Asymptotics for eigenvalues of a non-linear integral system

arXiv:0705.0192v1 [math.SP] 2 May 2007

Asymptotics for eigenvalues of a non-linear integral system D.E.Edmunds

J.Lang

February 1, 2008 Abstract Let I = [a, b] ⊂ R, let 1 < q, p < ∞, let u and v be positive functions with u ∈ Lp′ (I), v ∈ Lq (I) and let T : Lp (I) → Lq (I) be the Hardy-type operator given by Z x (T f )(x) = v(x) f (t)u(t)dt, x ∈ I. a

We show that the asymptotic behavior of the eigenvalues λ of the nonlinear integral system (f (x))(p) = λ(T ∗ (g(p) ))(x)

g(x) = (T f )(x)

( where, for example, t(p) = |t|p−1 sgn(t)) is given by ˆ n (T ) = cp,q lim nλ

n→∞

ˇ n (T ) = cp,q lim nλ

n→∞

„Z „Z

«1/r

, for 1 < q < p < ∞.

«1/r

, for 1 < p < q < ∞,

(uv)r dt I

(uv)r dt

I

Here r = 1/p′ +1/p, cp,q is an explicit constant depending only on p and q, ˆ n (T ) = max(spn (T, p, q)), λ ˇ n (T ) = min(spn (T, p, q)) where spn (T, p, q) λ stands for the set of all eigenvalues λ corresponding to eigenfunctions g with n zeros.

1

Introduction and preliminaries

Through this paper we shall assume I = [a, b], where −∞ < a < b < ∞, and let p, q ∈ (1, ∞), (x)(p) := |x|p−1 sgn (x), x ∈ R and 1/p′ = 1 − 1/p. Let u and v be positive functions on I, with u ∈ Lp′ (I), v ∈ Lq (I). Define the Hardy-type operator T : Lp (I) → Lq (I) by Z x f (t)u(t)dt, x ∈ I. (T f )(x) = v(x) a

Such maps have been intensively studied: see [4, Chapter 2]. 1

Since |I| = b − a < ∞, u ∈ Lp′ (I) and v ∈ Lq (I) then T is compact, see [5, chapter 2]. As more detailed information about the native of the compactness of a map is provided by its approximation, Kolmogorov and Bernstein numbers, much attention has been paid to the asymptotic behavior of these numbers for the map T . The analysis is decidedly easier when p = q, and an account of the situation in this case is given in [5]. For the case p 6= q we refer to [6], [7] in which a key role is played by the non-linear integral system: g(x) = (T f )(x)

(1.1)

(f (x))(p) = λ(T ∗ (g(q) ))(x),

(1.2)

and where g(q) is the function with value (g(x))(q) at x and T ∗ is the map defined Rb by (T ∗ f )(x) = u(x) x v(y)f (y)dy. The non-linear system (1.1) and (1.2) gives us the following non-linear equation: (f (x))(p) = λT ∗ ((T f )(q) )(x). (1.3) This is equivalent to its dual equation: (s(x))(q′ ) = λ∗ T ((T ∗ s)(p′ ) )(x).

(1.4)

And we have this relation: For given f and λ satisfying (1.3) we have s = (T f )(q) and λ∗ = λ(p′ ) satisfying (1.4), and for given s and λ∗ satisfying (1.4)we have f = (T ∗ s)(p′ ) and λ = λ∗(q) satisfying (1.3). By a spectral triple will be meant a triple (g, f, λ) satisfying (1.1) and (1.2), where kf kp = 1; (g, λ) will be called a spectral pair; the function g corresponding to λ is called a spectral function and the number λ occurring in a spectral pair will be called a spectral number. For the system (1.1) and (1.2) we denote by SP (T, p, q) the set of all spectral triples; sp(T, p, q) will stand for the set of all spectral numbers λ from SP (T, p, q). It can be seen that this non-linear system is related to the isoperimetric problem of determining sup kgkq ,

(1.5)

g∈T (B)

where B := {f ∈ Lp (I) : kf kp ≤ 1}. Moreover, this problem can be seen as a natural generalization of the p, q−Laplacian differential equation. For if u and v are identically equal to 1 on I, then (1.1) and (1.2) can be transformed into the p, q−Laplacian differential equation:  ′ − (w′ )(p) = λ(w)(q) , (1.6)

with the boundary condition

w(a) = 0. 2

(1.7)

If g, f and λ satisfy (1.1) and (1.2) then, the integrals being over I, Z Z Z |g(x)|q dx = g(g)(q) dx = T f (x)(g)(q) dx Z Z = f (x)T ∗ (g)(q) dx = λ−1 f (x)(f )(p) Z = λ−1 |f (x)|p dx. From this it follows that λ−1 = kgkqq /kf kpp and then for (g1 , λ1 ) ∈ SP (T, p, q) −1/q

we have λ1 = kg1 kq . Given any continuous function f on I we denote by Z(f ) the number of o

distinct zeros of f on I, and by P (f ) the number of sign changes of f on this interval. The set of all spectral triples (g, f, λ) with Z(g) = n (n ∈ N0 ) will be denoted by SPn (T, p, q), and spn (T, p, q) will represent the set of all correˆ n = max spn (T, p, q) and λ ˇn = min spn (T, p, q). sponding numbers λ. We set λ ˆn can be determined Our main result is that the asymptotic behavior of the λ when 1 < q < p < ∞: we show that ˆ n (T ) = cp,q lim nλ

n→∞

Z

I

r

(uv) dt

1/r

,

where r = 1/p′ + 1/q and cpq is a constant whose dependence on p and q is given ˇn when 1 < p < q < ∞. Moreover, explicitly. A corresponding result holds for λ ˆn = λ ˇ n = λn say, spn (T, p, p) contains exactly one element, so that in this case λ and the asymptotic behavior of the λn is given by the formula above. We now give some results to prepare for the major theorems in §2 and §3. Lemma 1.1 Let f 6= 0 be a function on [a, b] such that T f (a) = T f (b) = 0. Then P (f ) ≥ 1. Proof. This follows from the positivity of T and Rolle’s theorem. Lemma 1.2 Let (gi , fi , λi ) ∈ SP (T, p, q), i = 1, 2, 1 < p, q < ∞. Then for any ε > 0, P (T f1 − εT f2) ≤ P (T f1 − ε(p−1)/(q−1) (λ2 /λ1 )1/(q−1) T f2 ).

(1.8)

If the function f1 −εf2 has a multiple zero and P (T f1 −ε(p−1)/(q−1) (λ2 /λ1 )q/(q−1) T f2 ) < ∞, then the inequality (1.8) is strict.

3

Proof. We will use Lemma 1.1 and the fact that sgn(a−b) = sgn((a)(p) −(b)(p) ). P (T f1 − εT f2) ≤ Z(T f1 − εT f2) ≤ P (f1 − εf2 ) ≤ P ((f1 )(p) − εp−1 (f2 )(p) ) ( use (1.3) for f1 and f2 ), ≤ P (λ1 T ∗ ((g1 )(q) ) − εp−1 λ2 T ∗ ((g2 )(q) )) ≤ Z(λ1 T ∗ ((g1 )(q) ) − εp−1 λ2 T ∗ ((g2 )(q) )) ≤ P ((g1 )(q) − εp−1 (λ2 /λ1 )(g2 )(q) ) ≤ P (g1 − ε(p−1)/(g−1) (λ2 /λ1 )1/(q−1) g2 ) ≤ P (T f1 − ε(p−1)/(q−1) (λ2 /λ1 )1/(q−1) T f2 ).

Theorem 1.3 For all n ∈ N, SPn (T, p, q) 6= ∅. Proof. This essentially follows ideas from [3] (see also [8]), but we give the details for the convenience of the reader. For simplicity we suppose that I is the interval [0, 1]. A key idea in the proof is the introduction of an iterative procedure used in [3]. Let n ∈ N and define   Xn+1 n+1 |zi | = 1 On = z = (z1 , ..., zn+1 ) ∈ R : i=1

and

f0 (x, z) = sgn(zj ) for

Xj−1 i=0

|zi | < x
1, then dn (T ) ≥ λ i=0 ˆ = max{λ ∈ ∪n spi (p, q)}. The iteration process from Proof. Let us denote λ i=0 the proof of Theorem 1.3 gives us for each k ∈ N and z ∈ On a function gk (., z). By the Makavoz lemma we have dn (T ) ≥ max min kgk (., z)kq,I . k∈N z∈On

(1.9)

Let us suppose that we have min lim kgk (., z)kq, = max min kgk (., z)kq .

z∈On k→∞

k∈N z∈On

(1.10)

Then from (1.9) and (1.10) it follows that ˆ−1/q , dn (T ) ≥ min lim kgk (., z)kq ≥ λ z∈On k→∞

since limk→∞ gk (., z) ∈ SP (T, p, q). We have to prove (1.10). From the monotonicity of kgk (., z)kq,I we have max min kgk (., z)kq = lim min kgk (., z)kq, . k∈N z∈On

k→∞ z∈On

From max min ≤ min max it follows that l := lim min kgk (., z)kq, = max min kgk (., z)kq k→∞ z∈On

k∈N z∈On

≤ min max kgk (., z)kq = min lim kgk (., z)kq, =: h z∈On k∈N

z∈On k→∞

Denote Hk (ε) = {z ∈ On ; kgk (., z)kq ≤ h − ε} where 0 < ε ≤ h. 6

Since the mapping z 7→ gk (., z) is continuous, Hk (ε) is a closed subset of On , and from the construction of the sequence gk we see that H0 (ε) ⊃ H1 (ε) ⊃ .... If y0 ∈ ∩k∈N Hk (ε) 6= ∅ then h = minz∈On limk→∞ kgk (., z)kq ≤ limk→∞ kgk (., y0 )kq ≤ h − ε is a contradiction. Then there exist k0 ∈ N such that Hk (ε) = ∅ for k ≥ k0 and minz∈On kgk (., z)kq ≥ h − ε for k ≥ k0 . Then we have that h = l and (1.10) is proved. Next we define Bernstein widths which will help us in section 3. The Bernstein widths bn (T ) for T : Lp (I) → Lq (I) when 1 < p, q < ∞ are defined by: bn (T ) := sup

inf

Xn+1 T f ∈Xn+1 \{0}

kT f kq,I /kf kp,I ,

where the supremum is taken over all subspaces Xn+1 of T (Lp (I)) with dimension n + 1. Since u and v are positive functions, the Bernstein widths can be expressed as  P n+1 kT i=1 αi fi kq,I , inf bn (T ) = sup P k n+1 Xn+1 α∈Rn \{0} i=1 αi fi kp,I

where the supremum is taken over all (n + 1)-dimensional subspaces Xn+1 = span{f1 , ..., fn+1 } ⊂ Lp (I). Now we use techniques from Theorem 1.3 to obtain an upper estimate for the Bernstein widths. ˇ−1/q , where λ ˇ = min(spn (p, q)). Lemma 1.6 If n > 1 then bn (T ) ≤ λ Proof. Suppose there exists a linearly independent system of functions {f1 , ..., fn+1 } on I, such that:  P n+1 kT i=1 αi fi kq,I ˇ−1/q . >λ min Pn+1 α∈Rn \{0} k i=1 αi fi kp,I Let us define the n−dimensional sphere ) ! n+1 ( n+1 X X αi fi kp,I = 1 . αi fi , k On = T i=1

i=1

Let g0 (.) ∈ On and define a sequence of functions hk (.), gk (.) = gk (., g0 ), k ∈ N, according to the following rule: gk (x) = T hk (x),

hk+1 (x) = (λk T ∗ (gk (x))(q) )(p′ ) ,

where λk > 0 is a constant chosen so that khk+1 kp,I = 1. We denote On (k) = {hk (., h0 ), h0 (.) ∈ On }. As in the proof of Theorem 1.3 we have: kgk kq,I is a nondecreasing as k ր ∞. For each k ∈ N there exists gk ∈ On (k) with n zeros inside I; limk→∞ gk (., g0 ) is an eigenfunction and there exists g0 (.)

7

such that limk→∞ gk (., g0 ) is an eigenfunction with n zeros. Moreover λk is monotonically decreasing as k ր ∞. P  n+1 Let α ∈ Rn+1 be such that: g0 (.) = α f is a function for which i i i=1 limk→∞ gk (., g0 ) is an eigenfunction with n zeros. Then we have the following contradiction:  P n+1 kT i=1 αi fi kq,I ≤ kg0 (.)kq,I min Pn+1 α∈Rn \{0} k i=1 αi fi kp,I ˇ−1/q , ≤ lim kgk (., g0 (.))kq,I ≤ λ k→∞

In the next two sections we obtain an upper estimate for Kolmogorov numbers and a lower estimate for Bernstein numbers. We shall need the approximation numbers an (T ) of T , defined by an (T ) = inf kT − F k, where the infimum is taken over all linear operators F with rank at most n − 1.

2

The case q ≤ p

We recall Jensen’s inequality (see, for example [9], p.133) which will be of help in the next lemma. Theorem 2.1 If F is a convex function, and h(.) ≥ 0 is a function such that R h(t)dt = 1, then for every non-negative function g, I Z Z F ( h(t)g(t)dt) ≤ h(t)F (g(t))dt. I

I

The following lemma give us a lower estimate for eigenvalues. b−1/q , where λ b = max(spn (p, q)). Lemma 2.2 If n > 1 then an (T ) ≤ λ

Proof. For the sake of simplicity we suppose that |I| = 1. b ∈ SPn (T, p, q). Denote by {ai }n the set of zeros of gb (with Let (b g , fb, λ) i=0 b a0 = a) and by {bi }n+1 i=1 (with bn+1 = b) the set of zeros of f . Set Ii = (bi , bi+1 ) for i = 1, ..., n and I0 = (a0 , b1 ), and define Tn f (x) :=

n X

χIi (.)v(.)

i=0

Z

ai

u(t)f (t)dt. a

Then the rank of Tn is at most n. We have (see [4, Chapter 2]) dn (T ) ≤ an (T ) ≤ supkf kp ≤1 kT f − Tn f kq . Let us consider the extremal problem: sup kT f − Tn f kq . kf kp ≤1

8

(2.1)

We can see that this problem is equivalent to sup{kT f kq : kf kp ≤ 1, (T f )(ai ) = 0 for i = 0 . . . n}

(2.2)

Since T and Tn are compact then there is a solution of this problem, that is, the supremum is attained. Let f¯ be one such solution and denote g¯ = T f¯. We can choose f¯ such that g¯(t)b g (t) ≥ 0, for all t ∈ I. We have k¯ gkq,I ≥ kb gkq,I Note that for any f ∈ Lp (I) such that T f (ai ) = 0 for every i = 0, ..., n we have T f (x) = T + f (x) for each x ∈ I, where +

T f (x) :=

Z

K(x, t)f (t)dt = I

and K(x, t) :=

n X

χIi (.)v(.)

i=0

n X

Z

x

u(t)f (t)dt a

χIi (x)v(x)u(t)χ(ai ,x) sgn(x − ai ).

i=0

ˆ q , where λ ˆ = kˆ Set s(t) = |ˆ g(t)|q λ gkq,I . Then, all integrals being over I, we have Z

q

|¯ g (t)| dt

1/q

ˆ −1/q =λ

Z

1/q g¯(t) q dt s(t) gˆ(t)

(use Jensen’s inequality, noting that

Z

s(t)dt = 1)

1/p g¯(t) p dt s(t) gˆ(t) 1/p + Z T f¯(t) p ˆ −1/q dt =λ s(t) gˆ(t) R 1/p Z K(t, τ )f¯(τ )dτ p ˆ −1/q dt =λ s(t) gˆ(t) Z !1/p Z K(t, τ )fˆ(τ ) f¯(τ ) p −1/q ˆ s(t) =λ dτ dt gˆ(t) fˆ(τ )

ˆ −1/q ≤λ

Z

(use Jensen’s inequality, noting that ! Z K(t, τ )fˆ(τ ) K(t, τ )fˆ(τ ) ≥ 0 and dτ = 1 gˆ(t) gˆ(t) p !1/p Z Z ˆ(τ ) f¯(τ ) K(t, τ ) f ˆ −1/q s(t) ≤λ dτ dt fˆ(τ ) gˆ(t)

9

!1/p Z Z ¯ p K(t, τ )s(t) f (τ ) ¯ =λ dtdτ f (τ ) fˆ(τ ) gˆ(t) !1/p Z Z ¯ p q K(t, τ )|ˆ g (t)| f (τ ) ˆ ˆ −1/q λdtdτ =λ f¯(τ ) fˆ(τ ) gˆ(t) !1/p Z Z ¯ p f (τ ) ¯ −1/q ˆ ˆ g(q) (t)dtλdτ =λ f (τ ) K(t, τ )ˆ fˆ(τ )   Z q ∗ ˆ ˆ ˆ use K(t, τ )ˆ g(q) (t)dtλ = λT (ˆ g(q) )(t) = f(p) (t) ˆ −1/q

ˆ −1/q

=λ

!1/p Z ¯ p f (τ ) ˆ ˆ f (τ )f(p) (τ )dτ fˆ(τ )

( use fˆ(t)fˆ(p) (t) = |fˆ(t)|p ) Z 1/p p −1/q ¯ ˆ ˆ −1/q . f (τ ) dτ =λ =λ

ˆ−1/q . From this it follows that an (T ) ≤ λ Theorem 2.3 If 1 < q ≤ p < ∞, then ˆ −1/q = cpq lim nλ n

n→∞

Z

1/r

|uv|

dt

I

r

ˆn = max(spn (p, q)) and where r = 1/p′ + 1/q, λ ′

cpq =

(p′ )1/q q 1/p (p′ + q)1/p−1/q 2B(1/q, 1/p′)

(2.3)

(B denotes the Beta function). Proof. From [7] we have lim nan (T ) = lim ndn (T ) = cpq

n→∞

n→∞

Z

1/r

|uv|

I

dt

r

and since dn (T ) ≤ an (T ), an (T ) ց 0 and dn (T ) ց 0 then from Lemma 2.2 follows: Z r ˆ −1/q , cpq |uv|1/r dt ≤ lim inf nλ n n→∞

I

and from Lemma 1.5 we have

ˆ −1/q ≤ cpq lim sup nλ n n→∞

which finishes the proof. 10

Z

I

1/r

|uv|

dt

r

3

The case p ≤ q

ˇ −1/q , where Lemma 3.1 Let 1 < p ≤ q < ∞ and n > 1. Then bn (T ) ≥ λ ˇ λ = min(spn (p, q)). ˇ from SPn (T, p, q) Proof. We use the construction of Buslaev [2] Take (ˇ g , fˇ, λ) and denote by a = x0 < x1 < ... < xi < ... < xn < xn+1 = b the zeros of gˇ. Set Ii = (xi−1 , xi ) for 1 ≤ i ≤ n + 1, fi (.) = fˇ(.)χIi (.) and gi (.) = gˇ(.)χIi (.). Then T fi = gi (.) for 1 ≤ i ≤ n + 1. Define Xn+1 = span{f1 , ...fn+1 }. Since the supports of {fi } and {gi } are disjoint, then we have  P n+1 P kT i=1 αi fi kq,I k n+1 i=1 αi gi kq,I bn (T ) ≥ inf = inf . P P n+1 n+1 α∈Rn \{0} α∈Rn \{0} k k i=1 αi fi kp,I i=1 αi fi kp,I We shall study the extremal problem of finding P k n+1 i=1 αi gi kq,I . inf Pn+1 α∈Rn \{0} k i=1 αi fi kp,I

It is obvious that the extremal problem has a solution. Denote that solution by α ¯ = (¯ α1 , α ¯ 2 , ...). Since p ≤ q, a short computation shows us that α ¯i 6= 0 for every i, moreover we can suppose that the α ¯ i alternate in sign. Label P k n+1 ¯ i gi kqq,I i=1 α γ¯ := Pn+1 ; ¯ i fi kpp,I k i=1 α Pn+1 then the solution of the extremal problem is given by g¯ = ¯ i gi , f¯ = i=1 α Pn+1 ¯ ¯i fi where kf kp = 1. i=1 α Pn+1 Let us take the vector β = (1, −1, ...). Define the functions g˜ = i=1 βi gi , Pn+1 f˜ = i=1 βi fi . Then P q k n+1 i=1 βi gi kq,I −1 . λn := Pn+1 k i=1 βi fi kpp,I

It is obvious that γ¯ ≤ λ−1 ¯ < λ−1 . n . Suppose that γ −1 Since α ¯i 6= 0, |βi | = 1 and γ¯ < λ then 0 < ε∗ := min1≤i≤n+1 (βi /α ¯ i ) < 1. From Lemma 1.2 follows 1/(q−1) γ /λ−1 T (f¯)). P (T (f˜) − ε∗ T (f¯)) ≤ P (T (f˜) − ε∗ (p−1)/(q−1) (¯ n )

By repeated use of Lemma 1.2 with the help of (ε∗ )(p−1)/(q−1) ≤ ε∗ < 1 and γ¯ /λ−1 < 1 we get ¯ ≤ P (T (f˜)) = n. P (T (f˜) − ε∗ T (f)) On the other hand we have from Lemma 1.1 and the definition of ε∗ that n+1 X

P (T (f˜) − ε∗ T (f¯)) ≤ P (f˜ − ε∗ f¯) = P (

i=1

11

βi f i − ε ∗

n+1 X i=1

α ¯ i fi ) ≤ n − 1,

which contradicts γ¯ < λ−1 . Theorem 3.2 If 1 < p ≤ q < ∞ then ˇ −1/q = cpq lim nλ n

n→∞

Z

r

|uv| dt

I

1/r

ˇn = min(spn (p, q)) and cpq as in (2.3). where r = 1/p′ + 1/q, λ Proof. From [6] we have lim nbn (T ) = cpq

n→∞

Z

|uv|r dt I

1/r

and since bn (T ) ց 0 then from Lemma 1.6 it follows that cpq

Z

|uv|r dt

I

1/r

ˇ −1/q . ≤ lim inf nλ n n→∞

Moreover, from Lemma 3.1 we have ˇ −1/q ≤ cpq lim sup nλ n n→∞

Z

r

|uv| dt

I

1/r

which finishes the proof. When p = q the following lemma follows from Theorem 2.3 and Theorem 3.2 (we can find this result in a sharper form in [1]). Remark 3.3 When p = q then lim nλn

n→∞

−1/q

= cpq

Z

I

r

|uv| dt

1/r

where r = 1/p′ + 1/q, cpq as in (2.3) and λn is the single point in spn (p, q).

References [1] C. Bennewitz, Approximation numbers = Singular values, Journal of Computational and Applied Mathematics, to appear. [2] A.P. Buslaev, On Bernstein-Nikol’ski˘ı inequalities and widths of Sobolev classes of functions, Dokl. Akad. Nauk 323 (1992), no. 2, 202-205. [3] A.P. Buslaev and V.M. Tikhomirov, Spectra of nonlinear differential equations and widths of Sobolev classes. Mat. Sb. 181 (1990), 1587-1606; English transl. in Math. USSR Sb. 71 (1992), 427-446. [4] Edmunds, D.E. and Evans, W.D., Spectral theory and differential operators, Oxford University Press, Oxford, 1987. 12

[5] D.E. Edmunds and W.D. Evans, Hardy operators, function spaces and embeddings, Springer, Berlin-Heidelberg-New York, 2004. [6] D.E. Edmunds and J. Lang, Bernstein widths of Hardy-type operators in a non-homogeneous case, to be published in Journal of Mathematical Analysis and Applications. [7] D.E. Edmunds and J. Lang, Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case, Mathematische Nachrichten, 279 (2006), no. 7, 727-742. [8] Nguen, T’en Nam, The spectrum of nonlinear integral equations and widths of function classes, Math. Notes 53 (1993), no. 3-4, 424-429. [9] A. Kufner, O. John and S. Fucik, Function spaces, Noordhoff International Publishing, Leyden, (1977). Address of authors: D.E. Edmunds School of Mathematics Cardiff University Senghennydd Road CARDIFF CF24 4YH UK e-mail: [email protected] J. Lang Department of Mathematics The Ohio State University 100 Math Tower 231 West 18th Avenue Columbus, OH 43210-1174 USA e-mail: [email protected]

13