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Hindawi Publishing CorporationBoundary Value Problems Volume 2007, Article ID 65825, 15 pages doi:10.1155/2007/65825 Research Article On the Sets of Regularity of Solutions for a Class o

Hindawi Publishing Corporation

Boundary Value Problems

Volume 2007, Article ID 65825, 15 pages

doi:10.1155/2007/65825

Research Article

On the Sets of Regularity of Solutions for a Class of Degenerate

S Bonafede and F Nicolosi

Received 24 January 2007; Accepted 29 January 2007

Recommended by V Lakshmikantham

We establish H¨older continuity of generalized solutions of the Dirichlet problem, asso-ciated to a degenerate nonlinear fourth-order equation in an open bounded setΩ⊂ R n, withL1data, on the subsets ofΩ where the behavior of weights and of the data is regular enough

Copyright © 2007 S Bonafede and F Nicolosi This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In this paper, we will deal with equations involving an operator A : W ◦ 1,2,q p(ν,μ,Ω) →

(W ◦ 1,2,q p(ν,μ,Ω)) of the form

| α |=1,2 (−1)| α | D α A α

x, ∇2u

whereΩ is a bounded open set ofRn,n > 4, 2 < p < n/2, max(2p, √

n) < q < n, ν and μ are

positive functions inΩ with properties precised later,W ◦ 1,2,q p(ν,μ,Ω) is the Banach space

of all functions u :Ω→ R with the properties| u | q,ν | D α u | q,μ | D β u | p ∈ L1(Ω),| α | =1,

| β | =2, and “zero” boundary values;∇2u = { D α u : | α | ≤2}

The functionsA αsatisfy growth and monotonicity conditions, and in particular, the following strengthened ellipticity condition (for a.e.x ∈ Ω and ξ = { ξ α:| α | =1, 2}):



| α |=1,2

A α(x, ξ)ξ α ≥ c2

 

| α |=1

ν(x)ξ αq

| α |=2

μ(x)ξ αp



− g2(x), (1.2) wherec2> 0, g2(x) ∈ L1(Ω)

We will assume that the right-hand sides of our equations, depending on unknown function, belong toL1(Ω)

A model representative of the given class of equations is the following:

| α |=1

D α



ν



| β |=1

D β u 2

(q −2)/2

D α u

| α |=2

D α



μ



| β |=2

D β u 2

(p −2)/2

D α u

= −| u | σ −1u + f inΩ,

(1.3) whereσ > 1 and f ∈ L1(Ω)

The assumed conditions and known results of the theory of monotone operators allow

us to prove existence of generalized solutions of the Dirichlet problem associated to our operator (see, e.g., [1]), bounded on the setsG ⊂Ω where the behavior of weights and of the data of the problem is regular enough (see [2])

In our paper, following the approach of [3], we establish on such sets a result on H¨older continuity of generalized solutions of the same Dirichlet problem

We note that for one high-order equation with degenerate nonlinear operator satisfy-ing a strengthened ellipticity condition, regularity of solutions was studied in [4,5] (non-degenerate case) and in [6,7] (degenerate case) However, it has been made for equations with right-hand sides inL twitht > 1.

2 Hypotheses

Letn ∈ N,n > 4, and letΩ be a bounded open set ofRn Let p, q be two real numbers

such that 2< p < n/2, max(2p, √

n) < q < n.

Letν : Ω → R+be a measurable function such that

ν ∈ L1loc(Ω), 1ν 1/(q −1)∈ L1loc(Ω) (2.1)

W1,q(ν,Ω) is the space of all functions u ∈ L q(Ω) such that their derivatives, in the sense of distribution,D α u, | α | =1, are functions for which the following properties hold:

ν1/q D α u ∈ L q(Ω) if| α | =1;W1,q(ν,Ω) is a Banach space with respect to the norm

u 1,q,ν =



Ω| u | q dx + 

| α |=1



ΩνD α uq

dx

1/q

◦

W1,q(ν,Ω) is the closure of C ∞

0(Ω) in W1,q(ν,Ω).

Letμ(x) :Ω→ R+be a measurable function such that

μ ∈ L1 loc(Ω), 1μ 1/(p −1)∈ L1

W2,1,p q(ν,μ,Ω) is the space of all functions u ∈ W1,q(ν,Ω), such that their derivatives,

in the sense of distribution,D α u, | α | =2, are functions with the following properties:

S Bonafede and F Nicolosi 3

μ1/ p D α u ∈ L p(Ω),| α | =2;W2,1,p q(ν,μ,Ω) is a Banach space with respect to the norm

u u 1,q,ν+



| α |=2



ΩμD α up

dx

1/ p

◦

W1,2,q p(ν,μ,Ω) is the closure of C ∞

0(Ω) in W1,q

2,p(ν,μ,Ω).

Hypothesis 2.1 Let ν(x) be a measurable positive function:

1

ν ∈ L t(Ω) with t >

nq

q2− n,

ν ∈ L t(Ω) with t > nt

qt − n .

(2.5)

We putq= nqt/(n(1 + t) − qt) We can easily prove that a constant c0> 0 exists such

that ifu ∈ W ◦ 1,q(ν,Ω), the following inequality holds:



Ω| u | qdx ≤ c0



suppu

1

ν

t

dx

 q/qt 

| α |=1



Ων | D α u | q dx

q/q

We setν = μ q/(q −2p)(1/ν)2p/(q −2p)

Hypothesis 2.2. ν ∈ L1(Ω)

Hypothesis 2.3 There exists a real number r > q(q −1)/( q(q −1)(p −1)− q) such that

1

For more details about weight functions, see [8,9]

LetΩ1be a nonempty open set ofRnsuch thatΩ1⊂Ω

Definition 2.4 It is said that G closed set ofRn is a “regular set” if G is nonempty and ◦

G ⊂Ω1

Denote byRn,2the space of all setsξ = { ξ α ∈ R:| α | =1, 2}of real numbers; if a func-tionu ∈ L1

loc(Ω) has the weak derivatives D α u, | α | =1, 2 then∇2u = { D α u : | α | =1, 2} Suppose thatA α:Ω× R n,2 → Rare Carath´eodory functions

Hypothesis 2.5 There exist c1,c2> 0 and g1(x), g2(x) nonnegative functions such that

g1,g2∈ L1(Ω) and, for almost every x∈ Ω, for every ξ ∈ R n,2, the following inequalities



| α |=1



ν(x)−1/(q −1) A α(x, ξ)q/(q −1)

+ 

| α |=2



μ(x)−1/(p −1) A α(x, ξ)p/(p −1)

≤ c1





| α |=1

ν(x)ξ αq

+ 

| α |=2

μ(x)ξ αp

 +g1(x),

(2.8)



| α |=1,2

A α(x, ξ)ξ α ≥ c2

 

| α |=1

ν(x)ξ αq

+ 

| α |=2

μ(x)ξ αp



− g2(x). (2.9)

Moreover, we will assume that for almost everyx ∈ Ω and every ξ,ξ ∈ R n,2,ξ ξ ,



| α |=1,2



A α(x, ξ) − A α(x, ξ )

ξ α − ξ α 

LetF :Ω× R → Rbe a Carath´eodory function such that

(a) for almost everyx ∈ Ω, the function F(x, ·) is nonincreasing inR;

(b) for everyx ∈ Ω, the function F( ·,s) belongs to L1(Ω)

LetA : W ◦ 1,2,q p(ν,μ,Ω) →(W ◦ 1,2,q p(ν,μ,Ω)) be the operator such that for everyu, v ∈ W ◦ 1,2,q p(ν,

μ,Ω),

 Au, v  =

 Ω

 

| α |=1,2

A α



x, ∇2u

D α v



We consider the following Dirichlet problem:

(P) =

⎧

⎨

⎩

D α u =0, | α | =0, 1, on∂ Ω. (2.12)

Definition 2.6 A W-solution of problem (P) is a function u ∈ W ◦2,1(Ω) such that (i)F(x, u) ∈ L1(Ω);

(ii)A α(x, ∇2u) ∈ L1(Ω), for every α :| α | =1, 2;

(iii) Au, φ  =  F(x, u), φ in distributional sense

It is well known that Hypotheses2.1–2.3,2.5, and assumptions onF(x, s) imply the

existence of aW-solution of problem (P) (see [1]) Moreover, a boundedness local result for such solution has been established in [2] under more restrictive hypotheses on data and weight functions

More precisely, the following holds (see [2, Theorem 5.1])

Theorem 2.7 Suppose that Hypotheses 2.1 – 2.3 and 2.5 are satisfied Let q1∈(q, q(q −

1)/q), τ > q/( q− q1) Assume that restrictions of the functionsν q1/(q1− q) ,ν, g1, g2, and | F( ·, 0)| q1/(q1−1)on G belong to L τ(G), for every “regular set” G.

Then there exists u W-solution of problem (P) such that for every G, ess Gsup| u | ≤ M G <

+∞ , with M G positive constant depending only on known values.

S Bonafede and F Nicolosi 5

3 Main result

In the sequel of paper,G will be a “regular set.” In order to obtain our regularity result on

G, we need the following further hypotheses.

Hypothesis 3.1 There exists a constant c  > 0 such that for all y ∈ G and for all ρ > 0, with ◦ B(y, ρ) ⊂ G, we have ◦



ρ − n



B(y,ρ)

1

ν

t

dx

 1/t

ρ − n



B(y,ρ) ν τ dx

 1/τ

With regard to this assumption, see [3]

Hypothesis 3.2 There exist a real positive number σ and two real functions h(x)( ≥0),

f (x)(> 0) defined on G, such that

F(x, s)  ≤ h(x) | s | σ+f (x), for almost everyx ∈ G and every s ∈ R (3.2) Moreover, we assume that

withτ defined as above.

Using considerations stated in [1], following the approach of [3], we establish the fol-lowing result

Theorem 3.3 Let all above-stated hypotheses hold and let conditions of Theorem 2.7 be satisfied Then, the W-solution u of Dirichlet problem (P), essentially bounded on G, is also locally H¨olderian on G.

More precisely, there exist positive constant C and λ (0 < λ < 1) such that for every open setΩ,Ω ⊂ G, and every x, y ◦ ∈Ω

u(x) − u(y)  ≤ C

d

Ω,∂ G ◦− λ

| x − y | λ, (3.4)

where C and λ depend only on c1, c2, c0, c  , n, q, p, t, τ, σ, M G , diam G, meas G, f L τ(G) ,

h L τ(G) , g1 L τ(G) , g2 L τ(G) , ν L τ(G) , and 1/ν L t( Ω).

Proof For every l ∈ N, we define the functionF l:Ω× R → Rby

F l(x, s) =

⎧

⎪

⎨

⎪

⎩

− l ifF(x, 0) − F(x, s) < − l, F(x, 0) − F(x, s) ifF(x, 0) − F(x, s)  ≤ l,

l ifF(x, 0) − F(x, s) > l,

(3.5)

and the function f l:Ω→ Rby

f l(x) =

⎧

⎨

⎩

F(x, 0) ifF(x, 0)  ≤ l,

By Lebesgue’s theorem and property (b) ofF(x, s), we have that f l(x) goes to F(x, 0) in

L1(Ω)

Next, inequalities (2.6), (2.8)–(2.10), property (a) ofF(x, s), and known results of the

theory of monotone operators (see, e.g., [◦ 10]) imply that for anyl ∈ N, there existsu l ∈

W1,2,q p(ν,μ,Ω) such that

 Ω

 

| α |=1,2

A α

x, ∇2u l

D α v + F l

x, u l

v



dx =



for everyv ∈ W ◦ 1,2,q p(ν,μ,Ω).

From considerations stated in [1, Section 3], we deduce that there exists aW-solution

u of problem (P) such that

Moreover, see proof ofTheorem 2.7,

ess

G supu l  ≤ M G, for everyl ∈ N (3.9)

We setn = q2/(q −2p), a =(1/n)(q − n/t − n/τ).

Let us fixy ∈ G, ρ > 0 and B(y, 2ρ) ◦ ⊂ G Let us put ◦

ω1,l = ess

B(y,2ρ)infu l, ω2,l = ess

B(y,2ρ)supu l,

We will show that

osc

u l,B(y, ρ)

withc∈]0, 1[ independent ofl ∈ N

To this aim, we fixl ∈ Nand we set

Φl = 

| α |=1

νD α u lq

+ 

| α |=2

μD α u lp

,

ψ(x) = ρ − an

1 +f (x) + h(x) + g1(x) + g2(x) +ν(x)+ρ − q ν. (3.12)

Obviously, we will assume that

ω l ≥ ρ a (otherwise, it is clear that (3.11) is true). (3.13)

We introduce now the following functions:

F1,l(x) =

⎧

⎪

⎪

2eω l

u l(x) − ω1,l+ρ a ifx ∈ B(y, 2ρ),

(3.14)

S Bonafede and F Nicolosi 7

ϕ ∈ C ∞0(Ω): 0≤ ϕ ≤1 inΩ, ϕ =0 inΩ\ B(y, 2ρ) and satisfying

D α ϕ  ≤ cρ −| α |, | α | =1, 2, (3.15) where the positive constantc depends only on n.

Let us fixs > q and r ≥0 and define

v l =lgF1,lr

F1,q − l1ϕ s,

z l = − 1

2eω l



r

lgF1,lr −1

+ (q −1)

lgF1,lr

F1,q l ϕ s (3.16)

FromHypothesis 2.2and (3.15), we have thatv l ∈ W ◦ 1,2,q p(ν,μ,Ω) and the next

inequal-ities are true:

D α v l − z l D α u l  ≤ csϕ s −1

lgF1,l

r

F1,q − l1ρ −1 if| α | =1 a.e inB(y, 2ρ), (3.17)

D α v l − z l D α u l  ≤5 2s(r + 1)2

lgF1,l

r

F1,q − l1ϕ s

 

| β |=1

| D β u l |2



u l − ω1,l+ρ a 2



+ 2nqs2c2ρ −2

lgF1,l

r

F1,q − l1ϕ s −2 if| α | =2 a.e inB(y, 2ρ).

(3.18)

Sinceu l(x) satisfies (3.7), forv = v l, we obtain



Ω





| α |=1,2

A α

x, ∇2u l

D α v l+F l

x, u l

v l



dx =



Ωf l v l dx. (3.19) From this, taking into account (3.9) andHypothesis 3.2, we have



Ω



| α |=1,2

A α

x, ∇2u l

D α v l dx ≤3 +M σ

G



Ω



1 +f (x) + h(x)

v l dx. (3.20)

Hence



Ω



| α |=1,2



A α

x, ∇2u l

D α u l

− z l

dx ≤3 +M σ G

Ω



1 +f (x) + h(x)

v l dx + I1+I2,

(3.21) where

I i =

 Ω



| α |= i

A α

x, ∇2u lD α v l − z l D α u ldx, i =1, 2. (3.22) UsingHypothesis 2.5and definition ofz l, we have

(q −1)c2

2eω l



ΩΦl

lgF1,lr

F1,q l ϕ s dx ≤3 +M G σ

Ω



1 +f (x) + h(x)

lgF1,lr

F1,q − l1ϕ s dx

+



Ωg2(x)

− z l



dx + I1+I2.

(3.23)

Note that

F1,q − l1≤(diamG) a

2eω lq −1

ρ − aq,

− z l ≤(q −1)(r + 1)

2eω lq −1

ρ − aq ϕ s

lgF1,l

r

a.e inB(y, 2ρ), (3.24)

consequently, from (3.23), we obtain

c2

2eω l



B(y,2ρ)Φl



lgF1,l

r

F1,q l ϕ s dx

≤ c3(r + 1)

2eω lq −1 

B(y,2ρ) ρ − aq

1 +f (x) + h(x) + g2(x)

lgF1,lr

ϕ s dx + I1+I2,

(3.25) wherec3=(q −1)(3 +M σ

G)(diamG + 1).

Let us fix| α | =1 Let > 0, then, applying Young’s inequality and using (2.8) and (3.17), we establish

I1≤ c1

2eω l



B(y,2ρ)Φl F1,q l

lgF1,l

r

ϕ s dx

+c12eω lq −1 

B(y,2ρ) ρ − aq g1(x)

lgF1,l

r

ϕ s dx

+1− q

2eω lq −1

n(cs) q



B(y,2ρ) ρ − q νlgF1,lr

ϕ s − q dx.

(3.26)

Let us fix| α | =2 and estimateI2 To this aim, it will be useful to observe that the following equalities are true:

p −1

2

q+

q −2

qp =1, q −1= p −1

p q +

q

Moreover,

ρ − aq −2p μ ≤ ρ − anν + ρ − q ν in Ω. (3.28) Furthermore, due to (2.8), (3.18), and Young’s inequality, we have

I2≤ c4

2eω l



B(y,2ρ)Φl F1,q l

lgF1,l

r

ϕ s dx

+c5



2eω lq −1



1 +1



n

s n(r + 1) n



B(y,2ρ)



ρ − an

g1(x) +ν(x)+ρ − q νlgF1,lr

ϕ s − q dx,

(3.29) wherec depends only onc1,n, q; and c depends only onc1,n, q, p, c, and diam G.

S Bonafede and F Nicolosi 9 From (3.25), (3.26), and (3.29), we get

c2

2eω l



B(y,2ρ)Φl



lgF1,l

r

F1,q l ϕ s dx

≤



c1+c4





2eω l



B(y,2ρ)Φl F1,q l

lgF1,l

r

ϕ s dx

+

2eω l

q −1

c6(r + 1) n s n

1 ++1



n+1

B(y,2ρ) ψ

lgF1,l

r

ϕ s − q dx,

(3.30)

where the constantc6depends only onc1,c, n, q, p, M G,σ, and diam G.

Setting

 = c2

2

c1+c4

from the last inequality, we deduce



B(y,2ρ)Φl



lgF1,l

r

F1,q l ϕ s dx ≤ c7



2eω lq

(r + 1) n s n



B(y,2ρ) ψ

lgF1,l

r

ϕ s − q dx, (3.32)

where the constantc7depends only onc1,c2,c, n, q, p, M G,σ, and diam G.

Now, if we chooseϕ such that ϕ =1 inB(y, (4/3)ρ), from (3.32), withr =0 ands =

q + 1, we get



B(y,(4/3)ρ)

 

| α |=1

νD α u lq



F1,q l dx ≤ c7 

2eω lq

(q + 1) n



B(y,2ρ) ψdx. (3.33) Moreover, if we take in (3.32) instead ofϕ the function ϕ1∈ C0∞(Ω) with the properties

0≤ ϕ1≤1 inΩ, ϕ1 =0 inΩ\ B(y, (4/3)ρ), ϕ1=1 inB(y, ρ), and | D α ϕ | ≤ cρ −| α |inΩ,

| α | =1, 2, we obtain that for everyr > 0 and s > q,



B(y,2ρ)

 

| α |=1

νD α u lq





lgF1,l

r

F1,q l dx ≤ c7



2eω lq

s n(r + 1) n



B(y,2ρ) ψ

lgF1,l

r

ϕ s1− q dx.

(3.34)

We fix arbitraryr > 0 and s > q, and let

z l =lgF1,lr/ q

By means ofHypothesis 2.1, we establish thatz l ∈ W ◦1,q(ν,Ω) and for | α | =1,

νD α z lq

≤2q −1

r



q

q

lgF1,l

 (r/ q−1)q

F1,l

q 1



2eω lqD α u lq

νϕ sq/ q 1

+ 2q −1

s



q

q

lgF1,lrq/ q

ϕ(1s/ q−1)q c q ρ − q ν.

(3.36)

Now, it is convenient to observe thatq/( q− q1)> nt/(qt − n), then τ > nt/(qt − n);

moreover,ψ(x) ∈ L τ(G) From (3.34) and (3.36), we deduce



ΩνD α z lq

dx

≤ c8s n(r +1) n+q



B(y,2ρ) ψ τ dx

1/τ

B(y,2ρ)



lgF1,l

r(q/ q)(τ/(τ −1))

ϕ(1s/ q−1)q(τ/(τ −1))dx

(τ −1)/τ

, (3.37)

where the constantc8depends only onc1,c2,c, n, q, p, M G,σ, and diam G.

We set

θ = q(τ −1)

and for everyr, s > 0, we define

I(r, s) =



B(y,2ρ)



lgF1,l

r

Consequently, last inequality can be rewritten in this manner:



ΩνD α z lq

dx ≤ c8s n(r + 1) n+q

B(y,2ρ) ψ τ dx

1/τ

I

r

θ,

s

θ − m

(τ −1)/τ

. (3.40) Due toHypothesis 2.1,

I(r, s) =



B(y,2ρ) z q ldx ≤ c0



B(y,2ρ)

1

ν

t

dx

q/qt  

| α |=1



ΩνD α z lq

dx

q/q

. (3.41)

Let us denote by

Gthe norm of (1 +f (x) + h(x) + g1(x) + g2(x) +ν(x)) in L τ(G) By

simple computation, we have



B(y,2ρ) ψ τ dx

1/τ

≤ ρ − q



B(y,2ρ) ν τ dx

1/τ

+

G ρ − an (3.42)

Now, it is convenient to observe that (q − n/t − n/τ)( q/q) = n(θ −1)

Then, from (3.40)–(3.42), usingHypothesis 3.1, we get

I(r, s) ≤ M(r + s) m ρ n(1 − θ)



I

r

θ,

s

θ − m

θ

, for everyr > 0, s > q, (3.43)

wherem =2(q + n) q and the positive constant M depends only on c 1,c2,c, c0,c ,n, q, p,

t, 1/ν L t( Ω),M G,σ, meas G, diam G, and

G

S Bonafede and F Nicolosi 11

We set fori =0, 1, 2, that

r i = tq

t + 1 θ

i, s i = mθ

θ −1



θ i+1 −1

Then by (3.43), it is trivial to establish the following iterative relation:

I

r i,s i

≤ Mc9ρ n(1 − θ) θ i m

I

r i −1,s i −1

θ

for everyi ∈ N, (3.45) wherec9depends only onn, q, p, t, and τ.

Using this recurrent relation, we obtain that for everyi ∈ N,

I

r i,s i

≤

Mc9+ 1 1/(1 − θ)

θ S m(diamG + 1) n ρ − n I

r0,s0

θ i

whereS is a positive constant depending only on n, q, t, and τ.

Now, we assume that

meas



x ∈ B

y,4

3ρ :u l(x) ≥ ω1,l+ω2,l

2



≥1

2measB

y,4

3ρ . (3.47)

We observe that ifx ∈ B(y, (4/3)ρ) satisfies u l(x) ≥(ω1,l+ω2,l)/2, then F1,l(x) ≤4e, so

by [11, Lemma 4], we deduce



B(y,(4/3)ρ)



lgF1,l

r0

dx ≤ cρ n+ cρr0

2eω l



B(y,(4/3)ρ)

 

| α |=1

D α u llgF1,lr0−1

F1,l



dx,

(3.48) wherec depends only on n.

Then, using Young’s inequality, we get



B(y,(4/3)ρ)



lgF1,l

r0

dx ≤ cr0ρ n+r0

cr

0ρ

2eω l

r0 

B(y,(4/3)ρ)

 

| α |=1

D α u lr0

F r0

1,l dx. (3.49)

Last inequality, using H¨older’s inequality and (3.33), gives



B(y,(4/3)ρ)



lgF1,lr0

dx ≤ cr0ρ n+r0 

cr0 r0

2r0−1 

c7(q + 1) nt/(t+1)

ρ r0

×



B(y,2ρ) ψdx

t/(t+1)

B(y,2ρ)

 1

ν

t

dx

1/(t+1)

.

(3.50)

Observe that due to (3.42) andHypothesis 3.1,



B(y,2ρ) ψdx

t/(t+1)

B(y,2ρ)

1

ν

t

dx

1/(t+1)

≤ c10(1 +M)ρ n − r0, (3.51) wherec10depends only on measure of the unit ball inRn

S Bonafede and F Nicolosi 3

μ1/ p D α... q; and c depends only on< i>c1,n, q, p, c, and diam G.

S Bonafede...



| α |=1



ν(x)−1/(q