Chipot m quittner p (eds ) handbook of differential equations stationary partial differential equations vol 3

Elliptic Equations with Anisotropic Nonlinearity and Nonstandard Growth S.. In the present chapter we study the equations and systems whosemain parts satisfy the anisotropic nonstandard

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Department of Applied Mathematics and Statistics, Comenius University,

Bratislava, Slovak Republic

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This handbook is volume III in a series devoted to stationary partial differential equations.Similarly as volumes I and II, it is a collection of self contained, state-of-the-art surveyswritten by well-known experts in the field

The topics covered by this handbook include singular and higher order equations,

problems near criticality, problems with anisotropic nonlinearities, dam problem, Γ

-con-vergence and Schauder-type estimates We hope that these surveys will be useful for bothbeginners and experts and speed up the progress of corresponding (rapidly developing andfascinating) areas of mathematics

We thank all the contributors for their clearly written and elegant articles We also thankArjen Sevenster and Andy Deelen at Elsevier for efficient collaboration

M Chipot and P Quittner

v

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Kichenassamy, S., Laboratoire de Mathématiques, UMR 6056, CNRS and Université

de Reims Champagne-Ardenne, Moulin de la Housse, B.P 1039, F-51687 Reims Cedex 2, France (Ch 5)

Lyaghfouri, A., Mathematical Sciences Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (Ch 6)

Mancebo, F.J., E.T.S.I Aeronáuticos, Universidad Politécnica de Madrid, Plaza del Cardenal Cisneros 3, 28040 Madrid, Spain (Ch 4)

Musso, M., Departamento de Matemática, Pontificia Universidad Católica de Chile, Avda Vicuña Mackenna 4860, Macul, Santiago, Chile and Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy (Ch 3) Peletier, L.A., Mathematical Institute, Leiden University, PB 9512, 2300 RA Leiden, The Netherlands (Ch 7)

Shmarev, S., Departamento de Matematicas, Universidad de Oviedo, Spain (Ch 1)

vii

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1 Elliptic Equations with Anisotropic Nonlinearity and Nonstandard Growth

S Antontsev and S Shmarev

A Braides

M del Pino and M Musso

J Hernández and F.J Mancebo

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Contents of Volume I

1 Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via

C Bandle and W Reichel

2 Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain 71

G.P Galdi

W.-M Ni

4 On Some Basic Aspects of the Relationship between the Calculus of Variations

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Contents of Volume II

T Bartsch, Z.-Q Wang and M Willem

2 Nonconvex Problems of the Calculus of Variations and Differential Inclusions 57

G Rozenblum and M Melgaard

S Solimini

xiii

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Elliptic Equations with Anisotropic Nonlinearity

and Nonstandard Growth Conditions

Stanislav Antontsev

Departamento de Matematica, Universidade da Beira Interior, 6201-001 Covilha, Portugal

E-mail: anton@ubi.pt

Sergey Shmarev

Departamento de Matematicas, Universidad de Oviedo, 33007, Oviedo, Spain

E-mail: shmarev@orion.ciencias.uniovi.es

Contents

1 Introduction 3

1.1 Assumptions and results 3

1.2 Physical motivation 6

1.3 Previous work 10

2 The Lebesgue and Sobolev spaces with variable exponents 12

2.1 Spaces L p(x) (Ω) and W01,p(x) (Ω) 12

2.2 Anisotropic spaces 14

2.3 Embedding theorems 15

3 Existence theorems 16

3.1 Generalized p(x)-Laplace equation 16

3.2 Generalized diffusion equation 24

3.3 Equations with convection terms 32

4 Uniqueness theorems 34

4.1 Uniqueness of solution of the generalized p(x)-Laplace equation 36

4.2 Uniqueness of solution of the generalized diffusion equation 40

5 Localization caused by the diffusion–absorption balance 43

5.2 The energy relation 47

5.3 The ordinary differential inequality 49

5.4 Equations with convection terms 55 HANDBOOK OF DIFFERENTIAL EQUATIONS

Stationary Partial Differential Equations, volume 3

Edited by M Chipot and P Quittner

1

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2 S Antontsev and S Shmarev

6 Directional localization caused by anisotropic diffusion 56

7 Problems on unbounded domains 72

8 Systems of elliptic equations 76

8.1 Existence of solutions 78

8.2 Localization properties 78

8.3 Systems of other types 81

9 Examples: localization in borderline cases 82

9.1 Illustrative examples 82

9.2 The ordinary differential inequality in the limit case 95

Acknowledgements 97

References 97

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1 Introduction

1.1 Assumptions and results

This chapter is a contribution to the theory of elliptic equations with nonstandard growthconditions and systems of such equations We study the Dirichlet problem for the class ofelliptic equations

Here and throughout the chapter we use bold characters to denote the vector-valued

functions u= (u ( 1) , , u (n) )and bold capitals for the matrix-valued functions The tion∇s is used to denote the matrices with the entries D j u (i) The coefficients a i , c, a ij

nota-and the exponents of nonlinearity p i , α i , σ are given functions of their arguments.

A prototype of the differential operators of the form (1.1) is the p(x)-Laplacian

 p(x) u≡ div|∇u| p(x)−2∇u

which generalizes the p-Laplacian By this reason, we term the equations of the ture (1.1) the generalized p(x)-Laplace equations Equations of the type (1.3) are called

struc-the generalized diffusion equations

We discuss the questions of existence, uniqueness and localization of weak solutions

to the formulated problems Anticipating the precise conditions on the structure of the

equations under study, let us notice here that the coefficients a i (x, u) are always sumed separated away from zero so that the possible degeneracy or singularity of equa-tions (1.1) and (1.3) is solely defined by the properties of the nonlinearity exponents

as-p i (x) and α i (x) The main feature of equations and systems of the type (1.1) and (1.2)

is the gap between the coercivity and monotonicity conditions Let us write (1.1) in theform

− div A(x, u, ∇u) = Φ(x, u),

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and system (1.2) as

− div A(x, ∇u) = Ψ (x, u),

where A : Ω× R × Rn→ Rnis a vector-valued function, and A is a matrix with the entries

a ij : Ω× Rn2 → Rn2 In the present chapter we study the equations and systems whosemain parts satisfy the anisotropic nonstandard growth conditions

For nonconstant p i (x) and p ij (x) these conditions are usually termed nonstandard because

of the existing gap between the coercivity and monotonicity assumptions Unless explicitlystated, we always assume that

Φ(x, u) = −c(x, u)|u| σ (x)−2u + F (x)

with a nonnegative function c(x, u) and a continuous exponent σ (x) > 1 In case of a

system, the function Ψ (x, s) = (Ψ ( 1) , , Ψ (n) )is assumed to satisfy similar growth ditions

with given exponents p(x), q(x), δ(x), γ (x), σ (x) > 1.

We prove that under suitable assumptions on the data the afore-formulated problemshave weak solutions The solutions are elements of the function spaces which generalizethe Lebesgue and Sobolev spaces The basic information about these spaces is collected

in Section 2 In Section 3 we prove the existence theorems for problems (1.1) and (1.3)

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The methods of proof are different for the generalized p(x)-Laplace equation and

gen-eralized diffusion equations The solvability of the Dirichlet problem for the gengen-eralized

p(x)-Laplace equation is proved via an adaptation of the Galerkin method The

applica-tion of Galerkin’s method becomes possible if the coefficients p i (x) and σ (x) are subject

to a specific regularity assumption Namely, it is requested that p i and σ are continuous

with the logarithmic module of continuity It is known that under this assumption the set ofsmooth functions is dense in the generalized Sobolev spaces A weak solution of the gener-alized diffusion equation is constructed as the limit of a sequence of regularized problems

It is requested that either α i (x) ∈ (−1, ∞) and c(x, u) is bounded away from zero, or that

of the weak solution is proved a posteriori in the case of generalized p(x)-Laplacian For

the generalized diffusion equation boundedness of the prospected solution is requested forthe proof of existence and is established a priori Systems of the type (1.2) are studied inSection 8

In Section 4 we establish uniqueness of bounded weak solutions of problems(1.1) and (1.3) In the case of equation (1.1) the proof relies on the monotonicity ofthe principle part of the differential operator and requires the presence of the lower or-

der term We claim therefore that c(x, u) c0>0 and distinguish two cases in

de-pendence of behavior of the lower-order term We either claim that σ (x) ∈ (1, 2] and

p i (x) ∈ (1, ∞), or that σ (x) ∈ (2, ∞) but p i (x) ∈ (2n/(n + 2), 2] Sticking to the

mechanical terminology we may characterize these two cases as the mathematical models

of the diffusion–absorption processes of the types (a) slow diffusion–strong absorption,(b) fast diffusion–weak absorption In the case of the generalized diffusion equation (1.3) it

is sufficient to claim that α i (x) are bounded away from minus one and infinity, α i ∈ L2(Ω),

and that either c(x, u) ≡ c(x) or that c(x, s) is monotone: for all s, r ∈ R, x ∈ Ω,

(c(x, s) |s| p−2s − c(x, r)|r| p−2r)(s − r) 0.

The localization properties of weak solutions is the next issue of the study We showfirst that the solutions of problems (1.1) and (1.3) possess the same localization propertiesthat are intrinsic for the solutions of nonlinear elliptic equations with constant exponents

of nonlinearity and isotropic diffusion The typical localization property consists in the

fol-lowing: if the right-hand side f of equation (1.1) (or (1.3)) is identically zero in a ball B rof

radius r > 0, then one may indicate a concentric ball B ρ of a smaller radius ρ such that the solution is zero in B ρ The radius ρ is defined in terms of the problem data Under an ad- ditional condition on the rate of vanishing of f near the boundary of the ball B ra stronger

localization result is established: the weak solution must be zero in the same ball B r calization of this type is always caused by a suitable balance between the principal part ofthe differential operator (the diffusion) and the low-order terms (the absorption) and holdsfor equations with isotropic and anisotropic diffusion

Lo-Such properties were studied first for “model” nonlinear equations of relatively simplestructure and the proofs relied on the possibility of comparison of the solution under studywith supersolutions of the same equation (see [26] for the details) For more complicatedequations, including (1.1) and (1.3) with constant exponents of nonlinearity, such proper-ties were established via the local energy method This method allows one to reduce thestudy of the localization properties of solutions to nonlinear PDEs with several indepen-

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dent variables to the analysis of the local energy functions which satisfy certain nonlinearordinary differential inequalities

We prove next that the solutions of equations with anisotropic diffusion possess a new

property of localization caused by strong anisotropy Let u(x) be a nonnegative solution of

the equation

in an exterior domain Ω⊂ Rn with constant exponents of nonlinearity p > 1 and σ > 1.

For this equation one may formulate the following alternative:

1 < p σ ⇐⇒ the strong maximum principle holds [69],

1 < σ < p ⇐⇒ the compact support principle holds [62]. (1.6)

It happens, however, that this alternative is wrong for the solutions of equations withanisotropic diffusion operator We show that for equations with “strong anisotropy” thesolutions can be localized in a separate direction even in the absence of the absorptionterm The analysis of this effect is carried out in Section 6 Relying on this property onecan solve the Dirichlet problem for equations (1.1) and (1.3) posed on unbounded domainswithout conditions at infinity The conditions of solvability of these problems are formu-lated in terms of geometrical restrictions on the problem domain Roughly speaking, theseare conditions on the “asymptotic size” of the problem domain at infinity

The above-described results extend to the systems of elliptic equations of similar ture The methods used to study the systems are not specific and the results are obtainedvia suitable modifications of the arguments explained in the previous sections

struc-In the concluding section we discuss several borderline cases in which the variable ponent of nonlinearity is allowed to achieve its limit value In the case of equation (1.5)

ex-with p = 2 and variable σ this would mean that the function σ (x) − 2 is not necessarily

bounded away from zero Although in the case of constant exponents σ = p = 2 the

com-pact support principle is not valid, we show that for solutions of equations with variable

exponents the localization property may persist even if p = 2 and σ (x)−2 → 0 as x → x0.The presentation is partially based on the authors’ papers [14–16]

Let us consider first the problem of image recovery Suppose that the image u0, defined

on a domain Ω⊂ Rn , is the result of a linear transformation A of the true image u to which

a random noise n has been added,

u = Au + n.

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It is requested to recover u, knowing u0 The functions u and u0 usually are scales ofgray The method of total variation smoothing (see [21] and the further references therein)consists in solving the minimization problem

with a given Lagrangian multiplier λ = const It is assumed that the noise n has

zero mean, i.e., Ω (Au − u0) dx = 0, and that the standard deviation σ is given:

Ω |Au − u0|2dx = σ2 Another approach to the image recovery consists in minimizingthe energy integral Ω |∇u|2dx under similar restrictions on the random noise n, which

leads to the problem of minimizing the integral

The former method preserves the edges of the image where|∇u| is high However, the

flaw of the method is that it may also create edges due to the presence of the random noise.Using the latter method, one eliminates the noise effect by smoothing the input, but thedrawback is that it also destroys small details of the true image A combination of the twomethods consists in minimizing the energy

with the exponent p(x) close to 2 where there are likely no edges, and close to 1 where the

edges are expected The approximate location of the edges can be determined by lookingfor the zones where|∇u| is high The minimizer of the functional J [u] is a solution of the

p(x)-Laplace equation A detailed discussion of these and more complicated models in theimage restoration problems can be found in [22,57]

A special interest in the study of equations and systems of equations with nonstandardgrowth conditions is motivated by their applications to the mathematical modeling of non-Newtonian fluids in particular, the electrorheological fluids This kind of fluids is charac-terized by their ability to drastically change the mechanical properties under the influence

of an external electromagnetic field A mathematical model of electrorheological fluids

was proposed by Rajagopal and R˚užiˇcka in [63,64] Let v stand for the velocity of the

fluid, E denotes the external electromagnetic field and P be the pressure The system of

the modified stationary Navier–Stokes equations reads as [2,3,63,64]

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Once the field E is defined from Maxwell’s equations, the exponent p(E) becomes a tion of x The system for the components of the velocity vector v transforms then into the

func-system with the following nonstandard growth condition:

For p+> p−, this condition is known in the literature as the nonstandard growth condition

of (p+, p−) type We refer to the papers by Acerbi and Mingione [2,3] for a discussion ofthe regularity properties of weak solutions of the systems of equations with this type ofnonlinearity (see also the references therein to the previous work on this issue) The regu-larity of weak solutions to the parabolic counterpart of such systems is studied in [4].Equations and systems of equations with this type of nonlinearity appear also in themathematical modeling of stationary thermo-convective flows of non-Newtonian fluids

[11,9,71] In particular, in [11] the stress tensor S has the form

S= −pI +μ(θ ) + τ(θ) D(u) p(θ )−2

D(u),

where u is the velocity vector, p pressure, θ denotes the temperature and μ, p, τ are known

coefficients depending on the temperature This hypothesis leads to the system of equationswith nonstandard growth conditions

The principal parts of the equations and systems studied in the above-quoted papers

can be regarded as generalizations of the p(x)-Laplace equation which is formally elliptic wherever 0 < |D i u | < ∞, degenerates at the points x ∈ Ω where either |D i u| = 0 and

p i (x) >2 or|D i u | = ∞ and p i (x) <2, and becomes singular if either |D i u| = 0 and

p (x) <2 or|D u | = ∞ and p (x) >2

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As an example, let us consider the motion of a fluid in a porous medium and trace theinfluence of the hypotheses about the properties of the fluid and the medium on the com-

plexity of the mathematical model of this process Let us denote by ρ, v and p the density,

velocity and pressure of the fluid We assume first that the medium is homogeneous andisotropic and that there are no external forces and the mass sinks and sources The classicalDarcy law suggests that the fluid velocity is proportional to the gradient of pressure,

v= −k∇p, k = const. (1.8)

For the incompressible fluids the continuity equation holds, div v= 0 Combining these

two equations we conclude that the pressure p is a harmonic function For the compressible

fluids (gases) the continuity equation has the form div(ρv)= 0 In barotropic gases the state

equation is given by relation

p ≡ p(ρ) = ρ γ , γ = const, (1.9)which leads to the nonlinear elliptic equation for the pressure

Let now the medium be nonhomogeneous and anisotropic, i.e., the characteristics of the

medium may vary in dependence on the direction x i and the point of the medium: now λ i≡

λ i (x) , γ = γ (x), and instead of the constant k in (1.8) a diagonal matrix K(x) is used Let

us also admit the presence of the exterior mass forces and sources and sinks of mass which

may depend on the point x, the pressure p and its gradient ∇p Under these assumptions

the pressure in the incompressible fluid obeys the anisotropic generalized p(x)-Laplace

equation of the form

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The functions λ i (x) , γ (x), K i (x)can be either known function given a priori or may

implicitly depend on x For example, in the nonisothermic processes the exponent γ in the state equation is a function of the thermodynamic parameters, say, the temperature θ (x) Then p = ρ γ (θ (x)) , where the function θ (x) has to be defined from some complementary

conditions

The study of the qualitative properties of such equations is very important for cations and indispensable for understanding the mechanism of formation of the stagnationzones and the behavior of the flow around them We refer to the monographs [18,25,60] forthe derivation and thorough analysis of the mathematical models of continuum mechanics

of problem (1.11) is sought as an element of the generalized Sobolev space (also called

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Orlicz–Sobolev space or Musielak–Orlicz space) W01,p(x) (Ω) The theory of the ized Sobolev spaces is developed in [30,33,35,40,41,46,55,61,65,70,72,73] (see also thereview papers [45,66] for the relevant bibliography).

general-The solutions of equation (1.1) and system (1.2) belong to the anisotropic analogs of eralized Sobolev spaces In Section 2 we introduce the anisotropic generalized Lebesgue–Sobolev spaces and collect the already known results on the properties of function spaces

gen-W01,p(x) (Ω) and L σ (x) (Ω) The restrictions imposed on the regularity of ∂Ω and the claim

of logarithmic continuity of the exponents of nonlinearity in equations (1.1), (1.2) tee the fulfillment of the properties of density and embedding in the generalized Sobolevspaces

guaran-The localization properties of solutions of the p-Laplacian equation with absorption

terms and alternative (1.6) are given a detailed discussion in [62] The study relies on thestrong maximum principle for elliptic equations (with constant exponents of nonlinearity).The strong maximum principle asserts that if a nonnegative classical solution of equa-

tion (1.5) vanishes at a point x0∈ Ω, it must be identically zero in Ω, while the compact

support principle says that if Ω is an exterior domain inRn and u(x) is a classical tion of equation (1.5) such that u → 0 when |x| → ∞, then the support of u is compact.

solu-The compact support principle also holds for the weak solutions of general elliptic tions which contain (1.5) as a partial case This was proved by means of the method oflocal energy estimates – see [10], Chapter 1, and references therein for the history of thequestion

equa-An analog of the strong maximum principle for p(x)-Laplace equation is proved in the recent work [42] Let u(x) ∈ W 1,p(x) (Ω)be a weak supersolution of the equation

− p(x) u + c(x)|u| σ (x)−2u= 0

with p(x) ∈ C1( , σ (x) ∈ C( Ω) , c(x) ∈ L∞(Ω) and c(x) 0 in Ω Assume that the

exponents p(x) and σ (x) satisfy the condition

p(x) σ (x) < p∗(x)=

p(x)n

n −p(x) if p(x) < n,

∞ if p(x) > n.

If u 0 in Ω and u(x) ≡ 0 on Ω, then for every nonempty compact subset K ⊂ Ω there

is a positive constant C such that u(x) C a.e in K.

The possibility of directional localization in solutions of equations with anisotropic linearity in an infinite layer was described in [10], Chapter 2, for the solutions of a specialstructure

non-The validity of the compact support principle (alias the localization property) of weaksolutions suggests the possibility of existence of weak solutions to equations and systems

of the type (1.1)–(1.2) defined on the whole ofRn or on noncompact domains inRn Theexistence and multiplicity of solutions for the equation

 p(x) + c(x)|u| p(x)−2u = f (x, u) in R n

is studied in [34]

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In this chapter we do not discuss the regularity of the weak solutions of equations andsystems under study Relevant results can be found in [1–3,6,17,28,29,32,38–40,59] whichdeal with equations and systems of the type (1.1), (1.2) and their parabolic counterparts.The questions of existence, uniqueness and qualitative behavior of solutions of ellipticequations of the type (1.1) and (1.3) with constant exponents of nonlinearity, as well asparabolic equations with elliptic parts of similar form, were studied by many authors, see[10,19,23,24,26,27,31,43,49–51,58,68] and the literature cited therein

Parabolic equations with variable exponents of nonlinearity in the elliptic part were ied in papers [7,12,13], nonlinear parabolic equations with singularly disturbed exponents

stud-of nonlinearity near the critical values in the elliptic part were considered in [52–54]

2 The Lebesgue and Sobolev spaces with variable exponents

In this section we introduce the function spaces used throughout the chapter and describetheir basic properties The definitions of the function spaces and the sketch of their prop-erties presented in this subsection follow [48,55,61,67] (see also the works cited in theIntroduction)

2.1 Spaces L p(x) (Ω) and W01,p(x) (Ω)

It is always assumed that Ω is a bounded domain inRnwith Lipschitz-continuous

bound-ary and the function p(x) satisfies the conditions

1

becomes a Banach space

2 The following inequalities hold:

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4 According to (2.4), for every 1 q = const < p− p(x) < ∞,

It is straightforward to check that, for|Ω| < ∞,

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8 Sobolev’s inequality is valid in the following form: if p(x) satisfies conditions (2.1)–(2.2), then there exists a constant C > 0 such that, for every f ∈ W 1,p(x)

becomes a Banach space By V(Ω) we denote the dual space to V(Ω).

1 The spaces W01,p(x) (Ω) and L σ (x) (Ω)are reflexive and separable According to (2.5)

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For further convenience, we quote here the embedding and the trace-interpolation theorems

in Sobolev spaces which are repeatedly used further The thorough study of this issue can

be found, e.g., in [5] Let Ω⊂ Rn be a bounded domain with piecewise smooth

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3 For every u(x) ∈ W 1,p (Ω),

p

, p= p

3 Existence theorems

3.1 Generalized p(x)-Laplace equation

In this section we study solvability of the Dirichlet problem for generalized p(x)-Laplace

It is assumed that the data of problems (3.1) satisfy the following conditions

1 Ω⊂ Rn is a bounded domain with Lipschitz-continuous boundary Γ = ∂Ω.

2 The coefficients a i (x, r) and c(x, r) are Carathéodory functions (measurable in x for all r ∈ R and continuous in r for almost all x ∈ Ω) Unless explicitly stated, we

always assume that a i and c satisfy the conditions

0 < a0 a i (x, r) < ∞ ∀x ∈ Ω, r∈ R,

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with some positive constants a0, c0.

3 The functions p i (x) and σ (x) are bounded in Ω: it is assumed that there exist stants p−∈ (1, n), p+< ∞, σ−> 1, σ+< ∞ such that, for all x ∈ Ω,

The weak solution to this problem is understood as follows

DEFINITION 3.1 A locally integrable function u(x) is called weak solution of

3.1.1 A model equation. We begin with the study of the special situation when the

coef-ficients a i and c do not depend on u and the proof of existence of weak solutions is fairly

has at least one weak solution.

PROOF Let us introduce the operatorL : V(Ω) → V(Ω),

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It is obvious that the mapping L : V(Ω) → V(Ω) is continuous According to

(2.10)–(2.11) it is monotone

LEMMA3.1 The operator L : V(Ω)→ V(Ω) is coercive:

∀u ∈ V(Ω), (Lu, u) C min

Then there is i ∈ {0, 1, , n} such that λ i λ/(n + 1) (we do not loose the generality by

assuming that i= 1) It follows from (2.3) that

( Lu, u) CA p( ·) ( ∇u) + A σ ( ·) (u)

The space V(Ω) is separable, and the mapping L : V(Ω) → V(Ω)is monotone,

con-tinuous and coercive By the Browder–Minty theorem [20], Theorem 7.3.2, for every

Φ∈ V(Ω)the equationLu = Φ has at least one weak solution u ∈ V(Ω).

3.1.2 The general case.

THEOREM 3.2 Let conditions (3.2)–(3.4) be fulfilled, f ∈ L σ(x)

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Define the operatorL Ω : V(Ω)→ V(Ω),

Since the space V(Ω) is separable, there exists a fundamental system {φ k (x) } ⊂ V(Ω) Let

us search Galerkin’s approximations u (N )=N

k=1c kN φ k (x)as the solutions of the system

i=1v i φ i (x) we introduce the scalar product (v, w) P N =N

i=1v i w i and the norm

Ω

For every fixed u (N ) the left hand-side of this equality is a linear functional onP N By

Riesz’s theorem there exists g(u (N ) ) ∈ P N such that

(See the proof of Lemma 3.1.)

LEMMA3.3 The operator g( ·) : P N → P N is continuous.

PROOF Let {v k } ⊂ P N and|v k − v| N → 0 It follows that |v k − v| σ (x)+i |D i (v k−

v)|p i (x) → 0 almost everywhere in Ω Given an arbitrary ε > 0 one may choose a set

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On the other hand,

Ω ε (v k , η) − L Ω ε (v, η) 0 when k → ∞ for every fixed ε.

Thus, for every η ∈ P N (g(v k ) − g(v), η) N → 0, which means that g(v k ) → g(v) weakly

inP N Since the dimension ofP N is finite, the weak convergence g(v k ) → g(v) implies

LEMMA 3.4 For every N ∈ N, the solution u (N ) of problem (3.8) satisfies the estimate

(N )

V K with a finite constant K independent of N.

PROOF Let us take η = u (N )for the test-function in (3.8) and then apply Young’s ity:∀ε > 0,

LEMMA3.5 For every N ∈ N, the equation g(u (N ) ) = 0 has at least one solution in P N

PROOF Consider the family of operators

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It follows that (g τ (v), v) N > 0 for all τ V= |v| Nsufficiently large,say, for|v| N = R N n + 1 For τ = 0 the equation g0(v) = 0 has in the ball B N = {v ∈

P N: |v| N < R N } only the trivial solution, and for every τ ∈ [0, 1], the boundary of the ball

B N does not contain any solution of the equation g τ (v)= 0 According to Brouwer’s fixed

point theorem, the equation g1(v) ≡ g(v) = 0 has at least one solution in the ball B N

By Lemma 3.4, the sequence of Galerkin’s approximations contains a subsequence

{u (N ) } possessing the following properties: there exist functions u ∈ V(Ω), A i (x)∈

To complete the proof, we have to identify the limits A i (x) and v(x).

LEMMA3.6 For almost all x ∈ Ω,

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Letting N→ ∞ and using (3.9), we have that

Since ξ is arbitrary, we may take ξ = u ± εζ with ε > 0, ζ ∈ V(Ω) Simplifying and then

letting ε→ 0, we conclude that

By virtue of (3.9), we may pass to the limit when N→ ∞ in all three terms of this equality

The two terms on the left-hand side tend to zero, whence

3.1.3 Boundedness of weak solutions.

THEOREM 3.3 Let in the conditions of Theorem 3.2 ∞,Ω = K < ∞ and c0>0

Then the weak solution of problem (3.1) satisfies the estimate

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PROOF Let us define the constant μ from the relation K = c0μ σ−−1and then set M

ε=

max{1; μ + ε} with an arbitrary ε > 0 Set ζ ε = max{u − M ε ,0} and choose this function

for the test-function in the integral identity (3.5) Notice that

ζ ε=

u − M ε if u > M ε,and

∇ζ =

0 if u M ε,

∇u if u > M ε.Identity (3.5) becomes

Ω ∩(uM ε )

n

PROOF Fix an arbitrary k ∈ N and take the function ζ(x) = max{0, u − k} for the

test-function in Definition 3.1 Denote Ω k = Ω ∩ {x ∈ Ω: u(x) > k} and notice that

∇ζ(x) =

∇u for u > k,

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3.2 Generalized diffusion equation

Let us consider the problem

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(2) α i (x) and σ (x) are continuous functions satisfying the conditions

i , α±, σ±are known constants;

(3) unless specially indicated, Ω⊂ Rnis a bounded domain with Lipschitz-continuous

boundary Γ

The solution of problem (3.13) is understood in the following way

DEFINITION3.2 A locally integrable in Ω function u(x) is called weak solution of

prob-lem (3.13), if

(1) u ∈ L∞(Ω),|u| α i (x)/2|D i u | ∈ L2(Ω) , i = 1, , n,

(2) u = 0 on Γ in the sense of traces,

(3) for every test-function η ∈ W 1,2

0 (Ω) ∩ L σ (x) (Ω), the integral identity holds

with a constant Λ depending on

REMARK3.1 As is shown further, one can essentially relax condition (3.18) and claimits fulfillment in the following form:

D i α i (x)

2,Ω− C, Ω i−=x ∈ Ω: α i (x) 0.

P N

Riesz’s theorem there exists g(u (N ) ) ∈ P N such that

(See the proof of Lemma 3. 1 .)

LEMMA3. 3... 1 ,p (? ?),

p 

, p= p 

3 Existence theorems

3. 1 Generalized p( x)-Laplace... ε, 

∇u if u > M ε.Identity (3. 5) becomes

Ω ∩(u M ε )

 n 

PROOF Fix

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