0 ([,b]) u wekly for some u H 1 0 ([, b]). This u is our cndidte for the minimizer of F [u]. All is left to show is tht F [u] lim inf F [u k ] nd u A, i.e. u L = 1. These two conditions gurntee tht u is indeed minimizer in V. WLSC To develope this inequlity, we introduce the (nottion sving) symmetric, H 1 0 ([, b]) coervive, continuous, biliner form ssocited to F. And so, in the limit, (u, v) = b pu v + quv dx F [u k ] (u, u k ) (u, u). u is contined in V nd A α (u, u) (u, u) = F [u]. Since the u k re bounded in H 1 0 ([, b]), which is compctly embedded in L, we know of the existence of v L to which the u k converge strongly in L. We don t yet, however, know tht u nd v re the sme element. But for rbitrry w L, we hve (u v, w) (L ) (u, w) (u k, w) + (v u k, w), 3
where the left term cn be mde rbitrrily smll since wek convergence in H 1 0 ([, b]) implies wek convergnce in L nd the second term cn be mde rbitrrily by the Cuchy inequlity nd strong convergence in L. Since L is Hilbert spce, nd now (v u, w) = 0 for ny w L, we must conclude tht u = v. But this proves tht u A, becuse u k L v implies tht v = u L = 1. V is closed subspce for which we my pply Mzur s theorem which sserts tht closed, convex sets re wekly closed. It follows tht u V s well. Conclusion Combining the bove two results we find tht by WLSC F [u] lim F [u k ] = α nd becuse u is contined in A nd V, α = inf F [v] F [u]. v A,V We conclude tht F [u] = α nd tht u the minimizes F over A nd V. subsection*eigenpirs As discussed, the inductive minimizing of the functionl F over the subspces V i produces chin of functions nd their corresponding vlues under F, u 1, u,..., u i,... λ 1 λ... λ i = F [u i ]... We will show tht these u i nd λ i re ctully eigenpirs for the Sturm-Liouville opertor. To void lter confusion, we note tht the homogeneity of the functionl F llows us consider the following two problems s equivlent; min F [v] = λ i min v A,V i v V i F [v] v L = λ i. Thus, we will frequently see the inequlity F [u i ] λ i v dx, v V i, without hving to ssume tht v L = 1. Obtining Eigenpirs To obtin our eigenpirs, we inductively define sequence of closed subspces of H 1 0 ([, b]). Define A := {u H 1 0 ([, b]) : u L = 1}. Let V 1 := H 1 0 ([, b]) nd W 1 := V 1 A. 4
Minimizing F on W 1, we get u 1 W 1 nd Next, let λ 1 = inf {F (u) : u W 1 } = F (u 1 ). V :=< u 1 > nd W := V A. Minimizing F on W, we get u W nd Continue this process with λ = inf {F (u) : u W } = F (u ). V n :=< u 1 > < u >... < u n 1 > nd W n := V n A. Minimize F on W n to get the pir (λ n, u n ). Notice tht nd hence, Also, by construction, we get b Proof of eigenpirhood W 1 W W 3... W n W n+1..., λ 1 λ... λ n λ n+1... u n u k dx = δ n,k where δ n,k := { 1 if n = k 0 if n k If φ V i nd ɛ 0, then u i + ɛφ V i nd by definition of minimizer, (u i + ɛφ, u i + ɛφ) λ i (u i + ɛφ) dx. ɛ[(u i, φ) λ i u i φ dx] + ɛ [(φ, φ) λ i φ dx] 0. The second term is positive by definition of the minimum, nd by tking ɛ negtive nd smll enough, so tht the first term domintes the second, we must conclude (u i, φ) λ i u i φ dx = 0. Now, more generlly, for rbitrry ψ H0 1 ([, b]), ψ differs from φ in V i by liner combintion of u k for k = 1,..., i 1. But since (u i, u k ) = u i u k dx = δi k, the before lst equlity holds for ψ in plce of φ. This is exctly the wek formultion tht (u i, λ i ) form n eigenpir for the Sturm-Liouville opertor. 5
Properties of the eigenpirs Most obvious is tht the eigenvlues re ordered s result of the minimiztion process. More though is tht they pproch infinity. If this were not true, they would be bounded nd thusly the u i would be bounded. As result, the u i would L -converge strongly to n u with u dx = 1. (u u i ) dx 0. But (u u i ) dx = lim k (u k u i ) dx = when i k. This is contrdiction. Also, these eigenpirs re complete, in tht if (u, λ) is n eigen pir, then u = u i, λ = λ i for some i nd if λ i = λ j then i = j. These fcts follow from the following rgument. Suppose tht (u, λ) is n eigenpir. Then λ uu i dx = (u, u i ) = λ i uu i dx, (λ λ i ) uu i dx = 0 for ll i, so tht either u = 0 (the trivil cse) or λ = λ i for some i nd u u j for j i. Now suppose u / V i for the previously given i. Then u V j for some j < i nd thus we must conclude tht u minimizes F [u] over V j since we hve tht λ j λ i. But this mens tht there re two nontrivil minimizers of the function F over the sme spce, which, by the following clcultion, is contrdiction. Suppose u 1, u re L unit vectors both stisfing F [u 1 ] = F [u ] F [v] for ny v V. Then u1+u nd u1 u V, nd so F [u 1 ] F [ u 1 + u ] F [u ] F [ u 1 u ], p p(u 1) + qu 1 dx 4 (u 1 + u ) + q 4 (u 1 + u ) dx, p p(u ) + qu dx 4 (u 1 u ) + q 4 (u 1 u ) dx. Adding the two inequlities together, we find tht p p((u 1) + (u ) ) + q(u 1 + u ) dx ((u 1) + (u ) ) + q (u 1 + u ) dx, p ((u 1) + (u ) ) + q (u 1 + u ) dx 0. This is contrdiction since we ssume tht eigenvlues re nontrivil. 6
Eigenvectors Spn H 1 0([, b]) Not only does the Sturm-Liouville opertor hve n infinite spectrum of eigenvlues but the eigenvectors ctully form bsis for H0 1 ([, b]). For v H0 1 ([, b]), define n v n = (v, u i ) L u i. i=1 We first show tht v n form Cuchy sequence in H 1 0 ([, b]). Notice tht 0 (v v n, v v n ) = (v, v) n λ i c i (v, v) i=1 n λ i c i for ll n. Thus the sum converges, mking the til sum pproch zero. It follows from n C v n v m H 1 0 ([,b]) (v n v m, v n v m ) = λ i c i tht the v n re Cuchy sequence in H0 1 ([, b]). Now, it is not certin whether v n ctully converge to v or not, infct, ll we know is tht v n converge to some w in H0 1 ([, b]), but we will show tht the v n converge to v in L, which gives the desired result. Notice, we used the coercivity of the form, nd know we use the fct (u i, λ i ) re minimizing pir in V i. Notice tht (v v n )u i dx = 0 for i = 1,..., n so tht v v n V n+1. Thus, if we pick n lrge enough so tht λ n > 0 we hve (v v n ) dx 1 (v v n, v v n ) 1 (v, v). λ n λ n i=1 Since, λ n, the inequlity shows tht v n v in L. Eigenvlues re simple Suppose we hve two linerly independent eigenvectors u 1 nd u corresponding to some eigenvlue λ. Then, find rel numbers c 1 nd c not both zero so tht c 1 u 1() + c u () = 0. Define u(t) = c 1 u 1 (t) + c u (t). Then, u(t) is nontrivil solution to SLEP nd (λ, u) is n eigenpir due to linerity. From our boundry conditions, we see nd by choice of c 1, c, we hve u() = c 1 u 1 () + c u () = 0 u () = c 1 u 1() + c u () = 0. i=m 7
Since nd order liner initil vlue problems hve unique solutions, u 0. This contrdicts the liner independence of u 1 nd u. Hence, the eigenvlues re simple. Obtining simple eigenvlues depends on the Dirichlet Boundry conditions. This next exmple illustrtes how periodic boundry conditions cn yield eigenvlues tht re not simple. The following exmple cn be found in []. Exmple. Consider the following SLEP u = λu, (4) u( π) = u(π), u ( π) = u (π). (5) We let λ = µ where µ > 0. A generl solution of (4) in this cse is u(θ) = c 1 cos(µθ) + c sin(µθ), The first boundry condition gives us θ [ π, π]. c 1 cos(µπ) c sin(µπ) = c 1 cos(µπ) + c sin(µπ), which is equivlent to the eqution c sin(µπ) = 0. Hence the first boundry condition is stisfied if we tke µ = µ n := n, n = 1,, 3,. It is then esy to check tht the second boundry condition is lso stisfied. Hence λ n = n n = 1,, 3, re eigenvlues nd corresponding to ech of these eigenvlues re two linerly independent eigenfunctions given by u n (θ) = cos(nθ), v n (θ) = sin(nθ), θ [ π, π]. Finl comment The min reson we re interested in eigenpirs for the Sturm-Liouville opertor is we cn use them to solve equtions of the form We first solve the SLEP Lu = f, where f H 1 0 ([, b]). Lu = λu nd then write f = c i u i, which we cn do since the eigenvectors spn H 1 0 ([, b]). Then, define Then, u is solution to u := c i λ i u i for λ i 0. Lu = f. 8
References [1] G. Buttzzo, M. Giquint, nd S. Hildebrndt, One-dimensionl vritionl problems. An introduction. Oxford Lecture Series in Mthemtics nd its Applictions, 15. The Clrendon Press, Oxford University Press, New York, 1998. [] W. Kelley, A. Peterson, Topics in Ordinry Differentil Equtions. Prentice Hll, To be published. 9