Analysis of a singular convection diffusion system arising in turbulence modelling

We shall study some singular stationary convection diffusion system governing the steady-state of a turbulence model closely related to the k− ε one. We shall establish existence, positivity and regularity results in a very general framework.


Introduction
We shall first recall some basic ideas concerning the statistical turbulence modelling for fluids.The reader can consult [1,2] for a more detailed introduction.
Let u, , , and  be the velocity, pressure, density, and temperature of a Newtonien compressible fluid.Let also Ω ⊂ R 3 be a domain which is assumed to be bounded.Then the motion of the flow in Ω at a time  ∈ R + can be described by the compressible Navier Stokes equations (see system (C) page 8 in [3]).It is well known that direct simulation based on such a model is harder or even impossible at high Reynolds numbers.The reason is that too many points of discretization are necessary, and so only very simple configurations can be handled.
Thus, engineers and physicists have proposed new sets of equations to describe the average of a turbulent flow.The most famous one is the - model, introduced by Kolmogorov [4].We shall briefly present its basic principles in the following.Let V denote a generic physical quantity subject to turbulent (i.e., unpredictable at the macroscopic scale); we introduce its mean part (or its esperance) ⟨V⟩ by setting: ⟨V (, )⟩ = ∫ P V (, , ) P () , where the integral is taken in a probablistic context which we shall not detail any more here.Note, however, that the operation ⟨⋅⟩ is more generally called a filter.The probablistic meaning is one but not the only possible filter (see, for instance, [1] chapter 3).We shall then consider the decomposition: V = ⟨V⟩ + V  , where V  is referred to the noncomputable or the nonrelevant part and ⟨V⟩ is called the mean part (i.e., the macroscopic part).
The principle of the - model is to describe the mean flow in terms of the mean quantities ⟨u⟩, ⟨⟩, ⟨⟩, and ⟨⟩ together with two scalar functions  and , which contains relevant information about the small scales processes (or the turbulent processes).The variable  (SI: [m 2 /s 2 ]) is called the turbulent kinetic energy, and  [m 2 /s 2 ] is the rate of dissipation of the kinetic energy.They are defined by where ] is the molecular viscosity of the fluid.The model is then constructed by averaging (i.e., by appling the operator ⟨⋅⟩ on) the Navier-Stokes equations.Under appropriate assumptions (i.e., the Reynolds hypothesis in the incompressible case and the Favre average in the compressible case), we obtain a closed system of equations for the variables ⟨u⟩, ⟨⟩, ⟨⟩, ⟨⟩, , and  (see [1] pages 61-62 for the incompressible case and pages 116-117 in the compressible situation).
Here, we shall focus on the equations for  and , and we consider that the others quantities ⟨u⟩, ⟨⟩, ⟨⟩, and ⟨⟩ are known.Moreover, in order to simplify the readability, we do not use the notation ⟨⋅⟩; that is, in the sequel we will write u instead of ⟨u⟩ and  instead ⟨⟩ to represent the mean 2 International Journal of Partial Differential Equations velocity and density of the fluid.The equations for  and  are of convection-diffusion-reaction type: where (, ) := div u(, ), (, ) := (1/2)|∇u + (∇u)  | 2 − (2/3)(, ) 2 ≥ 0 (see Appendix B in the appendices), and  ] ,   ,  1 , and  2 are generally taken as positive constants (see (A.3) in the appendices).Note that (3) is only valid sufficiently far from the walls.In fact, in the vicinity of the walls of the domain Ω, there is a thin domain Σ called logarithmic layer in which the modulus of the velocity goes from 0 to O (1).In this layer we can use some wall law or a one equation model (see [1] chap. 1 and [5]) instead of (3).However, (3) can be considered even in the logarithmic layer if we allow the coefficients  ] ,   ,  1 , and  2 to depend appropriately on some local Reynolds numbers (see [1, pages 59-60 and page 115]).In this last situation the system is called Low-Reynolds number - model.
In the following, we focus on the study in the domain Ω := Ω \ Σ, and we assume that its boundary Ω is Lipschitz (see [6] page 127).We denote by n() the outward normal defined for almost all points  ∈ Ω.The boundary conditions for  and  on Ω are then on the form  =  0 ,  =  0 on Ω, where  0 and  0 are strictly positive functions which can be calculated by using a wall law (see [1, page 59]) or a one equation model (see [5]).In the following, we assume that  0 and  0 are given.Moreover, we can assume (see again [1, page 59]) that u ⋅ n = 0 on Ω.
We shall concentrate in this paper on a modified system obtained after introducing the new variables  [s] and  [m −2 ]  given by These variables have a physical meaning (see [2]):  represents a characteristic time of turbulence and  =  −1/2 is a characteristic turbulent, length scale.By using this change of variable in (3) and after considering some modelisation arguments for the diffusion processes (see the appendices), we obtain where the coefficients   ,   , and   are all positive.Problem () differs from the - one only by the diffusive parts, and it is attractive by some stronger mathematical properties.Another model closely related to these systems, and having some popularity, is the - one (with  =  −1 ; see for instance [2]).
In the papers [7,8], the authors have established the existence of a weak solution for () and a property of positiveness.This last feature takes the model useful in practice: it can be used directly or also as an intermediary stabilization procedure to the - one (see [9]).Another important property attempted for a turbulence model is its capability to predict the possible steady states.In the previous works (except in [10]) only the evolutive version of () was studied under very restrictive assumptions.In [10], however, the stationary problem is studied, but it is simplified by considering a perturbation of the viscosities that artificially cancel the singularity of the system.
Hence, in this paper we shall study the stationary version of () on a bounded domain Ω ⊂ R  ,  = 2 or 3, on which we impose the boundary conditions  = ,  =  on Ω.Remark that, by using (4) together with (6), we obtain Hence, we can assume that  and  are strictly positive given functions.
We shall establish existence, positivity, and regularity results in a very general framework.

Assumptions and Notations.
Let () denote the stationary system associated with ().For simplicity we introduce the new parameters  ind :=  ind where the subscript "ind" takes the integer values 3, 4, 5, 6, 7, and 8 or the letters  and .Then, our main model () has the following form: For physical reasons we are only interested in positive solutions (, ) for ().Note, however, that even with this restriction, the problem () may be singular (i.e., the viscosities ] + (  /) and ] + (  /) may be unbounded).Moreover, because we allow ] ≡ 0, the equations may degenerate (i.e., the viscosities may vanish).Hence, without additional restriction there may be various nonequivalent notions of weak solution (see for instance [11]).
In fact a good compromise between respect of the physics, simplification of the mathematical study, and obtention of significative results is to restrict  and  to be within the class S defined by In particular, if the parameters appearing in () are sufficiently regular and if we restrict  and  to be within the class S, then the notion of a weak solution for () is univocally defined: it is a distributional solution (, ) that satisfies the boundary conditions in the sense of the trace.
In this last situation we will tell that (, ) is a weak solution of () in the class S.
In order to be able to consider such a weak solution for (), we shall precise in the following some sufficient conditions of regularity for the data.
Let  = 2 or 3 denote the dimension of the domain Ω, and let  be a fixed number such that We then have the following continuous injection (see Lemma 5): , where  =  * > .(10) Recall that  = div(u).We will consider the following assumptions.
(i) Assumptions on Ω: bounded, and it has a Lipschitz boundary Ω.(11) (ii) Assumptions on the flow data (when  = 2, one assumption in ( 14) can be relaxed: u ∈ (  (Ω)) 2 with  > 2 (instead of  = 3) is sufficient, but this would not improve any result significantly.)u, , , , and ] are as follows: ,  : (iii) Assumptions on the turbulent quantities on the boundary are as follows: where  > 0 is a fixed number.
We will study problem () under themain assumption: Note that in the main situation ( 0 ) the assumption (12) made for ] allows the possibility that ] ≡ 0. In other words the molecular viscosity ] can be neglected in the model.This is often chosen in practice because the eddy viscosities   / and   / are dominant in the physical situations (see [1,5]).Remark also that the coefficients   are allowed to depend on , and the viscosity parameters   ,   may depend on , , .
For a given function  : Ω → R, we shall use the notations  + and  − to represent the positive and negative parts of , that is We will also consider some assumptions of low compressibility of the form for some  > 0 that will be precised.This last kind of condition seems to be necessary (see the Appendices) in order to obtain a weak solution for () in the three-dimensional case, whereas when  = 2, we shall use some particularities of the situation to obtain a weak solution without any assumption of low compressibility.Nevertheless in this case we will assume that in addition to ( 0 ) the following condition is satisfied: In all the situations, we have the following regularity result.
Theorem 3. Let (, ) be a weak solution of () in the class S, and assume that ( 0 ) is satisfied.We have the following.

Discussion on the Results
. Compared to previous works (see [1,7,12,13]) our basic assumption ( 0 ) made in Theorem 1 is very general.In particular, we do not artificially cancel the singularity in the model, and we only assume weak regularity on the data.For instance, the basic assumption we made on the mean flow is u ∈ ( 3 (Ω))  and div u ∈   (Ω) with some  > 3/2, whereas in the previous works it was assumed that u ∈ ( ∞ (Ω))  and div u ∈  ∞ (Ω).Our condition is more interesting from a practical point of view because it is satisfied when  is a weak solution of the Navier Stokes equations.Hence, our work could be used for a future analysis of the full coupled system Navier-Stokes plus ().
From a mathematical point of view the problem we study is a nonlinear, degenerate, and singular elliptic system.Several complications arise for its analysis.In particular, the balance between the increase/decrease of the source terms (i.e., the functions   , , and  appearing in the second members of ()), and the possible explosion/vanishing of the viscosities is difficult to control.The strategy followed here is to first carefully study some elliptic scalar problem (possibly degenerate and singular) of the form By using and developing some techniques due to Stampacchia, we are able to establish existence, positivity, and regularity results for problem ().These results (see Proposition 9), which also have an independent mathematical interest, are the key ingredients for proving Theorem 1.Under the additional assumption u ∈ ( ∞ (Ω))  , we give a Hölder continuity result for  and .Moreover, we establish an existence result for a classical solution under some smoothness assumptions on the data.

Organization of the Paper
(i) In Section 3, we shall recall some results concerning the truncature at a fixed level and the Stampacchia's estimates.This last technique takes an important role in our analysis; moreover, we shall need a precise control of the estimates.Hence, we shall present it with some details and developments.
(ii) In Section 4, we introduce a sequence (  ) of problems which approximate ().For  fixed (  ) is a PDE system of two scalar equations of the form (): one equation for the unknown  +1 and one for  +1 .The point is that the unknowns  +1 and  +1 are only weakly coupled.The coupling of the two equations is essentially realized through the quantities   and   calculated at the previous step.Hence, we shall firstly study carefully the problem ().The major tool used here is the Stampacchia's estimates.We next prove that (  ) is well posed.Hence, we obtain an approximate sequence of solutions (  ,   ) for problem ().Moreover, we prove that   and   are uniformly bounded from above and below, which are the key estimates.
(iii) In Section 5, we use the uniform bounds established in Section 4, in order to extract a converging subsequence from (  ,   ).We then prove that the limit (, ) is a weak solution of () in the class S.
Under the additional assumption u ∈ ( ∞ (Ω)) 3 , we are able to use the De Giorgi-Nash theorem, and we obtain an Hölder continuity result for , .By assuming in addition some smoothness properties for the data, we can iterate the Schauder estimates and prove Theorem 3.
(iv) In the appendices, we present the derivation of the - model from the - one.Moreover, we justify that the choice  =  2 / 3 is valid even in the compressible situation.The justification uses in particular a property of positivity of the function .We also discuss briefly the necessity of the low compressibility assumption when  = 3.Finally, we recall a generalized version of the chain rule for () where  is a Lipchitz function and  a Sobolev one.

Mathematical Background
In this section, we shall recall some results concerning the truncature at a fixed level and the Stampacchia's estimates.This last technique takes an important role in our analysis; moreover, we shall need a precise control of the estimates.Hence, we shall present here the technique with some details and developments.As in the rest of the paper we denote by Ω ⊂ R  a bounded open Lipschitz domain.These properties for Ω are always implicity assumed if they are not precised.

Truncatures and Related Properties. The technique of
Stampacchia is based on the use of special test functions which are constructed by using some truncatures.We shall recall some basic properties of the truncatures used in the paper.An important tool is the generalized chain rule (see Theorem D.1 in the appendices).Let  > 0; we denote by   the truncature function   : R → R defined by Note that   (⋅) truncates both the positive and the negative large values.In some cases, we need only to truncate the positive or the neghative side.For this reason, we introduce the semitruncatures  ,+ and  ,− defined by We then have the decomposition: For given V  ∈  1 (Ω) and  > 0, we shall also consider where we have used the notations: Let also  ,± be the functions defined above (30) while replacing   by  ,+ or by  ,− .It is easy to verify that  ,+ (resp.,  ,+ ) is in fact the positive (resp., the negative) part of   .In other words, we have The function   has the following properties: Where  :  1 (Ω) →  1/2 (Ω) denotes the trace function.

The Stampacchia Estimates.
The Stampacchia estimates is a general method which allows one to obtain an  ∞ -estimate for the weak solution of a large class of elliptic PDEs of the second order.The  ∞ -estimate presented in the original work [15] depends on various quantities related to the PDE problem studied, but the exact dependency is not established.In our analysis we need a precise control of the  ∞ -estimates with respect to some quantities (in particular with respect to the diffusion coefficient of the PDEs).Hence, in the following, we take over and detail the technique in order to obtain a more precise  ∞ -estimates.
The Stampacchia estimates are established by using the test functions   (or  ±  ) defined previously, where in this case V (resp., V  ) is a weak solution of the problem (resp., the sequence of problems) considered.
For technical reasons we need a classical result concerning some relationships between   functions and linear form on Sobolev spaces.
International Journal of Partial Differential Equations and there exists Ẽ ∈ (  (Ω)) 3 such that (34) Moreover, we have Proof.Property (33) is easy to prove by using the Sobolev injection Theorem; together with the Hölder inequality:  → ∫ Ω  is a linear form on  We next obtain (34) by using a classical result (see [16] Proposition IX.20).
The Stampacchia technique works in two steps: the first one is dependent on the problem (or the sequence of problems) studied, and the second one is independent of it.
Here, the purpose is to present the key ingredients of these two steps.Because the first one is dependent on the problem studied we cannot present it here in its entirety, but we will consider a simple problem which contains the main technical points (in fact this introductive presentation will be useful to treat a more complicated class of problems in Section 4).Let (V  ) ⊂  1 0 (Ω) be a sequence of functions satisfying where (]  ) is a given sequence of strictly positive bounded functions and (  ) ⊂   (Ω), with  > /2.Let also   ,   denote the bounds from above and below for ]  , that is, Step 1.By testing (36) with   =   (V  ), we obtain (40) This is the key estimate needed to pass at the second step which is independent of the problem studied.Note that with the particular sequence of problems (36) chosen here, the constants  and Φ are Hence, Φ does not depend on , .Moreover, if we assume that (  ) is uniformly bounded in the   -norm and that (  ) is uniformly bounded from above by a strictly positive constant, then  is also independent of , .This is an important point because we will see it hereafter; an estimate (40) with  and Φ independent of ,  leads to a uniform  ∞ bound for (V  ).
Step 2. Assume that we have obtained (40).We can obtain an  ∞ -estimate for V  as follows.
Let 2 * = 2/( − 2) be the Sobolev exponent associated to 2 in dimension .By using the Poincaré-Sobolev inequality we have Recall that we have assumed in (40) that Φ > 2/2 * .Hence, 2 * Φ/2 > 1, and by using Lemma 4.1 in [15] we obtain This property tells exactly that International Journal of Partial Differential Equations 7 In particular,  does not depend on  if the constants  and Φ appearing in (40) are independent of .For instance with the particular sequence of problems (36) the constants  and Φ are given by (41), and if we assume that ‖  ‖   ≤ ,   ≥  > 0, we obtain Remark 6. (i) If you are only interested in obtaining a uniform majoration or minoration for V  then instead of (40) it suffices to have In fact in this case we consider  ± () := | ± , | instead of (44).This function is decreasing, and we obtain again (46).But now this property tells that ±V  () ≤  ± a.e. in Ω,  ± = C± (Φ ± , , |Ω|) √  ± .(50) (ii) Let again (V  ) be a sequence of functions satisfying (36), and assume that   ≥  > 0.Then, we have The proof of (51) and ( 52) is obtained by taking over the first step of the technique of Stampacchia: we use the test function  ±  instead of   .

Approximate Sequence and Estimates
Let  ∈  1/2 (Ω); we denote by R its harmonic lifting that is: We define the functions  0 and  0 by the formula Hence, by the using the maximum principle (see [16, p. 189] and [17]) together with the condition ( 16), we obtain Let now  ≥ 0, (  ,   ) be given, and In order to construct an approximate solution ( +1 ,  +1 ) for problem (), we introduce the following system: where we used the notations For  ∈ N we denote by (  ) the following condition: Let  max be a fixed real number such that We shall also consider the condition: (  ) := (  ) + (  ≤  max ).
Note that (57) shows that the condition (  ) is satisfied for  = 0. We will prove in the sequel that, under condition (  ), we can obtain a weak solution ( +1 ,  +1 ) for problem (  ), with moreover ( +1 ,  +1 ) satisfying the condition ( +1 ).This last property ensures the right definition of an approximate sequence.More precisely, we have the following.
Moreover, ( +1 ,  +1 ) satisfies condition ( +1 ) and the estimates where  max was fixed in (60) and  min ,  min , and  max are positive numbers depending on DATA, but not on .
Remark 8. Proposition 7 is the key result that will be used later on to prove Theorem 1, whereas for Theorem 2 we shall establish and use a more simple version of this proposition (see Section 5.2).
In order to prove the proposition we establish intermediate results.
Remark that the system (  ) is composed of two coupled scalar elliptic equations in divergence form, with a possible singular and degenerate structure.Hence, the goal of this subsection is to study this last kind of scalar problem.
In order to do this, we first introduce a weight  : R + → R + which is assumed to be measurable and satisfying where  0 and  1 are two given reals.Let also  : Ω × R + → R be a Caratheodory function (i.e., for all  ∈ R + :  → (, ) is measurable, and for a.a. ∈ Ω :  → (, ) is continuous).
We want now to find sufficient additional conditions for  that guarantee the existence of a bounded positive weak solution for problem ().Hence, we shall consider where  1 ,  2 ∈   (Ω) and ℎ : R + → R + is continuous.In fact, more than establishing only the existence of a bounded positive solution for (), we are interested in obtaining some uniform (with respect to  1 ) bounds from above and below and some regularity results.We have the following.
Before proving Proposition 9, we establish an intermediate result.In a first step we consider the change of variable V = ln  in (), and for  ∈ N we introduce a truncated version (  ) of the system obtained: We then establish the following.Lemma 10. (i) Let  satisfy (62), and let  : Ω × R + → R be a Caratheodory function satisfying (63), (64).Then, for any  ∈ N, there exists a weak solution V = V  ∈  1 (Ω) ∩  ∞ for the problem (  ).
(ii) Let (V  ) be the sequence given in (i), and let  ≥ 1 be a fixed integer.Then, there exists   > 0 such that if the function  2 in (63) satisfies ‖ 2 ‖   ≤   then we have In particular,  is independent of , ], and .
Proof.(i) By using the divergence formula, we obtain, for all  ∈  1 (Ω), (68) Let V 0 := ln(R 0 ), and consider the change of variable where We now remark that ( 69) is a quasilinear equation in divergence form.Moreover, it is easy to see that  and  satisfy the classical growth assumptions and  satisfies also the classical coercivity condition.Note that (71) Hence,  is strictly monotonous in the third variable.We then conclude (see for instance [18,Theorem 1.5] or [14, Theorem 8.8 page 311]) that there exists a weak solution Ṽ ∈  1 0 (Ω) for (69).Consequently, V  := Ṽ + V 0 is a weak solution for (  ), that is, for all  ∈  1 0 (Ω), By applying Theorem 4.2 page 108 in [15], we obtain V  ∈  ∞ (Ω). (ii) Let   := 1 ℎ (  ) > 0, and assume now that      2      ≤   . ( With this additional assumption, we are able to obtain a useful estimation for ‖V  ‖  ∞ .Technically, we will detail a method due to Stampacchia.(See Section 3.2 for the notations and for an introduction of the method.Here only the first step of the technique will be developed further).Let  > | ln ‖ 0 ‖  ∞ (Ω) |; we consider the function   = V  −   (V  ).We have (see Lemma 4)   ∈  1 0 (Ω) ∩  ∞ (Ω), and by testing (72) with   , we obtain (a) We will now evaluate the terms I and II.The term I is simplified by writing one of its integrand factors, namely, ∇   (V  )   , as a gradient.More precisely we have ∇   (V  )   = ∇  , with   ∈  1 (Ω) ∩  ∞ (see Lemma D.2 in the Appendices).Hence, by applying the divergence formula, we see that I vanishes: We next estimate the term II: Remark now that on  − , we have V  ≤ − ≤ 0, which implies that    (V  ) ≤ 1. Consequently by using the assumption (63), we obtain The term II 1 is majorated as follows: Let  :=  V  .We have (see Theorem D.1 in the Appendices) On the other hand, V  ≥ − almost everywhere implies that  ≥  − a.e. in Ω.Hence, we obtain (65) by setting  min =  − and  max =   .
(a) The estimation (66) is obtained as follows.By using the test function  +  instead of   , we obtain This last estimation is only a first step in order to obtain the majoration for  max announced in (66).
By using next the Stampacchia technique (see again Section 3.2, Remark 6), we major the function  2 as follows: This leads to the majoration (66).

Proofs of the Theorems
We begin by a lemma.(101) Moreover by using (98) together with the property  +1 ≥  min > 0, we obtain Hence we can pass to the limit in the approximate problems (  ).We obtain a weak solution (, ) for problem ().That is for all  ∈ D(Ω): We will see that this last property allows one to obtain a weak solution for problem () under the assumptions ( 0 ) and ( 1 ) but without assuming a low compressibility condition of the form (23).
In order to prove this result, we take over the proof of Proposition 7 with slight modifications: if (  ,   ) is given and satisfies (  ) (it is not useful to consider (  ) here), then problem (  ) has at least one solution ( +1 ,  +1 ) satisfying in addition ( +1 ) and the estimates (61).
Step 3. By using (107) together with (108) we obtain the estimates (61).Hence, we have recovered the conclusions of Proposition 7. The remainder of the proof for Theorem 3 is exactly the same as for Theorem 1: we can extract a subsequence with the properties (98)-(99).These properties are sufficient to pass to the limit  → ∞ in (  ), and Theorem 2 follows.Hence, we can apply the first point in Proposition 9(ii) in each equation of ().We obtain ,  ∈ C 0, (Ω), for some  > 0. (ii) Assume that in addition we have the following:

Proof of
We remark now that the conditions in the second part of Proposition 9(ii) are satisfied for each equation of ().Hence, ,  ∈ C 2, (Ω), and it is a classical solution of ().

B. Positivity of the Function 𝐹
In this paragraph we will establish some properties of positivity for the function  appearing in the models.Let M  (R) denote the vector space of the -square matrix with real coefficients, equipped with the scalar product:  and by an easy calculation, we obtain This last expression is sometimes chosen (for instance in [1]) to equivalently define .
The important fact is that we always have  ≥ 0 but moreover, when  = 2, the stronger estimate  ≥ (1/3) 2 holds.These properties are established in the following lemma.
In the same way, we remark that

C. On the Low Compressibility Assumption
We will show here that, without any assumption of low compressibility of the form (23), problem () may be very hard to analyze when  = 3, and singular solutions or nonexistence of weak solution may occur.When the dimension equals two, we have seen in Theorem 2 that a condition of low compressibility is not necessary.The reason is related to the fact that a stronger property of positivity for  holds in this case; that is, we have  ≥  2 /3.When the dimension equals 3 such a property does not hold in general.
In fact, let for instance Ω =  R 3 (0, 1) and with  bounded from above and below.Hence, a contradiction can occur because the problem () may not have any weak solution (see for instance [21,22]).Note that in the considered example  and u satsify all the conditions needed in ( 0 ), except u ⋅ n = 0 on Ω, but this is not restrictive for the purpose here.In fact, we can consider the domain Ω 1 =  R 3 (0, 2) which contains Ω, and we can extend , u in Ω 1 in such a way that all the conditions in ( 0 ) are satisfied.Hence, we obtain an example within the main situation of the study, but the evocated problems remain the same.
The main result we have in mind is Theorem D.1 which is due to Stampacchia.In particular, we point out that the additional assumption (0) = 0 for the Lipschitz function  is only necessary if Ω is unbounded and  ̸ = ∞ or if we want () to have a vanishing trace on Ω when  has it (this last situation was in fact the case of interest of Stampacchia).Proof.See the appendix in [15] for the original proof or [19,Theorem 7.8] and [6,Theorem 4] for alternative proofs and additional comments.We also recall that the formula (D.1) may be interpreted in some critical points.In fact, let (  ) =1,.., denote the points of discontinuity of   , and let   := { ∈ Ω : () =   } be the associated level sets for the function .Let 1 ≤  ≤  be a fixed integer.If |  | > 0, then the formula (D.1) has a priori no sense in this last set which is not negligible.Nevertheless, it can be shown (see [15]) that /  = 0 on such a set.Hence, we interpret the right hand side of (D.1) as zero in the critical set   .
In this last condition the values for  6 ,  7 , and  8 are in fact their classical constant values (see (A.8) in the Appendices)In the sequel we denote by DATA some quantity depending only on the data under the assumption ( 0 ), that is, DATA = Const (Ω, , , ‖u‖ ( 3 )  ,   ,   , (              ) Assume that ( 0 ) holds.Then, there exists  > 0 such that, if ‖ + ‖   (Ω) ≤ , then problem () admits at least one weak solution (, ) in the class S.

Lemma 11 .
Under the assumptions of Proposition 7, we can extract a subsequence (still denoted by (  ,   )) such that By next using   − 0 as test function in (  ⋅ 1) and   −  0 as test function in (  ⋅ 2) we obtain a uniform bound for (  ) and (  ) in the  1 -norm.Hence, the second properties in (98) follow.Finally, property (99) is obtained by using the dominated convergence theorem.In fact, we have Hence, when  = 2, we obtain = ( 1  1 −  2  2 )  1 +  2  2 ) and the expression for  follows.