Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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(X(t), t ≥ 0), is given by

A F (X(t)) =

j∈ZZ

(j, X, t)(F (X(t) + δ

) − F (X(t)))

−

j∈ZZ

−

(j, X, t)(F (X(t) − δ

) − F (X(t))) for any t ≥ 0,

(5)

where, for any j ∈ ZZ

, δ

: ZZ

→ IN is given by

(i) =







1 if i = j

0 otherwise.

In particular, we have

AF (X(t)) = (L

F )(X(t)) for any (t, F ) ∈ (0, ∞) × B(S), (6)

where L

: F ∈ B(S) → L

F ∈ L(S, IR), for any t ≥ 0, is the time

dependent operator deﬁned by

F )(ξ) :=

j∈ZZ

(j, ξ, t)(F (ξ+δ

)−F (ξ))−

j∈ZZ

−

(j, ξ, t)(F (ξ −δ

)−F (ξ)).

(7)

3. Main results

Thanks to,

we know that the connection between the stochastic model and

(1) is given through the following nonlinear dynamic in l

(ZZ

) :







∂

ˆu + II

(ˆu) 3

f for t ≥ 0

ˆu(0) = 0,

(8)

where ∂II

denotes the sub-diﬀerential of II

in l

(ZZ

). Since

K is closed

and convex with S ⊆

K, then (cf.

) for a given

f ∈ BV (0, T ; l

(ZZ

)), the

evolution (8) has a unique solution ˆu ∈ W

1,∞

loc

(0, ∞; H) such that ˆu(0) = 0

and, for any t ≥ 0, ˆu(., t) ∈

K and



f(t, i) −∂

ˆu(t, i)



ˆu(t, i) −

ξ(i)



≥ 0 for any

ξ ∈

We have

Theorem 3.1. Assume that

f ∈ BV (0, T ; l

(ZZ

). Let ˆu be the solution of

(8) and (X(t), t ≥ 0) be the stochastic process generated by

f. Then, we

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have

i∈ZZ

(X(t, i) − ˆu(t, i))

≤

j∈ZZ

f(j, s)



ds, for any t ≥ 0. (9)

Thanks to,

recall that the proof of Theorem 3.1 is based on the following

estimates.

Lemma 3.1. Under the assumptions of Theorem 3.1, for any w ∈ S, we

have

j∈ZZ

(j, X, t)(X(t, j)−w(j)) ≤ ±

i∈ZZ

f(t, i)(X(t, i)−w(i)) ∀w ∈ S.

(10)

The estimates (10) are some kind of microscopic version of (2). To prove

(10) the authors of

prove some elementary intermediate estimates evolving

various types of sets. In these notes, we use essentially the following remark

and give a direct and short proof of this result. Then, the proof of Theorem

3.1 follows more or less the same step of.

Indeed, Lemma 3.1 gives the

connection between X(t) and (8). Then, we transform (8) into an evolution

problem in L

(IR

) governed by a sub-diﬀerential operator of the indicator

function of the set of rescaled conﬁgurations (see (14)). This transformation

allows us to give the connection between (1), (15) and the stochastic model.

Remark 1.

(1) For a given ξ ∈ S, if p

(i, j, ξ) > 0, then there exists at least one

staircase i

= i ∼ i

∼ . . . ∼ i

= j, such that ξ(i

) = ξ(i

p+1

) + 1 for

any p = 0, 1, . . . , m − 1. Let us denote this staircase by C

(i, j) ; i.e.

(i, j) = [i

, i

, . . . , i

m−1

, j],

and, for any k ∈ C

(i, j), we denote by

k, the adjacent side to k such

that u(k) = u(

k) + 1. It is clear that C(i, j) may not be unique, so

(it is not essential) to simplify the presentation, let us consider the

application:

: ZZ

× ZZ

→ C

(i, j)a staircase between i and j.

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(2) For a given ξ ∈ S, if p

−

(i, j, ξ) > 0, then there exists at least one upward

staircase i

= i ∼ i

∼ ... ∼ i

= j, such that ξ(i

) = ξ(i

p+1

) + 1 for

any p = 0, 1, . . . , m − 1. Let us denote this staircase by C

−

(i, j) ; i.e.

−

(i, j) = [i

, i

, ..., i

m−1

, j],

and, for any k ∈ C(i, j), we denote by

k, the adjacent side to k such that

u(k) = u(

k) −1. It is clear that C(i, j) may not be unique, so (it is not

essential) to simplify the presentation, let us consider the application :

−

: ZZ

× ZZ

→ C

−

(i, j) a staircase between i and j.

Proof of Lemma 3.1: Thanks to (4), we have

j∈ZZ

(j, X, t) (X(t, j) − w(j))

j,i∈ZZ

(i, j, X(t))

(t, i) (X(t, j) − w(j))

j,i∈ZZ

(i, j, X(t))

(t, i) (X(t, i) − w(i))

j,i∈ZZ

(i, j, X(t))

(t, i)



(w(i) −w(j)) − (X(t, i) − X(t, j))



= I

+ I

Since

j∈ZZ

(i, j, X(t)) = 1, for any (t, i) ∈ ZZ

× (0, ∞), then it is clear

that

i∈ZZ

(t, i) (X(t, i) − w(i)).

Let us prove that I

≤ 0. Thanks to Remark 1, we have

j,i∈ZZ

(i, j, X(t))

(t, i)

k∈C

(i,j)



(w(k)−w(

k))−(X(t, k)−X(

k, t))



≤

j,i∈ZZ

(i, j, X(t))

(t, i)

k∈C

(i,j)



(w(k) − w(

k)) − 1



≤ 0,

where we used the fact that |w(k) − w(

k)| ≤ 1 (since w ∈ K and k ∼

k).

The proof of

j∈ZZ

−

(j, X, t) (X(t, j) − w(j)) ≥

i∈ZZ

−

(t, i) (X(t, i) −w(i)),

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follows in the same way, we let the details to the reader.

To simplify the presentation of the proof of Theorem 3.1, we divide the

proof into two steps that we present in the following Lemmas.

Lemma 3.2. Under the assumptions of Theorem 3.1, we have

(1) For any w ∈ S and t ≥ 0, we have



i∈ZZ



X(t, i)−w(i)



≤

j∈ZZ

f(t, j)(X(t, j)−w(j))+

j∈ZZ

f(t, j).

(2) For any w ∈ W

1,∞

(0, T ; H) such that w(t) ∈ S, for any t ∈ [0, T ), we

have

i∈ZZ

(X(t, i) −w(t, i))

≤

i∈ZZ

∂w

∂s

(i, s) (w(i, s) −X(i, s))

j∈ZZ

f(j, s) (X(j, s) − w(j, s)) +

j∈ZZ

f(j, s)

+ M(t)

where



M(t)



t≥0

is a martingale satisfying

IE(M(t)) = 0 for any t ≥ 0.

Proof:

(1) Let us denote

I =



i∈ZZ



X(t, i) − w(i)



By deﬁnition of L, we have

I =

j∈ZZ

c(j, X, t)



i∈ZZ



(X(t))(i)−w(i)



−

i∈ZZ



X(t, i)−w(i)



So,

I =

j∈ZZ

c(j, X, t)



i∈ZZ

(X)(i) − X(i))(T

(X)(i) + X(i) −2 u(i))



j∈ZZ

c(j, X, t) (2 X(j) + 1 − 2 u(j))

j∈ZZ

c(j, X, t) (X(j) − u(j)) +

j∈ZZ

c(j, X, t).

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Thanks to (4), we deduce that

I =

j∈ZZ

c(j, X, t) (X(j) −u(j)) +

j∈ZZ

f(t, j),

and, using Lemma 3.1, the ﬁrst step of the lemma follows.

(2) As in,

we use the following stochastic integral equation : for any F :

S ×(0, ∞) → IR Lipchitz continuous in t and F (X(., 0), 0) = 0, we have

F (X(., t), t) =



∂F

∂s

+ L



(X(., s)) + M(t). (11)

Let F be given by

F (ξ, t) =

i∈ZZ



ξ(i) − w(t, i)



, for any (ξ, t) ∈ S × (0, T ).

Then,

∂F

∂s

(ξ, s) = −

i∈ZZ

∂w

∂s

(i, s)



ξ(i)−w(i, s)



, for any (ξ, t) ∈ S×(0, T ),

and (11) implies that, for any t ≥ 0,

i∈ZZ

(X(t, i) −w(t, i))



i∈ZZ

∂w

∂s

(i, s) (w(i, s) −X(i, s))

+ L

(F (X(., s), s) + M(t).

Then, by using the ﬁrst step, the result follows.

Proof of Theorem 3.1: Using the fact that ˆu is a solution of (8) and

X(t) ∈

K, for any t ≥ 0, we have

i∈ZZ

∂ ˆu

∂s

(i, s)(ˆu(i, s)−X(i, s))+

j∈ZZ

f (j, s)(X(j, s)−ˆu(j, s)) ≤ 0 for any t ≥ 0.

Then, using the second part of Lemma 3.2 with ˆu, we deduce (9).

At last, let us come back to the continuous model of Prigozhin (1) and

assume that

f ∈ BV

loc

(0, ∞; L

(IR

)) and there exists R > 0 such that spt(f) ⊆ B(0, R),

(12)

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here B(0, R) denotes the a ball with center 0 and radius R. Thanks

to,

we know that (1) has a unique solution u in the sense that u ∈

1,∞

loc

(0, ∞; L

(IR

)), u(0) = 0 and, for any t ≥ 0, u(., t) ∈ K and



f(t, x) −∂

u(t, x)



u(t, x) − ξ(x)



≥ 0 for any ξ ∈ K.

For the connection with the stochastic model, we rescale the source term





. We set

f(t, i) = f





for any (t, i) ∈ [0, ∞) × ZZ

and we consider (X(t))

t≥0

the Markov processus generated by

f as ex-

plained in the previous section.

Theorem 3.2. Under the assumption (12), we have





u(t, x) −

X(N t, [N x])





→ 0 as N → 0. (13)

Proof: Let ˆu be the solution of (8) and consider

(t, x) =

ˆu



N t, [N x]



and f

(t, x) = f



[Nx]



for any (t, x) ∈ [0, T ) × IR

. It is not diﬃcult to see that u

∈

1,∞

loc

((0, ∞); L

(IR

)) and, for any t ≥ 0, u

(., t) ∈ K

, where

z ∈ L

(IR

) ; |u(x) −u(y)| ≤

for |x − y| ≤

. (14)

In addition, for any ξ ∈ K

, we have



(t, x) − ∂

(t, x)



(t, x) − ξ(x)





f(N t, i) − ∂

ˆu(Nt, i)



ˆu(Nt, i) −

ξ(i)



where

ξ(i) = N

ξ(x) dx and I

= {z ∈ IR

; [Nz] = i}.

Now, since ξ ∈ K

and ˆu is the solution of (8), then

ξ ∈

K and



(t, x) − ∂

(t, x)



(t, x) − ξ(x)



≥ 0 for any ξ ∈ K

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In other words u

is the solution of the nonlinear dynamic







∂

+ ∂II

) 3

ˆu

(0) = 0.

(15)

Thanks to,

we know that

→ u in C([0, T ); L

(IR

)). (16)

Indeed, ∩

N∈IN

= K so that, letting N → ∞, we have ∂II

converges

in the sense of graph to ∂II

in L

∞

(IR

). On the other hand, as N → ∞,

→ f in L

loc

((0, ∞); L

(IR

)).

Then, by using classical perturbation result for nonlinear semigroup (cf.

(16) follows. At last, thanks to Theorem 3.1, we have





Nt, [Nx]



− u

(t, x)





= IE

i∈ZZ



Nt, i



− N u

(t, i)|

= IE

i∈ZZ



Nt, i



− ˆu



Nt, i



≤

i∈ZZ



s, i



≤



[Nx]



dsdx

≤



τ,

[Nx]



dτdx

→ 0 as N → ∞.

so, by using (16), we deduce (13).

Acknowledgements

This work was partially supported by the French A.N.R. Grant JC05-41831

and by ”FLUPARTI” project (supported and funded by ”Conseil R´egional

de Picardie”).

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References

1. F. Andreu, J. M. Mazon, J. D. Rossi and J. Toledo, The limit as p → ∞ in

a nonlocal p−Laplacian evolution equation. A nonlocal approximation of a

model for sandpiles. To appear in Cal. Var. Partial Diﬀ. Equ.

2. G. Aronson, L. C. Evans and Y. Wu, Fast/Slow diﬀusion and growing sand-

piles. J. Diﬀerential Equations, 131:304–335, 1996.

3. J. W. Barrett and L. Prigozhin, Dual formulation in Critical State Problems.

Interfaces and Free Boundaries, 8, 349-370, 2006.

4. H. Br´ezis, Op´erateurs maximaux monotones et semigroups de contrac-

tions dans les espaces de Hilbert. (French). North-Holland Mathematics

Studies, No. 5. Notas de Matem`atica (50). North-Holland Publishing Co.,

Amsterdam-London; American Elsevier Publishing Co., Inc., New York,

1973.

5. S. Dumont and N. Igbida, Back on a Dual Formulation for the Growing

Sandpile Problem. To appear in Eur. Journal Appl. Math..

6. L. C. Evans and F. Rezakhanlou, A stochastic model for sandpiles and its

continum limit. Comm. Math. Phys., 197 (1998), no. 2, 325-345.

7. N. Igbida, Evolution Monge-Kantorovich equation. submitted.

8. L. Prigozhin, Variational model of sandpile growth. Euro. J. Appl. Math. ,

7, 225-236, 1996

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On radial solutions for Navier boundary eigenvalue problem

with p-biharmonic operator

Siham El Habib

∗

and Najib Tsouli

†

University Mohamed I, Faculty of Sciences

Department of Mathematics and Computer

Oujda, Morocco

E-mails:

∗

s-elhabib@hotmail.com,

†

tsouli@hotmail.com

This work deals with the existence of radial solutions for a nonlinear eigenvalue

problem involving the p-biharmonic operator.

Keywords: p-biharmonic; Radial solutions.

1. Introduction

Let us consider the nonlinear eigenvalue problem

)



∆

u = λ|u|

p−2

u inΩ,

u = ∆u = 0 on∂Ω,

where 1 < p < +∞, ∆

u := ∆(|∆u|

p−2

∆u) is the operator of fourth order

called p-biharmonic operator and Ω = B

is the unit ball of IR

, N ≥ 2.

The spectrum of the p-biharmonic operator was recently studied by

many authors (cf.,

and the references therein).

P. Dr´abeck and M.

Otani

studied the spectrum of the problem (P

) in

a bounded and smooth domain Ω of IR

(N ≥ 1).

In one dimensional case and Ω = [0, 1],J. Benedikt (cf.

) gave the spectrum

of the p-biharmonic operator under Dirichlet and Neumann boundary con-

ditions.

In his thesis, M. Talbi (cf.

) studied a variety of problems with p-

biharmonic. He has included there many important techniques and tools

concerning such problems.

A. El Khalil, S. Kellati and A. Touzani (cf.

) showed that the spectrum

of the p-biharmonic operator with weight and with Dirichlet boundary con-

ditions contains at least one non-decreasing sequence of positive eigenvalues.

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279

Our study deals with the existence of radial solutions for the problem

). We would like to mention some works which were of main importance

to achieve this work: The ﬁrst one is of F. De Thelin

where he showed that

the problem

−∆

u = λ|x|

|u|

p−2

u, u ∈ W

1,p

), u ≥ 0, u 6= 0, (1)

has at least one positive radial solution. The second one is of A.Anane

where he determined all the eigenvalues associated with a radial eigenfunc-

tion for the problem (1) and some properties of the eigenfunctions.

We show similar results as those of A.Anane.

We prove that the prob-

lem (P

) has at least a sequence of eigenvalues associated with a radial

eigenfunction, any eigenvalue λ

(n ≥ 1) associated to a radial eigenfunc-

tion is given according to the ﬁrst eigenvalue λ

and any radial eigenfunction

associated to λ

(n ≥ 1) has a ﬁnite number of components of the nodal set

deﬁned by {x ∈ Ω : u(x) 6= 0}.

2. Preliminaries

In this section, we give some results and notations that will be used through-

out this paper:

For 0 < R < 1, we denote by:

• B

= {x ∈ IR

: |x| < R},

• C

= {x ∈ IR

: R < |x| < 1},

• λ

1,R

(resp. µ

1,R

) the ﬁrst eigenvalue of the problem (P ) in B

(resp.

• ϕ

(resp ψ

) the eigenfunction associated with λ

1,R

(resp µ

1,R

• λ

= λ

1,1

Deﬁnition 2.1. For any radial function u, we associate a real valued func-

tion u deﬁned by u(x) = ϕ(|x|) where |x| is the Euclidian norm of x in

The problem (P

) is equivalent to the following problem (for more details

see

)



Find (v, λ) ∈ (L

(Ω) \ {0}) × IR

∗

such that:

(v) = λg

(v),

where

(v) =

kvk

, g

(v) =

kΛvk