Graphs and Combinatorics (2018) 34:501–508 https://doi.org/10.1007/s00373-018-1889-5 ORIGINAL PAPER
Upper Bounds of the Eigenvalues Related to a Weighted Fractional p-Laplacian on Metric Graphs Mohammed El Aïdi1
Received: 13 September 2017 / Revised: 19 March 2018 / Published online: 28 March 2018 © Springer Japan KK, part of Springer Nature 2018
Abstract The main result of the article states explicit upper bounds of eigenvalues for a weighted fractional p-Laplacian operator on a connected metric graph with a finite total length. Keywords Fractional p-Laplacian operator · Metric graph · Sobolev spaces Mathematics Subject Classification Primary 35P15 · 58J50 · 35R02; Secondary 47B65 · 81Q35 · 46G12
1 Introduction Let us recall a result on upper bounds of the eigenvalues for a weighted p-Laplacian operator on G, a connected metric graph with a finite total length, i.e., |G| = |e| < ∞ where E(G) stands for the set of edges of G. An element of e∈E (G) e ∈ E(G) is identified to the real segment [0, |e|], where |e| represents the length of e. The topological definition of connected means path connectedness. Let V(G) be the set of vertices of G and Ev (G) be the set of all edges incident to v ∈ V(G), and the cardinal of Ev (G) is denoted by degv (G) and its called the degree of v, e.g., degv (G) = 1 when v ∈ ∂G. Therefore, the present author studied upper bound of the eigenvalues for the following nonlinear problem endowed with vertices conditions.
B 1
Mohammed El Aïdi
[email protected] Departamento de Matemáticas, Universidad Nacional de Colombia, sede Bogotá, Avenida carrera 30 número 45-03, Edificio 404, Bogotá, D.C., Colombia
123
502
Graphs and Combinatorics (2018) 34:501–508
−λ p u = V |u| p−2 u on E(G) (∗)G, p : p−2 u e (v) = 0, for all v ∈ V(G), e∈Ev (G) |u e (v)| such that λ is a positive number, p ∈ (1, ∞), V is a positive integrable potential on G, and u e (v) is the outward oriented derivative (also it is called the oriented evaluation) of u e (v) at the vertex v. We recall that on an edge e ∈ E(G), the p-Laplacian operator − p u := −∇ · (|∇u| p−2 ∇u) takes the form −(|u | p−2 u ) . Subsequently, he furnishes the following result on an explicit upper bound of λk ((∗)G, p ), the kth variational eigenvalue, for (∗)G, p , see Theorem 1.1 in [5]. Theorem 1.1 [5] If V is a positive integrable function on G and p ∈ (1, ∞), then we have |G| p−1 λk ((∗)G, p ) ≤ V (x)d x, for all k ∈ N\{0}, (1.1) kp G where d x is induced by the Lebesgue measure on the associated edges of G. The proof of this theorem is essentially based on using the following approximation result, see part i) of Theorem 1.2 in [9]. Lemma 1.2 [9] Let G be a metric compact graph and μ be a Borel finite sigma measure on G and p ∈ (1, ∞), and wr (x) = [r (x)]−1/ p , where r (x) is a measurable positive function on G. Then for any n ∈ N, there exists Pn , a Step(G)-valued linear operator on L 1, p (G) such that the rank of Pn is at most n and u − Pn u L p (G ,dμ) ≤
wr L p/( p−1) (G) [μ(G)]1/ p n+1 for all u ∈ L 1, p (G),
∇u L p (G ,r (x)d x) , (1.2)
such that u L p (G ,dμ) = ( G |u(x)| p dμ(x)). Step(G) represents the linear space of step functions, i.e., functions taking a finite number of different values, each one on a connected subset of G. L 1, p (G) represents the Banach Sobolev space constituted by C(G), the set of continuous functions u on G such that their distributional derivatives belong to L p (e)—the space of p-integrable functions on e. Theorem 1.1 generalizes a result of Solomyak, see Theorem 1.1 in [10]. Precisely, the author gives upper bounds of the eigenvalues of −λu = V u under Dirichlet conditions in one vertex and natural transition conditions in all remaining vertices. Consequently, the purpose of the present article is to furnish an analogous result of Theorem 1.1 for the case of a weighted fractional p-Laplacian operator, see below the announcement of Theorem 1.3. Precisely, for V a real-valued positive integrable function on G, λ a real positive number, and p ∈ (1, ∞), we consider the following nonlinear problem: λ(− p )s u = V |u| p−2 u in G (∗)G, p,s : u=0 in G\G,
123
Graphs and Combinatorics (2018) 34:501–508
503
such that (− p )s is the fractional p-Laplacian operator with the restricted condition s ∈ ( 1p , 1), λ is a positive number, and G is the graph with infinite total length. When the graph G is an edge identified to a bounded interval and G is an edge with infinite length and identified to the real line, then we fall on the eigenvalue problem for (− p )s in a real interval with a zero condition in the complement of the interval. As we observe that the study of upper bounds of eigenvalues associated to (∗)G, p,s the vertices conditions are not required. Our main result states Theorem 1.3 Let s ∈ ( 1p , 1) for p ∈ (1, ∞). If V is a positive integrable function on G, then we have |G| p( ps−1) Ds, p λk ((∗)G, p,s ) ≤ kp where Ds, p = 4
8( ps+1) ps−1
p
V (x)d x, for all k ∈ N\{0},
(1.3)
G
.
Remark 1.4 When G is a bounded edge, p = 2 and V = 1, we have an accurate upper bound, in contrast of that given, e.g., by the author Kwa´snicki [7] which furnishes asymptotic behaviour for eigenvalues. Also, our upper bound of the first eigenvalue is given explicitly compared with that furnished in Lemma 2.4 and Remark 3.2 of [3], as expected when G is reduced to an edge. The proof of Theorem 1.3 is furnished in the third section.
2 Preliminaries Without loss of generality, we assume that G is a simple graph, i.e., it has neither parallel nor loops edges [1]. Let Ls, p (G) be the Banach Sobolev space comprised of C(G), the set of continuous functions u on G, such that u e , the restriction of u on an edge e, belongs to W p,s (e), the fractional Sobolev space endowed with the classical Gagliardo semi-norm [·]e, p,s such that p
[u]e, p,s = e×e
|u(x) − u(y)| p d xd y < ∞. |x − y|1+sp
This semi-norm is identically equal to zero on C, the set of constant functions. Then, on the quotient space W s, p (e)/C, [·]e, p,s is a norm. As expected there is another way p p to obtain a norm, e.g., by taking (u L p (e) + [u]e, p,s )1/ p . We say that u ∈ L p (G) if u e belongs to L p (e) for e ∈ E(G), and the L p (G)-norm 1/ p 1/ p p = u . of u is defined by u L p (G) = G |u(x)| p d x e e∈E (G) L p (e) Consequently, for e an edge of G, the corresponding energy functional of (∗)e, p,s p is defined by Je (u) = λp e×e |u(x)−u(y)| d xd y − 1p e |u(x)| p V (x)d x for u ∈ |x−y|1+sp C(e) ∩ W s, p (e) = Ls, p (e). Accordingly, by using this restriction energy functional and the definition of L p (G)-norm, we get JG (u), the global energy functional
123
504
Graphs and Combinatorics (2018) 34:501–508
s, p (e). So, associated to (∗)G, p,s for u ∈ C(G) ∩ W s, p (G) = e∈E (G) C(e) ∩ W − p eigenvalues of (∗)G, p,s are the critical values of the C 1 -functional [u]G, p,s such that p u L p (G,V (x)d x) = G |u(x)| p V (x)d x = 1. Let G be the compactification of G, (we recall that G is not necessarily compact) and k we consider W s, p (G) = {u ∈ Ls, p (G) s.t. (u(xl ) = 0)1≤l≤k for (xl )1≤l≤k ∈ G }. For defining the kth eigenvalue for (∗)G, p,s , we need the following closed subspace s, p (G) := {u ∈ W s, p (G) s.t. u W L p (G,V (x)d x) = 1}. The Rayleigh quotient associated to (∗)G, p,s is equal to RG, p,s (u) = [u] p1 for G, p,s
s, p (G). u∈W Now, by using the minimax principle for a nonlinear eigenvalue problem, e.g., see [2] and Chapters 0 and 4 in [8], the kth variational eigenvalue for (∗)G, p,s is formally defined by
λk ((∗)G, p,s ) = inf
F∈Wk
max RG, p,s (u) ,
u∈F\{0}
such that s, p (G) s.t. K is a symmetric compact set, and γ (K ) ≥ k}, Wk = {K ⊂ W
and γ (K ) stands for the Krasnoselskii genus [4], i.e., γ (A) = inf{l ≥ 1, s.t. there is an odd continuous map from A to Sl−1 }, where A is a symmetric subset of a Banach space and Sl−1 is the unit sphere in Rl . As we observe that Wk+1 ⊂ Wk thus (λk ((∗)G, p,s ))k≥1 is an increasing sequence.
3 Miscellaneous Lemmas and the Proof of Theorem 1.3 The following theorem is the counterpart version of Lemma 1.2 for approximating a function in Ls, p (G) by a step function. Theorem 3.1 Let G be a compact graph, s ∈ ( 1p , 1) such that p ∈ (1, ∞). Then for any n ∈ N, there exists Pn , a Ls, p (G)-valued linear operator on Step(G), such that the rank of Pn is at most n and p
u − Pn u L p (G ,dμ) ≤ where Ds, p = 4
8( ps+1) ps−1
p
Ds, p |G| p(sp−1) μ(G)[u]G , p,s , for all u ∈ Ls, p (G), (3.1) (n + 1) p and μ(G) = G V (x)d x.
The proof of Theorem 3.1 is based on two relevant lemmas, see below Lemmas 3.2 and 3.6.
123
Graphs and Combinatorics (2018) 34:501–508
505
Lemma 3.2 Let (ξ, x) be a pair of vertices of an edge in G and s ∈ ( 1p , 1). Then for u ∈ Ls, p (G), we have p
|u(x) − u(ξ )| p ≤ Ds, p |G| p(sp−1) [u]G, p,s ,
(3.2)
such that Ds, p is an explicit constant depending on s and p. Proof Let e be an edge linked two vertices (ξ, x) ∈ G 2 such that b = |e| is the length of e, thus e is identified to the interval [0, b]. It has been shown, see Lemma in [6], that for v, a continuous function on [0, 1], we have
1
|v(1) − v(0)| ≤ 8
g
−1
0
4B y2
dh(y),
(3.3)
such that g, h are positive continuous increasing functions on [0, ∞) and B is a positive number at least equal to
[0,1] [0,1]
g
v(ζ ) − v(σ ) dζ dσ. h(ζ − σ )
For our purpose, let us choose g(y) = y p , h(y) = y
s+ 1p
(3.4)
, thus g −1 ( 4B )= y2
41/ p B 1/ p y 2/ p
1
1 s+ p −1 dy. Whence, by considering z 1 = bζ, z 2 = bσ and p )y p z1 1 )−u(z 2 )| v( b ), (3.4) becomes bsp−1 [0,b]2 |u(z dz 1 dz 2 . Now, by |z 1 −z 2 |sp+1
and dh(y) = (s +
putting u(z 1 ) = assuming that the vertex x (resp. ξ ) is identified to b (resp. 0), and using a simple integration taking into account that s > 1p , inequality (3.3) changes to |u(x) − u(ξ )| p ≤ 4
8bsp−1 (sp + 1) sp − 1
p e×e
|u(z 1 ) − u(z 2 )| p dz 1 dz 2 . (3.5) |z 1 − z 2 |sp+1
Therefore, we have p
p
|u(x) − u(ξ )| p ≤ Ds, p |e| p(sp−1) [u]e, p,s ≤ Ds, p |G| p(sp−1) [u]G, p,s , where Ds, p = 4
8(sp+1) sp−1
p
.
(3.6)
Apropos of the statement of the second lemma, first we need the definition of the limit of a family of subgraphs, i.e., closed connected subsets of G, see §2 in [10]. Definition 3.3 Let (E j )1≤ j≤l be a family of mutually disjoint subgraphs of G, and let (·) be a positive real-valued function on C (G), the set of subgraphs of G. We say that is super-additive if lj=1 (E j ) ≤ (∪1≤ j≤l E j ). Definition 3.4 Let S(G) be the set of super-additive functions on C (G), we say that a family (Er )r ∈N of increasing (resp. decreasing) in G converges w.r.t. to , subgraphs if limr →∞ (Er ) = (∪n∈N E n ) (resp. n∈N E n ), and by convention we put ({x}) = 0 for all x ∈ G.
123
506
Graphs and Combinatorics (2018) 34:501–508
Definition 3.5 Let ( l )1≤l≤r be a family from C (G), we say that ( l )1≤l≤r is an almost partition of V ∈ C (G), if V = ∪1≤l≤r l and ∩1≤l≤r l = ∅. Now, the second lemma is based on a known result, see Theorem 2.1 in [9], and stating. Lemma 3.6 [9] Let G be a compact metric graph and ∈ S(G). Then for any n ∈ N there exists a partition (E j )1≤ j≤k of G where each E j ∈ C (G) such that (E j ) = (E j ) ≤ (n + 1)−1 (G) for all j ∈ {1, 2, . . . , k} where k ≤ n and inf max1≤l≤r ( l, j ) is the infimum over all almost partitions ( j,l )1≤l≤r of E j . The proof of Lemma 3.6 is a little bit hard, also there is a version for a compact metric tree, see Theorem 2.1 in [10]. Then thanks to these two previous lemmas of this section, we are able to show Theorem 3.1.
3.1 Proof of Theorem 3.1 Let be a positive function on C (G) defined by (E) = |E| p(sp−1) E V (x)d x for E ∈ C (G), we notice that belongs to S(G), because by considering (E j )1≤ j≤k1 a partition of E into a family of subsets from C (G), we have k1 j=1
(E j ) =
k1
|E j |
p(sp−1)
Ej
j=1
≤ |E| p(sp−1)
k1 j=1 E j
V (x)d x
V (x)d x
V (x)d x = (∪1≤ j≤k1 E j ).
= |E| p(sp−1) E
Now, let (E j )1≤ j≤k be a partition of G into a family of subsets from C (G) such that E j = ∪1≤l≤r j,l and ∩1≤l≤r j,l = ∅, i.e., ( j,l )1≤l≤r constitutes an almost partition of E j . Thus, by considering ξ ∈ ∩1≤l≤r j,l = ∅ and applying (3.2) to each j,l , we obtain |u(x) − u(ξ )| p V (x)d x ≤ Ej
≤
r
sup |u(x) − u(ξ )| p
l=1 x∈ j,l r
Ds, p | j,l |( ps−1) [u] j,l p
p
l=1 p
≤ max ( j,l )Ds, p [u] E j . 1≤l≤r
123
j,l
V (x)d x
j,l
V (x)d x (3.7)
Graphs and Combinatorics (2018) 34:501–508
507
Now, by taking the minimum over all ξ ∈ ∩1≤l≤r j,l = ∅ and the infimum over all almost partitions of E j , we get a point x E j ,μ ∈ E j such that (3.7) becomes
p
Ej
|u(x) − u(x E j ,μ )| p V (x)d x ≤ inf max ( j,l )Ds, p [u] E j ,
(3.8)
1≤l≤r
(E j ) := inf max1≤l≤r ( j,l ) where the infimum is taken over all almost such that partitions of E j . We recall that the family (E j )1≤ j≤k forms a partition of G, thus let us consider the step function v(x) = kj=1 u(x E j ,μ )X E j where X E j stands for the characteristic function of E j . Whence, by applying inequality (3.8) and Lemma 3.6 to (and there is a partition with k ≤ n), we obtain G
|u(x) − v(x)| p V (x)d x ≤
k
(E j )Ds, p [u] p E j , p,s
j=1
(E j )Ds, p [u] p ≤ max G , p,s 1≤ j≤k
≤
p Ds, p |G|( ps−1) p [u] V (x)d x. G , p,s (n + 1) p G
(3.9)
The proof of Theorem 3.1 is complete by taking Pn (u) = v. 3.2 Proof of Theorem 1.3 Let (G m )m∈N be a family of increasing compact subgraphs of G, such that G = u s, p (G) such that u = ∪∞ 1 m=1 G m . Let u be an element of W [u]G, p,s belongs to the s, p (G), i.e., [u ] unit sphere in W 1 G, p,s = 1. Now, since that for each j ∈ {1, . . . , k}, x j ∈ G, then there is a sequence (x j,m )m∈N such that limm→∞ x j,m = x j such that x j,m ∈ G m and the fact that u 1 is a continuous function, we have |u 1 (x j,m )| = |u 1 (x j,m ) − u 1 (x j )| < k 1/ε p for ε a small positive number. Therefore, by applying inequality (3.1) to each G m and taking Pk u 1 = kj=1 u 1 (x j,m )X E j , we have
|u 1 (x) − Pk u 1 (x)| p V (x)d x ≤ Gm
Ds, p |G| p( ps−1) (k + 1) p
V (x)d x.
(3.10)
G
Whence, by using the Minkowski inequality, we obtain
1/ p |u 1 (x)| V (x)d x) p
1/ p
≤
|u 1 (x) − Pk u 1 (x)| V (x)d x p
Gm
Gm
+
1/ p |Pk u 1 (x)| p V (x)d x
,
(3.11)
Gm
123
508
Graphs and Combinatorics (2018) 34:501–508
the last integral of (3.11) is bounded by ε(
G
V (x)d x)1/ p and tends to zero when m u(x )
goes to infinity—this due to the fact that u 1 (x j ) = [u]G,jp,s = 0 for all j ∈ {1, . . . , k}. Therefore, by taking the limit w.r.t. m in the two sides of (3.11), and using (3.10), we get p
1 = u L p (G,V (x)d x) ≤
Ds, p |G| p( ps−1) p [u]G, p,s kp
V (x)d x.
(3.12)
G
Then, particularly for u ∈ F\{0} such that F ∈ Wk , we have RG, p,s (u) ≤
Ds, p |G| p( ps−1) kp
V (x)d x.
(3.13)
G
The right hand side of (3.13) does not rely neither on u nor on any F ∈ Wk , thus by taking, successively, the maximum on u and the infimum on F, we get inequality (1.3). The proof of Theorem 1.3 is complete. Remark 3.7 With regard to future related problems, it is interesting to look for on the Weyl-type laws for the fractional eigenvalue problem (∗)G, p,s , and also whether there is a similar version of Theorem 1.3 when s > 1, e.g., by considering s = 1 + η such that η ∈ (0, 1).
References 1. Berkolaiko, G., Kuchment, P.: Introduction to Quantum graphs, vol. 186. American Mathematical Society, Providence (2013) 2. Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional p-Laplacian. Discret. Contin. Dyn. Syst. Ser. A 36(4), 1813–1845 (2016) 3. Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem. Interfaces Free Bound. 16, 419– 458 (2014) 4. Coffman, C.V.: A minimum–maximum principle for a class of nonlinear integral equations. J. Anal. Math. 22, 391–419 (1969) 5. El Aïdi, M.: On the eigenvalues for a weighted p-Laplacian operator on metric graphs. Complex Var. Elliptic Equ. (2018). https://doi.org/10.1080/17476933.2018.1434630 6. Garsia, A.M., Rodemich, E., Rumsey, H.J.: A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. 20, 565–578 (1971) 7. Kwa´snicki, M.: Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262, 2379–2402 (2013) 8. Perera, K., Agarwal, P.R., O’Regan, D.: Morse Theoretic Aspects of p-Laplacian Type Operators, vol. 161. American Mathematical Society, Providence (2010) 9. Solomyak, M.: On approximation of functions from Sobolev spaces on metric graphs. J. Approx. Theory. 121, 199–219 (2003) 10. Solomyak, M.: On eigenvalue estimates for the weighted Laplacian on metric graphs. In: Birman, M.S., Hildebrandt, S., Solonnikov, V.A., Uraltseva, N.N. (eds.) Nonlinear Problems in Mathematical Physics and Related Topics I. International Mathematical Series, vol. 1, pp. 327–347. Springer, New York (2002)
123
