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ISSN 2074-1871 Уфимский математический журнал. Том 10. № 4 (2018). С. 123-129.
ON INVERSE SPECTRAL PROBLEM AND GENERALIZED STURM NODAL THEOREM FOR NONLINEAR BOUNDARY VALUE PROBLEMS
Ya. IL'YASOV, N. VALEEV
Abstract. In the present paper, we are concerned with the Sturm-Liouville operator
C[q\u := —u" + q(x)u
subject to the separated boundary conditions. We suppose that q e L2(0, ж) and study a so-called inverse optimization spectral problem: given a potential q0 and a value Xk, where к = 1, 2,..., find a potential q closest to q0 in the norm of L2(0, ж) such that the value Xk coincides with fc-th eigenvalue Xk(q) of the operator C[q\.
In the main result, we prove that this problem is related to the existence of a solution to a boundary value problem for the nonlinear equation
—u'' + q0(x)u = Xk и + ой3
with a = 1 or a = —1. This implies that the minimizing solution of the inverse optimization spectral problem can be obtained by solving the corresponding nonlinear boundary value problem. On the other hand, this relationship allows us to establish an explicit formula for the solution to the nonlinear equation by finding the minimizer of the corresponding inverse optimization spectral problem. As a consequence of this result, a new method of proving the generalized Sturm nodal theorem for the nonlinear boundary value problems is obtained.
Keywords: Sturm-Liouville operator, inverse optimization spectral problem, nodal theorem for the nonlinear boundary value problems.
Mathematics Subject Classification: 34L05, 34L30, 34A55
1. Introduction
In the present paper, we are concerned with the relations between the existence of solutions to the so called inverse optimization spectral problem ([5, 6]) for the Sturm-Liouville operator
C[q]u := —u" + q(x)u (1.1)
subject to the separated boundary conditions
w(0) cos a + w'(0) sin a = 0, (1.2)
и(ж) cos a + u'(-k) sin a = 0, (1.3)
and the existence of weak solutions to the nonlinear boundary value problems
—u" + q0(x)u = \u + 5u3, x e (0,^), u(0) cos a + u'(0) sin a = 0, (NPS)
и(ж) cos a + u'sin a = 0,
Ya. Il'yasov, N. Valeev, On inverse spectral problem and generalized Sturm nodal theorem
for nonlinear boundary value problems.
© Il'yasov, Ya., Valeev, N. 2018. Поступила 19 сентября 2018 г.
The second author was partially supported by RFBR grant no. 18-51-06002 Az-a.
with 8 = 1 and 8 = — 1.
We suppose that q E L2 := L2(0,n). Under these eonditions C[q] defines a self-adjoint operator on the Hilbert space L2(0,n) (see, e.g., [3, 8, 10]), so that its spectrum consists of an infinite sequence of eigenvalues ap(C[q]) := (Ai(g)}°=1 which can be ordered as follows:
\i{q) < X2(q) <____Furthermore, to each eigenvalue Xk(q), there corresponds a unique (up to
a normalization constant) eigenfunetion 0fc(q) with exactly k — 1 zeros in (0,t).
The inverse spectral problem consisting in recovering of the potential q(x) from a knowledge of the spectral data is a classical problem and, beginning with the celebrated papers by Ambartsumvan [1] in 1929, Borg in 1946 [2], Gel'fand & Levitan [4] in 1951, it received a lot of attention.
It is well known (see, e.g., [2, 4]) that the inverse spectral problem with only finitely many given eigenvalues {Xi}r[=1j m < have infinitely many solutions and in general is meaningless. However, if one assume that a certain information about the potential q is known in advance, for instance, an approximate function q0 of the potential q is given, then it is natural to consider the following m-parametric inverse optimization spectral problem: given a potential q0 and {Aj}™1; m < find a potential q closest to q0 in a prescribed norm such that A» = Xi(q) for all i = 1,... ,m.
In the present paper, we study the following 1-parametric variant of this problem: for k ^ 1 we consider the problem
P(k): given A E R and q0 E L2(0,n), find a potential q E L2(0,n) such that A = Xk(q) and
1^0 — q\\i? = inf{||®) — q\\L2 : A = Xk(q), q E L2(0,n)}.
Our main result is as follows
Theorem 1.1. Let q0 E L2(0,n) be a given potential, k ^ 1. Then
(1°) for any A E R, there exists a solution q to the inverse optimization spectral problem
P (k).
Furthermore,
(2°) for A < Afc (q0), there exists a non-zero weak solution us of {NP$) \$=1, and for A > A^ (q0), there exists a non-zero weak solution u$ of {NP$) |«s=-1 so that the following explicit formula holds
q = q0 — 8uj a.e. in (0,^).
(3°) The solution us(%) of {NP&), 8 = ±1 possesses exactly k — 1 roots in (0,^).
The case k = 1 with the zero Dirichlet boundary conditions has been studied in our recent papers [5], [6], where we proved the existence and uniqueness of solution q in the case A > \]_(q0). Furthermore, in this case, stronger result holds, namely: the uniqueness theorem for (NP$)\^=-1 is satisfied so that (NP$)\^=-1 possesses a unique positive solution for A > X]_(q0).
Remark 1.1. For X = (q0), problem P(k) becomes trivial since in this case q = q0.
Remark 1.2. Each eigenfunetion <pk(q) of C[q\, as well as each weak solution u E W^'2(0,'n) of (NPs), 8 = ±1, obeys AC[0, ^-regularity (see e.g. [10]J.
It is worth pointing out the following result, which in itself is notably important.
Lemma 1.1. Let k ^ 1. Then
(i): for each X ^ Xk(qo), problem (NPs)\^=1 has no non-zero weak solution with k — 1 or less roots in (0,^);
(ii): for each, X ^ Xk (qo), problem (NPs) \s=-1 has no non-zero weak, solut ion with k — 1 or more roots in (0,^) .
Theorem 1,1 and Lemma 1,1 imply the following corollary.
Corollary 1.1. Assume q0 E L2(0,n). Then
(1°) For each X E ap(C[q\), nonlinear boundary value problem (NPs)|<s=i has no non-zero weak solution with k — 1 or less roots in (0, n), and problem (NPs) |<s=-1 has no non-zero weak solution with, k — 1 or more roots in (0,^).
(2°) For ea ch X E (Xk-i(qo),Xk(qo)), k ^ 1, nonlinear boundary value problem (NPs)|^=i possesses an infinite sequences of distinct weak solutions (uls)r=k. Moreover, uls, I = k,... has exactly I — 1 roots in (0,^).
(3°) For ea ch X E (Xk (q0), Xk+1(q0)), k ^ 1, nonlinear boundary value problem (NPs) |^=-1 possesses at least k distinct weak solu,Hons (uls)f=v Moreover, uls, I = 1,... ,k has exactly I — 1 roots in (0,^).
Here we assumed that X0(q0) = — ro.
We emphasize that this result is nothing more than a generalization of the well known Sturm nodal theorem to the nonlinear problem. To the best of the authors' knowledge, such method of proving this statement was not explored previously.
The paper is organised as follows. Section 2 contains some preliminaries and the proof of Lemma 1,1, In Section 3, we prove Theorem 1,1,
2. Preliminaries
In what follows, we denote by (-, •) and || ■ \\L2 the scalar product and the norm in L2(0,n), respectively; W1,2(0,-k),W2,2(0,^) are usual Sobolev spaces with the norms
i
fK P'K \ 2 / i'^ i'^
|2J_ , / l„./|2 7 \ II__II _ / / I„.|2J„ , / l„.//|2.
/ n-K p-K \ 2 / V
|M|i =U \u\2dx + J \u'\2dx) , = f y \u\2dx + J \u"\2dx)
W^2 := W01,2(0,n) is the closure of CJ^(0,^) in the norm
Ml =[[ \Vu\2dx^J
In what follows, we assume that ||0fc(q)Hh2 = 1 k =1, 2,____
Proposition 2.1. Let k ^ 1 and the sequences (qj)°°=1 in L2(0,n) and (\A&(qj)\)°=^n R are bounded. Then the sequence (fa(qj)) is bounded in W2,2 (0,n).
Proof. We observe that the equation £[q]<frk (Qj ) = Xk (qj )fa (qj ) is equivalent to the following integral equality
fa (q3 )(x) = Xk (q3 ) Go (x,Ofa (q3 )(£ K - G0 (x,Ç )q3 (Qfa (q3 №<%, (2.1) Jo Jo
where G0(x,£) is the integral kernel of operator (£[0])-1. Since G0 G C[0,^] x C[0,^], identity (2.1) implies
Hfa (Qj) H ^ (Afc max\G0 (x,^)\ +max\G0 (x,^)\Hqj ^(o,*) ) Hfa (Qj ) H l2(O,^) ,
\ i,X J
where Ak = sup^ \Xk(qj)\. Now taking into account that the set Ho_j lk2(o,^) is bounded, we get
Hk(Qj)Hc[o,*j < cHfa(qj)HL2(o,n) for some C < independent of j = 1, 2,.... Since C[q]<frk(Qj) = Xk(qj)fa(qj), we hence get
/ \fak(q3)\2dx ^ \q3(x)<j)k(q3)\2dx + Ak \fa(q3)\2dx Jo Jo Jo
C'K P'K
(qj)llc[0M \0j(x)\2dx + AJ \fa(q3)\2dx<Ci < / 0 Jo
where C1 is independent of j = 1, 2,.... Hence, in view of that \\$k(qj)\\l2 = 1, we obtain
\\^k(qj)\\w2,2 <C2 <
where C2 does not depend on j = 1, 2,... □
Lemma 2.1. If B is a bounded set in L2, then the family of operators £[q] is uniformly below semi-bounded on L2 with respect to q E B, i.e.,
2 -1}.
— ^ < ^ ^ inf inf { (£[q\^>,^>) : ^ E L2
L2
Proof. We follow an approach proposed by Shkalikov in [9]. We write £[q]y = £[0}y + Qy, where £[0]y = —y"(x), Qy = q(x)y(x). Let a > 0 be a sufficiently large number. We introduce R(a) := (£[0] + aI)-1/2. Let us estimate the norm of the operator R(a)QR(a). For arbitrary f,g E L2(0,n), we have
(R(a)QR(a)f,g) = (QR(a)f,R(a)g). Denote by = (/)2, ^i(x) = sin(/x), I = 1, 2,... the eigenvalues and eigenfunctions of the operator £[0]. Then f = ^1= fi^i(x), g = 52^ 9j^j(x) in L2 and
9j fi
■Z] = 12 jrjv oo oo
(QR(a)f,R(a)g) =
i=1 3=1
si(a)sj (a)
(Q^i^j)
where si(a) = ^J+ a, I = 1, 2,... We observe that
)\ ^ max \^i(x)\ max (x)\ \q(s)\ds < C\\q\\L2, i,j = 1, ^ where C < <x> does not depend on i, j = 1, 2,... Hence,
oo oo
\(R(a)QR(a)f,g)\
Si(a)sj(a)
i=1 j=1
\(QMs )\
<
c m* EE ^
\g3 U
=1 si(^)si(a)
<
L2
\
3=1
\
£\ \ 2 fe wW
-C p(a)
L2 \\J \\L2
where
Therefore,
p(v) = E
=1 (*(«))2'
\ (R(a)QR(a)f,g) \ ^ Cp(a)\\q\\L2\\f \\L2 for all f,g E L2. (2.2)
We denote h = R(a)v for v E L2(0, n). Since \\q\\L2 is bounded on B and p(a) ^ 0 as a ^ we obtain that for sufficiently large a the relations
((£[0] + al + Q)h, h) = (v, v) + (R(a)QR(a)v, v) > 0
hold for all v E L2. Hence, for sufficiently large a and for any ^ E L2 such that \[^\\l2 = 1, there holds
(£[q]ip, ip) > —a (ip, ip) = —a > — <x, for all q E B. This completes the proof. □
Lemma 2.1 implies immediately the following corollary.
Corollary 2.1. If B is a bounded set in L2, then X1(q) ^ p, > —<x for all q E B, where ^ is independent of q E B.
1
Lemma 2.2. For k ^ 1, the map Xk(■) : L2(0,n) ^ R is continuously differentiable with the Frechet-derivative
1 r
D\k (g)(h)=„^, fak (q)hdx, for all q,h e L2. (2.3)
k(q)\\L2 jo
Proof. Since Xk(q) is isolated, Corollary 4.2 in [7] implies that Xk(q) is Frechet differentiable and (2.3) holds. By the analyticity property (see [8]), the map fak(■) : L2(0,n) ^ W2'2(0,n) is analytic. Due to the Sobolev theorem, the embedding W2'2(0,n) C L4(0,n) is continuous. Hence, the map fak(■) : L2(0,n) ^ L4(0,^) is continuous and therefore the norm of the derivative DX\(q) depends continuously on q e L2(0,n). This implies that Xk(q) is continuously differentiable in L2(0,n). □
Proof of Lemma 1.1. We shall give the proof only for (i). The proof of (ii) is similar.
Let A ^ Xk(q0), k ^ 1 and 8 = 1. We prove by contradiction, namely, we assume that (NP$) has a non-zero weak solution u with k — 1 or less roots in (0,^). We consider the identities
u" + (—q0 + A + u2)u = 0, fa + (—qo + Xk )fa = 0
We observe that —q0 + A + u2 > —q0 + Xk. However, by the Sturm Comparison Theorem this yields that u should has more than fak roots in (0,^) that is more than k — 1 roots in (0,^). This is a contradiction. □
3. Proof of the main result
We give the proof only for the case k = 2; the other cases can be proved in the same way. For the proof in the case k =1 and A > Ai(g0) see also [5], [6]. Let A* e R. We consider the following minimization problem
Qa* = inf[Q(q) : A* = \2(q), q e L2(0,n)}, (3.1)
where Q(q) := W^ — qWh, q e L2(0,n).
Let qj e L2(0, n), j = 1, 2,... be a minimizing sequence for this problem, i.e., A2(qj) = A* and
Q(qj) ^ Q\*. We observe that if Wqj Wt2 ^ then W?0 — qj W^2 ^ i.e., Q(q) is a coercive functional. Hence, the sequence qj is bounded in L2(0,n), and by the Banach-Alaoglu theorem there exists a subsequence, which we again denote by (qj), such that qj ^ q as j ^ weakly in L2(0,n) for some q e L2(0,^).
We consider the sequences of eigenfunctions (fa(qj)) and (fa(qj)). By assumption, A* = A2(qj) for all j = 1, 2,.... Furthermore, in view of that qj is bounded in L2(0, n), by Corollary 1.1 we infer that the sequence X1(qj) is bounded below. Therefore, since X1(qj) < A2(qj) = A*, for j = 1, 2,..., we conclude that IX1(qj )| is bounded. It follows from Proposition 2.1 that the sequences fa1(qj) and fa (qj) are bounded in W 2'2(0,^). In view of this, by the Sobolev embedding theorem there exist subsequences, which we again denote by fa(qj) and fa(qj), such that
Mqj) ^ fa2(qj) ^ fa* as j ^ (3.2)
strongly in W 1>2(0,^) and C 1[0,^] for some fa*,fa* e W01,2 (0,t) fl C 1[0,^]. We notice that, since Wfa1(qj))WL/2 = 1, Wfa(qj)) WL/2 = 1, for every j = 1, 2,..., it follows that fa\, fa2 = 0. Furthermore, we may assume, by passing to a subsequence if necessary, that X1(qj) ^ A* as j ^ <x> for some A* e R. Let m = 1, 2. Then
fam(q3 )=Xm(q3) G0(x,0(fam(q3 )(£) — M — / G0(x,e )q3 (0(fam(qj )(£) — M 00
P'K P'K
+ Xm(q3) Gc(x,eWm(Odt — Ga(x,Oq3(£R, Jo Jo
for each j = 1, 2,.... Hence, strong convergences (3.2) and the weak convergence qj ^ q in L2(0,n) imply
= Knl Go(x,Ofm(Odt - Go(x,OmWrn (№, m =1, 2, (3.3) oo
and therefore,
d2
- dx2 ^*m(x)+ V(x)^*m(x) = Am0m(X), X G (O^), m =1, 2. (3.4)
This means that (A*, 01) and (A2,0*) coincide with some eigenpairs of the operator C[q], i.e.,
Kn = ^ (q), fm = (q), m = 1,2, (3.5)
for some ii,i2 G N. Let us show that im = m for m =1, 2. By the Sturm comparison theorem (see e.g.,[10]) for each j = 1, 2,..., every eigenfunction (qj)(x) , m =1, 2 has exactly m — 1 roots. By strong convergences (3.2) in Cl[0,^] this yields that the limiting function has at most m — 1 roots. Hence, we get that i2 ^ 2 and il = 1, i.e., Al = Xl(q) is the principal eigenvalue of C[q].
Since (<frl(qj), <fi2(qj)) = 0 for all j = 1, 2,..., by passing to the limit we have (<frl, $2) = 0. Hence, = $2 and therefore,
«2 = 2, A2 = X2 (q).
Thus, q is an admissible point for minimization problem (3.1). Taking into consideration that the weak convergence qj ^ q in L2 imply
Q(q) < Qx*.
we obtain that Q(q) = Qx*. Therefore, q is a solution of (3.1). This concludes the proof of assertion (1°) in Theorem 1.1.
Let us prove (2°). Assume that A* = X2(q0).
Since Q and X2(q) are Cl-functionals in L2, the Lagrange multiplier rule implies
PlDQ(q)(h) + V2DX2(q)(h) = 0, Vh G L2, (3.6)
where such that + l^2l = 0. By (2.3) we therefore get
(—2^l(qo — q)+ V24>2(q))hdx = 0 for all h G L2, (3.7)
Jn
where \\^2(q)\\h2 = 1. Hence,
2^l(qo — q) = ^202($) a.e. in Q.
We observe that ^l = 0,^2 = 0. Indeed, if ^l = 0, then (q) = 0 a.e. in Q, which is a contradiction. Suppose = 0, then q0 = q a.e. in Q and consequently A* = A2(q0), which contradicts our assumption A* = A2(q0). Hence, we have
q = q0 — vtfrl(q) a.e. in Q, (3.8)
with some constant v = 0. Substituting this into the identity
—02 (q) + #2 (q) = \*<h (q),
we obtain
— <j>l(q) + qoh (q) = A* h (q) + v4>l(q). (3.9)
This means that u = lu12(q) satisfies (NP$) and q = q0 — 8u2 a.e. in Q with #=sign(^). In view of Lemma 1.1, we infer that 8 = 1 if A < Xk(q0), and 8 = —1 if A > Xk(q0). This concludes the proof of (2°).
The proof of (3°) follows immediately since u(x) = (q)(%) • W^Wl2 and by Sturm nodal theorem the eigenfunction <^k(q)(%) of C[q] has exactly k — 1 roots.
Proof of Corollary 1.1. For A = Xk (go), k = 1,..., by (i) in Lemma 1.1 we conclude that nonlinear boundary value problem (NP$)|<s=i has no non-zero weak solution with k — 1 or less roots in (0,^), whereas by (ii) in Lemma 1.1, problem (NP$^^^ has no non-zero weak solution with k — 1 or more roots in (0,^) and this implies (1°). □
Since X1(q) < X2(q) < ..., Assertions (2°), (3°) immediately follow Assertions (2°), (3°) in Theorem 1.1.
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Yavdat Ilyasov
Institute of Mathematics,
Ufa Federal Research Center, RAS,
450008, Ufa, Russia
Instituto de Matematica e Estatlstica.
Universidade Federal de Goias,
74001-970, Goiania, Brazil
E-mail: [email protected]
Nurmukhamet Valeev
Institute of Mathematics,
Ufa Federal Research Center, RAS,
450008, Ufa, Russia
Bashkir State University,
450076, Ufa, Russia
E-mail: [email protected]
