UNIQUENESS AND EXISTENCE OF SOLUTIONS IN THE BV t ( Q ) SPACE TO A DOUBLY NONLINEAR PARABOLIC PROBLEM

In this paper we present some results on the uniqueness and existence of a class of weak solutions (the so called BV solutions) of the Cauchy-Dirichlet problem associated to the doubly nonlinear diffusion equation b(u)t−div(|∇u−k(b(u))e|p−2(∇u−k(b(u))e))+g(x, u) = f(t, x). This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media, gases flowing in pipelines, etc. The solvability of this problem is established in the BVt(Q) space. We prove some comparison properties (implying uniqueness) when the set of jumping points of the BV solution has N -dimensional null measure and suitable additional conditions as, for instance, b−1 locally Lipschitz. The existence of this type of weak solution is based on suitable uniform estimates of the BV norm of an approximated solution.

where Q :=]0, T [×Ω and Σ :=]0, T [×∂Ω.We also assume some structure conditions such as the ellipticity of the diffusion operator (which is implied by the monotonicity of the power vectorial function with p > 1) and the monotonicity of the real continuous function b.In fact, we shall assume that (1.5) b is continuous strictly increasing and b(0) = 0.
Another typical example is given by a turbulent gas flowing in a pipeline.If we assume that the gas is perfect and the pipe has uniform cross sectional area, we arrive to the system ρ t + (ρv) x = 0 where ρ, P and v are the density, pressure, and velocity of the gas which are unknown functions depending on the scalar x (the distance along the pipe) and time t.Using asymptotic methods, it was shown in Díaz-Liñán [DLi89] that for large values of time the term ρv t +ρvv x can be neglected and so the second equation may be replaced by P x = − λ 2 ρ|v|v.Then, if we define u = |P |P , u satisfies equation (1.1) with b(u) = u 1/2 sign(u), k = g = f = 0 and p = 3/2.We notice that in this case b −1 is locally Lipschitz.
Previous results on the mathematical treatment of problem (1.1), (1.2) and (1.3) can be found in the references of the paper Díaz-de Thelin [DT94] where the authors pay an special attention to the stabilization question.The main goal of this paper is to improve the uniqueness results of Díaz-de Thelin [DT94] where the weak solutions are assumed such that b(u) t ∈ L 1 (Q).
Based in the works [Vo67], [VoHu69] and [J92], we shall prove in this paper a comparison result in a class of weak solutions such that b(u) t is a bounded Radon measure on Q (i.e.b(u) t ∈ M b (Q)).A preliminar version of our comparison result was shortly presented in [DPa93].The present version also contains an enlarged presentation of Chapter 2 of the Ph.D. of the second author ( [Pa95]).In [GM92a], [GM92b], and [BeGa95] the authors prove some comparison results in the class of weak solutions such that b(u) ∈ BV (0, T ; L 1 (Ω)) for some related nonlinear parabolic problems but always assuming p = 2. Recently, using some techniques raised by S. N. Kruzhkov for hyperbolic equations and inspired in Carrillo [C86], Gagneux and Madaune-Tort proved in [GM94] and [GM95] a uniqueness result for case p = 2.Some more general results for the case p = 2 avoiding the assumption b(u) t ∈ L 1 (Q) can be found in [P95], [PG96] and [U96] where the authors use that any weak solution satisfies the equation in a "renormalized way".
In Section 2, we introduce the assumptions on the data and the notion of bounded BV solution.Section 3 is devoted to recall several properties on bounded variation functions which will be important for the study of the uniqueness of BV solutions presented later in Section 4. Our main result, a comparison criterium depending on the initial data and the forcing terms, assume a condition on the Hausdorff measure of the set of points where the solutions are not approximately continuous.Finally, the existence of a BV solution is established in Section 5 under some extra information on u 0 and f .

Definition of BV solutions
Given a general Banach space B, its dual topological space will be denoted by B .By •, • B,B we denote the duality between B and B. We shall use the Sobolev space W 1,p 0 (Ω) and its dual W −1,p (Ω) where p > 1 and We define the space of bounded variation (with respect to variable t) functions by where M b (Q) denote the space of bounded Radon measures over Q.
In what follows we shall assume a series of conditions on the data: (2.1) and some increasing continuous function ω; (2.2) We start by introducing the notion of weak solution of problem (1.1), (1.2) and (1.3) inspired in [AL83] and [DT94]: Definition 1.We shall say that a function u defined on Q is a weak solution of problem (1.1), (1.2) and (1.3), if and moreover u satisfies the equality By assuming more regularity on b(u) t , we arrive to the following notion: Definition 2. Let u be a weak solution of problem (1.1), (1.2) and (1.3).We shall say that u is a BV solution of (1.1), (1.2) and (1.3) if in addition Notice that in that case b(u) t is a bounded Radon measure on Q and so, the duality between the spaces X and X can be also represented by the correspondent integral with respect to the measure b(u) t for all measurable Borel function v ∈ X , i.e.
In what follows, we shall adopt the integral representation of this duality if the test function is a measurable Borel function.

Treatment of discontinuous functions. Some technical lemmas.
In the development of next sections we shall need some properties of functions whose first generalized derivatives are bounded regular (signed) measures.The following notions and properties can be found in [Vo67], [VoHu85], [F69] and [EvG92].Here, and in what follows, L d (E) will denote the d-dimensional Lebesgue measure of a set We choose υ such that f υ (x 0 ) ≥ f −υ (x 0 ).Such a vector υ is called a defining vector.Vol'pert proved in [Vo67] that if x 0 is a regular point for f (x) and if υ is the defining vector for which f υ (x 0 ) = f −υ (x 0 ), then the associate approximate limit of f in x 0 exists and for any ω ∈ R d f ω (x 0 ) also exists and it is equal to f (x 0 ).A point verifying this, is called a point of approximate continuity.When f υ (x 0 ) = f −υ (x 0 ) the vector υ is uniquely determined (except for the sign of f υ (x 0 )).A point x 0 verifying this inequality is called a jump point of f in the direction υ.The set of jump points of a function f is denoted by Γ f .From Theorem 9.2 of [Vo67] follows that if f ∈ BV (G), G ⊂ R d , then any point of the G is either a point of approximate continuity or a jump point of f with exception of a set H d−1 -dimensional measure zero.For this class of functions, the inward and outward traces of the function f exist on Γ f H d−1 -almost everywhere (see e.g.[VoHu85]).Moreover these traces coincide with the approximate limits f υ and f −υ respectively and the defining vector υ is the outward normal at the point x 0 of Γ f .To extend the differentiation formulas and Green's formula to the class of BV functions it is necessary to define a certain borelian function f H d−1 -almost everywhere equal to a given function f .This borelian representant is the so called symmetric mean value of f .Let us indicate its connection with the inward and outward traces of f , and consequently with the approximate limits f υ and f −υ .We define f by the limit (when it exists) where the sequence {η ε } ε corresponds to an averaging kernel (see [VoHu85, Ch. 4, Section 5, Section 6, p. 181]).It can be shown that if x 0 is a regular point of this function, the above limit exists and does not depends on the averaging kernel.Besides at this point the equality holds, where υ is a defining vector.In particular, if α is a real continuous function, we can define the functional superposition by means of Remark 1.An important property is that ᾱ(f (x 0 )) = α(f (x 0 )) for any x 0 point of approximate continuity of f .Since any summable function f is approximately continuous L d -almost everywhere, then the above equality holds L d -almost everywhere.So, if f is H d−1 -approximately continuous function, then ᾱ(f ) = α(f ) H d−1 -almost every where in G.
However, d-dimensional measure is too large when we try to apply differentiation formulas to functions with measures as generalized derivatives.The generalizes of the classical formulas of differentiation by using the symmetric mean value to functions with measures as generalized derivatives are shown in [VoHu85, Chapter 5, Section 1].
By applying these notions to the case of G = Q and d = N + 1, we can obtain the following lemma which gives an important property of the functions whose generalized derivatives are summable functions.
So, for any (t, x) in Γ u − Λ, there exists an unique vector υ = (υ t , υ x ) (depending of the point (t, x) and where υ t ∈ R and υ x ∈ R N ) which is the inward normal and there exist the approximated limits u υ (t, x) and u −υ (t, x) (see [VoHu85]).Let S be a Borel subset of Γ u −Λ.Since u ∈ W 1,1 (Q), one has that L N +1 (S) = 0 (see [Vo67], [VoHu69]).From that, and as (χ S is the characteristic function of the subset S).Applying now Theorem 2, p. 203 of [VoHu85] we get The above equality implies that H N (S) = 0. Finally, as S is arbitrary, we conclude that The main result of the general theory of BV functions that we shall use later in order to prove our uniqueness theorem is given in the following lemma: approximately continuous function on Q, then by the above lemma and Remark 1 we have that v = v ∂u ∂t -almost every where on Q.

Proof of Lemma 2:
) is defined almost everywhere x ∈ Ω with respect to the Lebesgue measure and L N summable in A Ω (see [Vo67] and [EvG92]).Moreover since L N (A Ω ) = 0 and so the statement of the lemma holds.
Lemma 3. Assume u ∈ L ∞ (Q) and b as in (1.5).If in addition we assume that Proof: To prove this property, we show that u t is a bounded Radon measure on Q.Following Vol'pert and Hudajaev [VoHu85, Chapter 4, Section 2] it is enough to prove that there exists a positive constant In order to do that, we use the fact that of distributions.From the assumptions (3.1) and (3.2) we obtain the result.
Remark 3. Condition (3.2) sometimes is verified in an implicit way.For instance, if

let be η a locally bounded borelian function in R. Define the function H given by
holds for all φ bounded borelian function on In particular, we have the following "chain rule formula" then the relations given in b) and c) are also true replacing η(u), H(u) and v by η(u), H(u) and v respectively.
Proof: a) since H is a locally Lipschitz continuous function the conclusion comes from Lemma 3. b) is consequence of a), the rule chain for the one-dimensional case (Theorem 13.2 of [Vo67]) and Theorem 4.5.9 of [F69].c) is proved using the integration by parts formula for BV function [VoHu85] and the above mentioned theorem of [F69].Finally, we obtain d).From the fact that u and v are H N -absolutely continuous functions and the properties of functional superposition we obtain that the borelian representatives η(u), H(u) and v are equal to the functions η(u), H(u) and v H N -almost everywhere where in Q t respectively (see Remark 1).And finally, applying Lemma 2 (see Remark 2) we conclude the proof.

Comparison and continuous dependence results
In this section, we give several results on the comparison and continuous dependence of BV solutions of the problem (1.1), (1.2) and (1.3) under the main condition (4.3).We shall use later the inequality where β = 2 if 1 < p ≤ 2 and β = p if p ≥ 2 which holds for any η and η in R N from Tartar's inequality (see e.g.Díaz-de Thelin [DT94]).
Theorem 1. Assume that b, k and g verify (1.5), (2.1), (2.2) and (2.3).Let (f 1 , u 01 ) and (f 2 , u 02 ) be a pair of data satisfying (2.4) and (2.5).Let u 1 and u 2 be two BV solutions of the problem (1.1), (1.2) and (1.3) associated to (f 1 , u 01 ) and (f 2 , u 02 ) respectively.We also suppose that and that Then, for any t ∈ [0, T ] we have Remark 5. i) The regularity (4.2) on the functions u i can be obtained by assuming some regularity properties on function b.In particular we note that if b −1 is a locally Lipschitz continuous function, condition b(u i ) ∈ BV t (Q) implies (4.2) (see Lemma 3).ii) Also, we can assume b as in (3.3) and if M is a positive constant such that u i L ∞ (Q) ≤ M , for i = 1, 2 and we suppose that b −1 1 and b 2 have Lipschitz constants L 1 and L 2 respectively on the interval [−M, M ] with (4.4) then (4.2) holds (see the Remark 3 and Lemma 3).
Remark 6.The case b locally Lipschitz continuous function was previously considered in [DT94].
and assumption (4.4) is not needed.Notice that in that case assumption (4.3) always holds due to Lemma 1.Some consequences of the above theorem are the following results: Corollary 1.Let u 1 and u 2 be two BV solutions as in Theorem 1 associated to the data (f 1 , u 01 ) and (f 2 , u 02 ).Assume that and so u 1 ≤ u 2 thanks to the monotonicity of function b.
Proof: Take f 1 = f 2 and u 01 = u 02 in Corollary 2. Notice that the above results are also true under the conditions of Remark 5 given in the case i) and in the case ii).
Arguing as before, we can obtain analogous results to Corollaries 1-3 for BV solutions which lie in [−M, M ].On the other hand, we can make explicit M for bounded data Lemma 5. Let u be a weak solution of (1.1), (1.2) and (1.3).Assume (1.5), (2.1), (2.2), (2.3), and for the data, we assume Thus, there exists a positive constant M > 0 such that Proof: See e.g.Benilan [Be81].
Proof of Theorem 1: For any n ∈ N, we define T n , approximation of the sign 0 + function (sign 0 + (s It is easy to see that (4.5) To simplify the notation, we set is an admissible test functions since u 1 and u 2 are BV solutions and T n is a regular function.As moreover, we are assuming (4.3), then Tn (u 1 − u 2 ) = T n (u 1 − u 2 ) H N -a.e. in Q.Thus, and thanks to Lemma 2, we can adopt the notation (2.11); that is Considering the relations (2.9) verified by u 1 and u 2 and subtracting, we obtain In order to pass to the limit we need some technical results Lemma 6.Under the assumptions of Theorem 1, we have Proof of Lemma 6: Since u 1 and u 2 are BV solutions, we have that b(u 1 ) and b(u 2 ) belong to BV t (Q) ∩ L ∞ (Q).On the other hand, by Lemma 4 and Remark 4, Moreover by assumption (4.3), for all n ∈ N , T n (u 1 −u 2 ) is also an H N -approximately continuous function (see [VoHu85, Theorem 2, p. 164]).Thus, using that b is strictly increasing Proof of Lemma 7: It is a slight improvement of Díaz-de Thelin [DT94].For the sake of completeness we give the detailed proof.For any n ∈ N, the integral term in (4.9) it can be written as the addition of the integrals Here we drop writing the t-dependence.We shall find an estimate on |I 2 (n)| in terms of I 1 (n).Due to the assumption (2.1) on k, we need to distinguish the cases 1 < p ≤ 2 and p > 2.
Case 1 < p ≤ 2: Applying Young's inequality we get for any ε > 0. Using the inequality (4.1) to the first term of the right hand side of (4.10), by assumption (2.1) on k and the properties of T n we obtain that Taking ε p = p /C, we get Now, since γp − 1 > 0, we have that and then (4.9) is proved.
Case p > 2: By Hölder inequality Using inequality (4.1)where we set β = p and η 1 = ξ 1 , η 2 = ξ 2 , the first multiplicative factor in (4.11) is bounded by Using again the Hölder inequality and the properties of T n , we obtain the estimate where .
For the second multiplicative factor in (4.11), we have from assumption (2.1) on k and (4.5).Combining both inequalities, we arrive to End of the proof of Theorem 1: By the previous two lemmas, we obtain the key inequality Using the assumption (2.3) on g, the conclusion of the theorem is immediate if C * is zero.More in general we set v j (t, x) = e −C * t b(u j (t, x)) for j = 1, 2. Then sign (uj )   ∂t are also bounded regular measures in Q. Choosing T n (v 1 − v 2 ) as test function and working as before, we obtain By assumption (2.3), one has that and thus, the conclusion holds.
Remark 8.The assumption (4.3) on the measure of the jump points set is merely needed in the proof of Lemma 6.This assumption could be replaced by any other condition implying the conclusion of Lemma 6.In particular, we have Corollary 6.Let u 1 and u 2 be bounded BV solutions of (1.1), ( 1

.2) and (1.3). Assume the hypotheses of Theorem 1 but replacing (4.3) by (4.12)
there exist two homeomorphisms on R, Ψ 1 and Ψ 2 , Then u 1 and u 2 verify the comparison criterium given in Theorem 1.

Existence of bounded BV solutions
In order to obtain the existence of bounded BV solutions for problem (1.1), (1.2), (1.3) we shall assume some additional conditions on functions f and u 0 :
Proof: We start by considering, a sequence of regular problems having a unique solution by the classical theory of partial differential equations.After that, we shall obtain suitable a priori estimates.Finally passing to the limit we shall find a bounded BV solution.In view of the structural assumptions, we shall distinguish two cases, according p satisfies 1 < p < 2 or 2 ≥ p.
Case 1 < p < 2: Regularization.We define a sequence of uniformly parabolic problems with coefficients and free term bounded regular functions.Consider the following regularized equation in Q where we define the vectorial function φ r by for any r ∈ N, and such that φ r ∈ C 1 (R N ) verifies with bm the Yosida approximation of b.We recall that bm converges uniformly on compacts sets to b, bm is a Lipschitz nondecreasing function such that | bm | ≤ |b| and that b m and b −1 m are Lipschitz nondecreasing functions; see [Be81].
We take a sequence of functions {k s } ∞ s=1 belonging to C ∞ (R) such that they verify (2.1) and k s converges to k uniformly on compacts of R.
For any integer n, we consider a function g n ∈ C ∞ (Ω × R) satisfying the assumptions (2.2) and (2.3) uniformly on n and such that g n (x, η) converges to g n (x, η) in L 1 (Ω) for any fixed η in R, for a.e.x ∈ Ω, as n → ∞.
Finally we regularize the initial condition.We consider u 0,q ∈ C ∞ 0 (Ω) such that u 0,q * u 0 in L ∞ (Ω) as q → ∞ and such that φ r (∇u The equation (5.4) is uniformly parabolic.So, by well-know result (see e.g.Ladyzenskaja, Solonnikov and Uralceva [LSU68, Chap.V]) there exists a unique classical solution û = u m,r,s,n,l,q of (5.4) satisfying In what follows, we denote by û the function u m,r,s,n,l,q .In order to study the convergence of the sequence û we shall need some uniform estimates in suitable functional spaces.
A priori estimates.By the maximum principle where M 1 is a positive constant independent of m, r, s, l and q.On the other hand, if we denote by v := b m (û) t and we differentiate equation (5.4) with respect to t, we obtain that For any η > 0, we define the function H η approximating the absolute value function in the following way: we first introduce and finally we define It is clear that Multiplying equation (5.8) by H η (v), and integrating on Passing to the limit when η → 0, we obtain the inequality from the properties of h η and the monotonicity of the vectorial function φ r .Now, by (2.3), we arrive |v| dx ds.Taking this into account, one verifies that Using the equation satisfied by ∂ ∂t b m (û(0, x)) and the uniform bounded-ness of the data, we get Applying Gronwall's lemma, we obtain that

Thus
(5.9) with M 2 = e C * T .From (5.7) and (5.3) (5.10) Now, we shall show that there exists M 4 > 0, such that (5.11) uniformly in t ∈ [0, T ], m, r, s, n, l and q.Firstly, we shall show that there exists an uniform positive constant M such that (5.12) where ξ := ∇û − k s (b m (û))e.To do that, we multiply (5.4) by û and we integrate on Ω: The In what following it will appear several positive constants denoted by C i , i = 2, 3, 4, . . .which are independent on t and the parameters m, r, s, n, l and q.They will dependent on the exponent p, the measure of Ω and the above estimates.Some of them are also function of some positive parameters ε and δ we shall introduce later.We shall only indicate the ε and δ dependence.
By estimates (5.7) and (5.9), the assumption (2.3) on g, (5.1) on f and the properties of g n and f l , there exists a positive constant C 2 uniform in m, r, s, n, l and q, such that for any t.By Young's inequality, we have where ε is an arbitrary positive real number.The last integral is uniformly bounded in view of (5.7) and the assumptions on the sequences {k s } and {b m }.By the properties of φ r , the first integral of the right hand side is bounded by Ω | ξ| p dx, for any t in [0, T ].Hence, (5.13) for some positive constant C 3 = C 3 (ε).Besides, from the properties of φ r , we have that On the other hand, applying Young's inequality to Ω | ξ| p (1+| ξ|) 2−p (1 + | ξ|) 2−p dx with exponents 2 p and 2 2−p we get Using the estimates (5.13) and (5.14) into (5.15),we obtain the inequality for some positive constant C 7 depending on ε, δ.To verify the estimate (5.12), it is enough now to choose 0 < δ 1 and ε 1 such that δ 2/p > 0. Now (5.12) implies (5.11) from the uniform boundedness of Finally, multiplying the relation (5.4) by v ∈ X and integrating on Q, we obtain, using the Hölder's inequality that The properties (5.5) and (5.6) of φ r , the assumptions (2.2) on g and (5.1) on f and the properties on g n and f l lead to estimate for some positive constant M 5 independent on m, r, s, n, l and q where we used estimates (5.7) and (5.12).In this way, we obtain the following uniform estimate in L p (0, T ; Passing to the limit.By the estimates (5.7), (5.9), (5.10), (5.11) and (5.16) we can find a bounded BV solution of problem (1.1), (1.2) and (1.3) as limit of some subsequence of {û} := {u m,r,s,n,l,q } (which we will denote again by {û}).Moreover, this solution belongs to C([0, T ], L 1 (Ω)).Indeed by the estimates (5.7), (5.10) and (5.11) and Corollary 4 of Simon [S87], there exists a subsequence of {û} (called again {û}) and a function u ∂t in the sense of distributions.Moreover, ∂bm(û) ∂t converges to ∂b(u) ∂t weakly in L p (0, T ; W −1,p (Ω)) + L 1 (Q) from (5.16).By usual argument, we obtain that g n (x, û) converges to g(x, u) in L 1 (Q).Since û is bounded in L ∞ (0, T ; W 1,p 0 (Ω)), then φ r (∇û − k s (b m (û))e) is also bounded in L ∞ (0, T ; (L p (Ω)) N ) and thus there exists a subsequence of {û} (again called {û} such that φ r (∇û − k s (b m (û))e) converges to Y weakly* in L ∞ (0, T ; (L p (Ω)) N ).Multiplying the equation (5.4) by a test function v ∈ L p (0, T ; W 1,p 0 (Ω)) ∩ L ∞ (Q) and integrating on Q, we obtain that (5.17) Let us see that u verifies (2.9).To do that, we pass to the limit in the variables in (5.17) when m, r, s, n, l and q → ∞.By the above convergences, we arrive to for all v ∈ L p (0, T ; W 1,p 0 (Ω)) ∩ L ∞ (Q).We have to prove now that which is not completely obvious due to the nonlinear character of the differential operator.We shall prove this by using Minty's type argument (see also [DT94]).We shall show that (5.20) Then, we can obtain (5.19) by taking ξ ∈ W 1,p 0 (Ω) arbitrary and the function χ = u − λξ with λ > 0 (λ < 0).To prove (5.20), we take 0 ≤ ϕ ∈ c ∞ c 0, T and for any χ ∈ W 1,p 0 (Ω) we use the decomposition where and Due to monotonicity of φ r , the integral I 2 is non negative.On the other hand (5.26) The integrals (5.25) and (5.26) converge to zero when m, r, s, n, l, q → ∞.The weak convergence of b m (û) t to b(u) t in X and the fact that uϕ(t) ∈ X , imply that the integral (5.23) (i.e.− ∂bm (û)  ∂t , uϕ(t) X,X ) converges to (5.27) .
We shall also show that the integral (5.24) converges to (5.27) as in [DT94].We define It is easy to see that B m (û) is bounded in Q and thus z û(t) L 1 (0,T ) is uniformely bounded.
As in Lemma 2 of Bamberger [Ba77] we get that a.e.t ∈]0, T [, in the sense of D (0, T ).Now, thanks to the convergence of û and b m (û) and the boundedness of z û in L 1 (0, T ) we obtain that (5.28) Passing to the limit, lim and so lim I 1 = 0.
Summarizing: we have proved that the limits of integrals I 1 , I 2 , I 3 , I 4 are non negative and thus (5.19) holds.Then, u satisfies the equation (2.9).By standard arguments we can see that u verifies also (2.8).
Case p ≥ 2: As in the case 1 < p < 2, we begin by defining a family of regular problems, we find suitable a priori estimates and finally we obtain u as the limit of the regular solutions associated to the family of regular problems.The family of regularized problems can be defined now by − ∆u + g n (x, u) = f l (t, x) in Q, (5.30) u(t, x) = 0 on Σ, (5.31) b m (u(0, x)) = b(u 0,q (x)) in Ω (5.32) with b m , k s , g n , f l and u 0,q as in the case 1 < p < 2 and > 0. The existence of a classical solution is again a well-known result (see [LSU68, Chapter V]).The rest of details follows the same arguments.
The above theorem proves the existence of BV solution of problem (1.1), (1.2) and (1.3).Nevertheless our uniqueness results on BV solutions we will need some additional assumptions.The following corollary gives an answer in this sense.Proof: We use the same technique that in the proof of Theorem 2. Due to that, we shall made mention only to the new arguments.Let û be the solution of the regularized problems.As before, we obtain the estimates (5.7), (5.11), (5.9).Now, we shall find an L 2 (Q) uniform estimate on By the estimates (5.7), (5.2) and (5.1) and since Φ r is non negative, we have with C 1 a constant independent on m, r, s, n, l, q.By Young's inequality, (5.36) ).Thus, On the other hand, as b verifies (3.5), we obtain for some positive constant L independent on m, r, s, n, l, q.Considering the above inequalities and estimate (5.11), from (5.36) we arrive to for some positive constant C 2 .Finally, since 1 < p < 2 and Q is bounded, applying the Hölder's inequality we get (5.37) with C 3 a positive constant independent on m, r, s, n, l, q.This new estimate jointly with the estimates given in the Theorem 2 allows us to show that the BV solution obtained as limit of the sequence {û} verifies (5.35).
Remark 9.The bounded BV solution u obtained in Theorem 2 belongs to W 1,1 (Q).Then, by Lemma 1, the Hausdorff N -dimensional measure of the set of jumping points of u is zero.Then, by Corollary 3, this solution is unique in this class of solutions.An other way to obtain the above conclusion is by applying Corollary 6 with Ψ 1 and Ψ 2 the identity, since any pair of solutions u 1 , u 2 obtained as in Theorem 2 are in the W 1,1 (Q) space.
and û → u a.e. in Q (except, perhaps, for a subsequence).By (5.7) û * u in L ∞ (Q) weakly* and û u in L p (0, T ; W 1,p 0 (Ω)) from (5.11).By (5.7) and the assumption on b m we can deduce the weak* convergence b m (û) * β in L ∞ (Q) for some β ∈ L ∞ (Q).The fact that û converges to u almost everywhere of Q and the properties of b m and b imply that b m (û) → b(u) a.e. point of Q.By Lebesgue's Theorem there is strong convergence of b m (û) to b

p,
for all ε > 0. Now, since û is uniformly bounded (see (5.7)) and since we have assumed (5.33), then there exists a positive constantL k•b such that for all s, m ∈ N L k•b ≥ lip(k s • b m , [−M 1 , M 1 ]) (:=the Lipschitz constant of k • b in the interval [−M 1 , M 1 ] [VoHu85]e limit it is equal to 0, the point x 0 is a point of F -rarefaction of the set E. Taking F as R d , we denote by E * the set of points of density of E and E * the set of points of rarefaction of E. Finally, the set ∂E = E * \ E * is called the essential boundary of the set E (in many cases, the essential boundary of a set E coincides with the boundary of E, however the boundary and the essential boundary of a set do not always coincide -for example the boundary and essential boundary of a disk minus a radius are not the same[VoHu85]).Let now f : R d → R be a Lebesgue measurable function.The real number is called an approximate limit with respect to the set E ⊂ R d of the function f as x → x 0 if for all ε > 0 the point x 0 is a point of E-density of the set {x ∈ R d : |f (x) − | < ε}.
a.e. in Q.