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

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December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

→ g

in V

and a.e inQ

, with kg

∞

≤ kg

∞

. (31)

Let us prove the existence of approximate solutions, to do this we show for

ﬁxed n ∈ IN existence of ordered lower and upper solutions to (27).

We begin by showing existence of an upper solution.

Let

∈ W

1,p

(Ω) ∩ C

(Ω) be the solution of the Torsion problem:

−∆

u = kµ

∞

+ n

in Ω, u = 0 on ∂Ω, (32)

and α

∈ W

1,p

(Ω) ∩ L

∞

(Ω) , see,

the solution of the problem

−∆

u + g(u)|∇u|

= −kµ

∞

in Ω, u = 0 on ∂Ω, (33)

By taking β

+ β where β is a large real number such that β

≥

max(α

, g

) almost everywhere in Q

. 

3.4. A priori estimates and passage to the limit

We have constructed a pair of ordered lower and upper solutions to (27),

now we use

to derive the existence of a solution u

∈ Q

∩L

∞

), u

∈

Q([0, T ]; L

(Ω)) for all s ≥ 1, and α

≤ u

≤ β

a.e. in Q

We collect some a priori estimates, we begin by

Lemma 3.1. (nT

− g

)

−

)

is bounded in L

Proof. Fix k > kg

∞

, let h > 0 and consider ϕ

= T

−

))exp(−G(u

)) where G(s) =

g(σ)dσ for all s in IR as a test func-

tion in (27), we obtain

|∇u

exp(−G(u

))χ

{k<|u

|<h+k}

−n

−g

)

−

but

exp(−kgk

(IR)

) ≤ exp(−G(s)) ≤ exp(kgk

(IR)

) for all s ∈ IR (34)

then (28) gives

−n

− g

)

−

− T

))exp(−G(u

)) ≤ Ch,

the choice of k gives by dividing on h and passing to the limit as h tends

to 0

− g

)

−

≤ C.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

The proof is then complete.

Proposition 3.3. For ﬁxed k > 0, let u

be solution to (27), then for all

n ∈ IN

|∇T

≤ Ck, (35)

|∇T

− g

≤ C(k + 1), (36)

lim

h→+∞

sup

n∈IN

|{|u

− g

| ≥ h}| = 0. (37)

and

lim

h→+∞







sup

h≤u

−g

≤h+k

|∇u

dxdt







= 0. (38)

Moreover there exists a measurable function u : Q

→ IR such that u ≥ g

a.e. in Q

and T

(u), T

(u −g

) belong to V

and, up to a subsequence

→ u a.e. in Q

, T

−g

) * T

(u −g

) in V

and a.e. in Q

. (39)

Finally

lim

h→+∞

h≤u−g

≤h+k

|∇u|

dxdt = 0. (40)

Proof. In all that follows, we set v

= u

− g

and v = u − g

. By taking

)exp(−G(u

)) as test function in (27) and using the periodicity of

, (34) we obtain

|∇T

≤

)exp(−G(u

))

− g

)

−

)exp(−G(u

)).

(41)

Then (28) and Lemma 3.1 gives (35).

For (36) we have

|∇T

) − g

) |

|∇(T

) − g

) |

{|v

|≤k}

≤ C

(|∇T

+ |∇g

) χ

{|v

|≤k}

≤ C(M

+ 1) ≤ C(k + 1)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

where M

= kg

∞

+ k. Now we prove (37), to do this we follow.

Fix h > 0, let ε ∈ ]0, h[ we have :

|{|v

| ≥ ε}| = |{|T

)| ≥ ε}| ≤

Then

|{|v

| ≥ ε}| ≤ C

k∇T

≤ C

h + 1

taking for example ε =

we obtain:



| ≥



| ≤ C

h + 1

but p ≥ 2, then (37) is proved.

Now we show the existence of a measurable function u such that

→ u a.e in Q

, and T

(u), T

(u −g

) ∈ V

for all k ∈ IN. (42)

We adapt to our case the method used in

(see also

Given ε > 0, ﬁx (by the help of (37)) k ∈ IN such that



| >



| < ε, independently of n. (43)

Let T

∈ C

(IR) nondecreasing function such that T

(s) = s for |s| ≤

and

(s) = sign(s)k for |s| > k.

Taking in (27) the test function T

)ϕ where ϕ ∈ C

∞

) we get

))

− div(T

)|∇u

p−2

∇u

) + |∇u

p−2

∇u

.∇v

)

+g(u

)|∇u

) − nT

−

)

= T

− div(G

)) + T

∇v

, div(G

) = g

. (44)

has a compact support, thanks to (31) and Lemma 3.1 the quan-

tities |∇u

p−2

∇u

.∇v

), nT

−

) and T

∇v

are

bounded in L

), so are g(u

)|∇u

) and T

because of

(19) and (30). Similarly, G

) and T

)|∇u

p−2

∇u

are bounded

in (L

))

, then from (44) (T

))

is bounded in V

+ L

). Since

) is bounded in V

a compactness result from

leads to compactness

of (T

))

in L

). Thus for a subsequence, it also a Cauchy sequence

in measure.

Take τ > 0, for n and m large we have

|{|T

) − T

)| > τ }| ≤ ε. (45)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

By the choice of T

|{|v

−v

| > τ }| ≤ |



| >



|+|



| >



|+|{|T

)−T

)| > τ }|.

(46)

Then from (43):

|{|v

− v

| > τ }| ≤ 3ε, (47)

so that v

= u

− g

is a Cauchy sequence in measure. Passing to subse-

quence and using (31) there exists a measurable function u such that u

converges to u almost everywhere and

− g

) * T

(u −g

) in V

and

) * T

(u) in V

Moreover, Lemma 3.1 and Fatou’s lemma give v

−

= 0 a.e. in Q

then

u ≥ g

a.e. in Q

Now we prove (40), it suﬃces to prove (38).

Take ψ(v

) = T

− T

)) as test function in (27) we obtain



∂v

∂t

, ψ(v

)  +

|∇u

p−2

∇u

∇v

{h≤v

≤h+k}

g(u

)|∇u

ψ(v

)−n

−

)ψ(v

) =

ψ(v

)

, ψ(v

the truncation function satisﬁes the sign condition and ψ is positive then

g(u

)|∇u

ψ(v

) ≥ 0 and n

−

)ψ(v

) = 0,

and due to the integration result in Proposition 3.2 we have

|∇u

{h≤v

≤h+k}

−

|∇u

p−2

∇u

∇g

{h≤v

≤h+k}

≤

≥h}

| +

||∇ψ(v

(48)

then

{h≤v

≤h+k}

|∇u

≤

{h≤v

≤h+k}

|∇G

≥h}

{h≤v

≤h+k}

|∇g

by (37) and the equi-integrability of the sequences (g

)

, (|G

)

and

(|∇g

)

in L

) we obtain (38), The boundedness of (T

h+k

))

and (38) gives (40).

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

Return to the proof of Theorem 3.1. The purpose in what follows is the

strong convergence

) → T

(v) in V

(49)

to this end we use almost the same techniques as in

and.

Recall ﬁrst the

following time regularization of T

(v) due to R. Landes in the stationary

case and adapted to the parabolic one in

and.

Let (z

)

be a sequence of positive function such that:

∈ W

1,p

(Ω) ∩ L

∞

(Ω), kz

∞

≤ k, ∀ν > 0,

→ T

(v)(T ) a.e. in Ω as ν tends to inﬁnity,

lim

ν→+∞

1,p

(Ω)

= 0,

then note T

(v)

the unique solution of the problem:







∂T

(v)

∂t

= ν(T

(v) − T

(v)

(0) = z

which has the form

(v)

(t) =

νexp(ν(s − t))v(s)ds + z

exp(−νt) (50)

Then we have (see

)

(v)

→ T

(v) in V

and a.e. in Q

(v)

∞

≤ k, ∀ν > 0.

We are ready to prove (49).

Let (u

)

be a sequence of solutions of (27) where µ

satisﬁes (28), and

let u given by Proposition 3.3. For h > 2k, we introduce as in

and

the

following function

= T

− T

) + T

) − T

(v)

)

we will note ε(n, ν, h) all the quantities (possibly diﬀerent) such that

lim

h→+∞

lim

ν→+∞

lim

n→+∞

ε, ν, h) = 0

and this will be the order in which the parameters we use will tends to

inﬁnity, that is, ﬁrst n, then ν and ﬁnally h. Choosing w

as test function

in (27) we have



∂v

∂t

, w

 +

|∇u

p−2

∇u

∇w

g(u

)|∇u

−n

−

+  g

, w



December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

remark that

= T

) −T

) + T

) − T

(v)

)

where M = h + 2k, then (w

)

is bounded in V

and

* T

(v − T

(v) + T

(v) −T

(v)

) in V

and a.e. in Q

and

lim

n→+∞

(v − T

(v) + T

(v) −T

(v)

lim

n→+∞

 g

, w

= g

, T

(v − T

(v) + T

(v) −T

(v)

) 

the fact that T

(v)

→ T

(v) in V

and a.e. in Q

gives

lim

ν→+∞

(v − T

(v) + T

(v) − T

(v)

) =

(v − T

(v)),

lim

ν→+∞

 g

, T

(v − T

(v) + T

(v) − T

(v)

) = g

, T

(v − T

(v)) 

by the Lebesgue theorem

lim

h→+∞

(v − T

(v)) = 0.

Moreover

 g

, T

(v−T

(v)) =

{h≤v≤h+2k}

∇v ≤ kG







{h≤v≤h+2k}

|∇v|







then, because of (40) we have



∂v

∂t

, w

 +

|∇u

p−2

∇u

∇w

− n

−

= ε(n, ν, h)

but

−

= n

−

− v

−

− T

− v

−

) + T

− v

−

) −T

(v)

)

= n

−

(−v

−

− T

(−v

−

) + T

(−v

−

) − T

(v)

)

= n

−

(−v

−

) − v

−

− (T

−

) + T

(v)

))

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

The positivity of v and (50) gives T

(v)

≥ 0 then T

−

) + T

(v)

≥ 0,

and clearly T

(−v

−

) − v

−

≤ 0 a.e. in Q

, then n

−

≤ 0 and

ﬁnally



∂v

∂t

, w

 +

|∇u

p−2

∇u

∇w

≤ ε(n, ν, h)

the rest of the proof is the same as

as far as the term 

∂v

∂t

, w

 is

concerned and in

for the others.

Periodicity condition passes to the limit in the following way:

By (44) one has

∂

S(u

− g

) converge strongly in V

+ L

but

S(u

− g

) converge strongly in V

Then by Lemma 3.2

S(u

− g

) → S(u − g

) strongly in C([0, T ]; L

(Ω)).

But S(u

− g

)(0) = S(u

− g

)(T ) then

S(u − g

)(0) = S(u − g

)(T ) in Ω.

And periodicity is now proved. 

References

1. N. Alaa, M. Iguernane, Weak periodic solutions of some quasilinear parabolic

equations with data measures , Journal of Inequalities in Pure and Applied

Mathematics. 3 (3),No. 46, (2002).

2. N. Alaa, M. Pierre, Weak solutions of some quasilinear elliptic equations with

data measures, SIAM J. Math.

Anal. 24 (1993), n. 1, 23-35.

3. P. Benilan, L. Boccardo, T. Gallou¨et, R. Gariepy, M. Pierre, J. L. V`azquez,

An L

theory of existence and uniqueness of nonlinear elliptic equations, Ann.

Scuola Norm. Sup. pisa CI. Sci., 22, (1995),641-655.

4. D. Blanchard, A. Porretta, Nonlinear parabolic equations with natural growth

terms and measure initial data. Annali della Scuola Normale Superiore di

Pisa - Classe di Scienze S´er. 4, 30 . 3-4 (2001), p. 583-622

5. D. Blanchard, F. Murat, Renormalised solutions of nonlinear parabolic prob-

lems with L1-data: existence and uniqueness, Proc. Royal Soc. Edinburgh

Sect. A 127 (1997), 1137-1152.

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6. L. Boccardo, S. Segura, C. Trombetti, Bounded and unbounded solutions for

a class of quasi-linear elliptic problems with a quadratic gradient term, J.

Math. Pures Appl., (9) 80 (2001), 919–940.

7. G. Dal Maso, F. Murat, L. Orsina, A. Prignet,Deﬁnition and existence of

renormalized solutions of elliptic equations with general measure data, C. R.

Acad. Sci. Paris S´er. I Math. 325 (1997), 481-486.

8. J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for

nonlinear equations, Potential Analysis, Volume 19, Number 2, September

2003, pp. 99-161(63)

9. A. El Hachimi And A. Lamrani Alaoui, Existence of stable periodic solutions

for quasilinear parabolic problems in the presence of well-ordered lower and

upper-solutions,proceeding of EJDE, 9 (2002), pp. 1-10.

10. P. Hess. periodic-parabolic boundary value problem and positivity, Pitman

Res. Notes Mathematics, Vol. 247, Longman Sci. Tech., Harlow, 1991.

11. R. Landes, On the existence of weak solutions for quasilinear parabolic bound-

ary value problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 217-237.

12. P. Polacik, Parabolic equations: asymptotic behaviour and dynamics on in-

variant manifolds, In: Handbook of Dyn. Syst., Vol. 2, North-Holland, Ams-

terdam, pp. 835–883.

13. A. Porretta, Existence results for nonlinear parabolic equations via strong

convergence of truncations , Ann. Mat. Pura ed Applicata (IV) 177 (1999),

pp. 143–172.

14. A. Porretta; Nonlinear equations with natural growth terms and measures

data, proceeding of EJDE, 9 (2002), pp. 183-202.

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to parabolic equations, J. Funct. Anal., Vol. 106, No. 2, pp. 1163-1175, 1992.

16. S. Segura de Leon, Existence and uniqueness for L

data of some elliptic

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(1987), 65-96.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

Existence and L

∞

-regularity results for some nonlinear elliptic

Dirichlet problems

Ahmed Youssﬁ

Department of Mathematics and Informatic

Faculty of Sciences Dhar El Mahraz

B.P 1796 Atlas, F`es, Morocco

E-mail: ahmed.youssﬁ@caramail.com

We summarize recent lectures devoted to the study of the existence and L

∞

regularity results of some Dirichlet problems associated to equations having

degenerated coercivity in the principal part.

Keywords: Orlicz spaces; Nonlinear elliptic equations; Degenerate coercivity;

A priori estimates; L

∞

-estimates; Rearrangements.

1. Introduction

During the 19th century, H¨older regularity results to the Dirichlet prob-

lem for linear elliptic equations has been object of much research and has

developed by many authors. Far from being complete, the story goes that

motivated in solving the Hilbert’s 19th conjecture, E. DeGiorgi

in 1957,

the ﬁrst, proved his famous regularity result which asserts that local weak

solutions for the linear problem

(

i,j

)

= 0 weakly in Ω,

u∈W

1,2

(Ω),

(1)

where the coeﬃcients x → a

i,j

(x), i, j = 1, ..., N are only bounded and

measurable and satisfying the uniform ellipticity condition

i,j

≥ α|ξ|

, a.e. in Ω, ∀ξ ∈ IR

, for some α > 0, (2)

are H¨older continuous by studying local pointwise estimates. The global

bound appears in the 1960’s in the works of G. Stampacchia

15,16

and

Ladyzhenskaja-Ural’tseva.

Following the methods of DeGiorgi, Ladyzhen-

skaja and Ural’tzeva established that weak solutions of quasilinear elliptic

equations are H¨older continuous. Since, several L

∞

−regularity results to

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

the Dirichlet problem, linear or nonlinear, more general than (1) have been

established.

Let us consider the problem

(

−div(a(x, u)∇u) = f in Ω,

u∈H

(Ω) ∩ L

∞

(Ω),

(3)

with the degenerate coercivity

a(x, s) ≥

(1 + s)

with α > 0

where Ω is a bounded open subset of IR

, N ≥ 2, and

f ∈ L

(Ω) with m > N/2.

The existence of bounded solutions for (3) was proved in Alvino et al.,

Boccardo-Brezis

and in Boccardo et al.

The result was then extended in

Alvino et al.

for the nonlinear elliptic problem

(

−diva(x, u, ∇u) = f in Ω,

u∈W

1,p

(Ω) ∩ L

∞

(Ω),

(4)

p > 1 with degenerate coercivity

a(x, s, ξ) · ξ ≥

(1 + s)

θ(p−1)

|ξ|

with α > 0 (5)

and where

f ∈ L

(Ω) with m > N/p. (6)

When we replace in (4) f by −divg, where

|g| ∈ L

(Ω) with m > N/(p − 1), (7)

Benkirane-Youssﬁ

have proved that problem (4) has bounded solutions.

Strongly nonlinear elliptic problems are also treated. It is so the problem

(

A(u) + H(x, u, ∇u) = f − divg in Ω,

u∈W

1,p

(Ω) ∩ L

∞

(Ω),

(8)

where A(u) = −diva(x, u, ∇u) has uniform coercivity and f and g satisfying

(6) and (7) respectively, has at least one solution Boccardo et al.

and

Ferone et al.

In Boccardo et al.

the authors have considered the problem

(8) with A(u) = −div(a(x, u)∇u) having degenerate coercivity and g =

0 and have shown that it has at least one solution in H

(Ω) ∩ L

∞

(Ω).

The general case 1 < p < +∞ with g = 0 and a(x, s, ξ) satisfying (5)

Trombetti

established that (8) has at least one solution.