Asymptotic behaviour of a class of degenerate elliptic-parabolic operators : a unitary approach

Vol. 13, N 4, 2007, pp. 669–691 www.esaim-cocv.org

DOI: 10.1051/cocv:2007029

ASYMPTOTIC BEHAVIOUR OF A CLASS OF DEGENERATE ELLIPTIC-PARABOLIC OPERATORS: A UNITARY APPROACH

Fabio Paronetto

Abstract. We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equa- tions ∂t(rhu)−div(ah·Du) with rh(x, t) 0,rh ∈ L^∞(Ω×(0, T)). The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain G-convergence for elliptic operators (rh≡0),G-convergence for parabolic operators (rh≡1), singular perturbations of an elliptic operator (ah≡aandrh→r, possiblyr≡0).

Mathematics Subject Classification. 35J15, 35K10, 35M10, 45J45.

Received May 26, 2005. Revised January 5, 2006 and February 17, 2006.

Published online July 20, 2007.

1. Introduction

In the papers [5, 13, 14] De Giorgi and Spagnolo introducedG-convergence for a class of elliptic operators, precisely for a class of elliptic operators in divergence form defined by an elliptic and simmetric matrix with bounded coefficients. Tartar extended this convergence to the non-simmetric (and then non-linear) case (see, for instance, [16] and [17]).

Later in [3] Colombini and Spagnolo defined G-convergence for a class of parabolic operators in divergence form still defined by a simmetric matrix with bounded coefficients depending, in this case, also on time. Before introducing the aim of this paper we recall the definition ofG-convergence in both cases, denoting the conver- gence by EGin the elliptic case and by P G in the parabolic one, as extended to non-simmetric operators by Tartar (for a book containing results about bothEGandP G convergences we refer to [7]).

Considern∈N fixed. Moreover, forλ₀Λ₀andM positive real numbers, denote byM_U(λ₀,Λ₀, M), with U open set ofR^k,k∈N, the class ofn×nmatrices defined as follows:

a= [a_ij(y)]ⁿ_i,j=1 ∈L^∞(U) such that λ₀|ξ|²

a(y)·ξ, ξ

Λ₀|ξ|² a(y)·ξ, ηM

a(y)·ξ, ξ_1/2

a(y)·η, η_1/2

(1)

for everyξ, η ∈Rⁿ, fora.e. y∈U.

Keywords and phrases. G-convergence, PDE of mixed type, linear elliptic and parabolic equations.

1 Dipartimento di Matematica “Ennio De Giorgi”, Universit`a del Salento, via per Arnesano, 73100 Lecce, Italy;

[email protected]

c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-cocv.org or http://dx.doi.org/10.1051/cocv:2007029

Definition 1.1. Let Ω be a bounded open set ofRⁿ andT >0. Consider a sequence (a_h)_h⊂ MΩ(λ₀,Λ₀, M), a_h=a_h(x) (referring to (1), in this casek=n). Givena=a(x)∈ MΩ(λ₀,Λ₀, M) we say that

a_h −→^EG a in Ω if for everyf ∈H⁻¹(Ω) it results that

u_h→u in L²(Ω)

a_h·Du_h→a·Du in L²(Ω)ⁿ-weak, whereu_h andudenote respectively the solutions (see Def. 2.5 with r≡0) of

−div(a_h·Dv) =f in Ω

v= 0 in∂Ω

−div(a·Dv) =f in Ω

v= 0 in ∂Ω.

For a sequence (a_h)_h⊂ M_Ω×(0,T)(λ₀,Λ₀, M),a_h=a_h(x, t) (referring to (1), in this casek=n+ 1), and given a=a(x, t)∈ M_Ω×(0,T)(λ₀,Λ₀, M) we say that

a_h −→^{P G} a in Ω×(0, T) if for everyf ∈L²(0, T;H⁻¹(Ω)) and ϕ∈L²(Ω) it results that

u_h→u inL²(0, T, L²(Ω)) a_h·Du_h→a·Du inL²(0, T, L²(Ω)ⁿ)-weak, whereu_h andudenote respectively the solutions (see Def. 2.5 withr≡1) of

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂v

∂t −div(a_h·Dv) =f inΩ×(0, T)

v= 0 in∂Ω×(0, T)

v=ϕ inΩ× {0}

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂v

∂t −div(a·Dv) =f in Ω×(0, T)

v= 0 in ∂Ω×(0, T)

v=ϕ in Ω× {0}.

In [3] the authors studied the connection between EGand P Gconvergence: in particular they proved that if (a_h)_h∈ M_Ω×(0,T)(λ₀,Λ₀, M) satisfies

τ→0limsup

h I ω|a_h(x, t+τ)−a_h(x, t)|dxdt= 0 ∀I×ω⊂⊂(0, T)×Ω, (2) then

a_h(·, t) −→^EG a(·, t) in Ω fora.e. t∈(0, T) iff a_h −→^{P G} a in Ω×(0, T) (3) and showed with a counterexample that this is not always true.

In this paper we consider strongly degenerate parabolic, or elliptic-parabolic, operators like P u= ∂

∂t(ru)−div(a·Du) withr=r(x, t)0 (4)

and study the limit behaviour of the sequence of Cauchy-Dirichlet problems (for the existence result we refer to [8], but see also [11])

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

∂

∂t(r_hu)−div(a_h·Du) =f in Ω_h,+(t) ×(0, T)

−div(ah·Du) =f in Ω_h,0(t) ×(0, T)

u= 0 in∂Ω×(0, T)

u=ϕ in Ω_h,+(0)× {0}

(5)

where (the initial condition on Ω_h,+(0) will be clarified at the end of Sect. 2) Ω_h,+(t) :={x∈Ω|r_h(x, t)>0}

and Ω_h,0(t) :={x∈Ω| r_h(x, t) = 0}, (ah)_h is a sequence in M_Ω×(0,T)(λ₀,Λ₀, M, N), the class of matricesa satisfying (1) and

a(x, t)−a(x, s)N|t−s|

for a.e. x∈Ω and every s, t ∈[0, T], r_h belonging to a suitable class F defined in (6). Arising from (3) the aim of the present paper is to give a more general definition ofG-convergence, for problems (5) (see Def. 4.3), which is independent of the sequence (r_h)_h and a compactness result with respect to it (see Th. 4.5). This in particular justifies (3) and includes other phenomena, as singular perturbations (in which the result is stronger, see Prop. 5.1), but we refer to the lastgli Studi di Lecce section for examples. We want to stress that, since r in (4) may be equal to zero on some region with positive measure, a difficulty in this situation is that the natural compactness result (see Th. 2.7) is not guaranteed. Only for the sequence of the solutions (the solutions u_hto the problems (5)), we are able to obtain the compactness via a regularity result (see Th. 2.8).

We recall that, in the general situation, a first study in this direction was already made, in the periodic case and withr=r(x), in [9].

Elliptic-parabolic operators like those in (4) were already studied, as regards the existence of the solution, probably first by Showalter (see, for instance, [10] for one of the first papers and [11] for a recent book) and recently in [8] for a more general class of operators (nonlinear and possibly forward, backward and stationary).

The interest to study such problems lies on the fact that many diffusion problems lead to differential equations like

∂

∂t(r(x, t)u)−div(a(x, t)·Du) =f

which may be also of mixed type (see for example [1], Chap. 3, and the references therein),i.e. partially elliptic and partially parabolic. For some applications see also the examples in the last section.

The scheme of the paper is the following: Section 2 is dedicated to existence of the solution to an equation P u=f and to the position of the problem. In Section 3 there are some compactness results: since a “classical type” compactness result (see Th. 2.7, withr_h≡1 for the classical case) does not hold in general, we pass through a regularity result (Th. 2.8) to obtain it. In Section 4 we defineG-convergence and prove a compactness result in two steps: Theorem 4.1 furnishes, given a sequence of operatorsP_hu= _∂t^∂ (r_hu)−div(a_h·Du) in a suitable class, the existence, up to a subsequence, of a limit operatorP u= _∂t^∂(ru)−div(a_r·Du), Theorem 4.5 states that a_risindependent ofr. In the last section we give some examples, including also singular perturbations.

2. Elliptic-parabolic equations and statement of the problem

From now onT,λ₀, Λ₀,M will be fixed positive constants,N, C₁, C₂non-negative constants,µ₀ a non-positive constant (indeedµ₀could also be positive, but in that case it is sufficient to considerµ₀= 0) and Ω a bounded open set ofRⁿ with Lipschitzian boundary. We will denote for brevity

V:=L²(0, T;H₀¹(Ω)), H:=L²(0, T;L²(Ω)), V:=L²(0, T;H⁻¹(Ω)).

We denote byF(C₁, C₂, µ₀) the class of measurable functionsrsatisfying (i) r∈L^∞(Ω×(0, T)), r0,

(ii) r_L^∞_(Ω×(0,T))C₁, (iii) t→

Ω

u(x)v(x)r(x, t)dx absolutely continuous on [0, T], (iv) d

dt _Ωu(x)v(x)r(x, t)dxC₂ u_H0¹_(Ω)v_H¹_0(Ω) fora.e. t∈[0, T], (v) d

dt _Ωu²(x)r(x, t)dxµ₀ u²_H1

0(Ω) fora.e. t∈[0, T]

(6)

for everyu, v∈H₀¹(Ω). For ar∈ F(C1, C₂, µ₀) we define Ω^r₊(t) :=

x∈Ωr(x, t)>0

, Ω^r₀(t) :=

x∈Ωr(x, t) = 0

. (7)

Remark 2.1. The class just defined is compact, i.e. if (r_h)_h is a sequence in F(C1, C₂, µ₀), there is a subse- quence (r_hj)_j and a function r∈ F(C₁, C₂, µ₀) such that r_hj →rin L^∞(Ω×(0, T))-weak∗. In fact, there is a subsequence (r_hj)_j and a functionrsuch thatr_hj →rin L^∞(Ω×(0, T))-weak∗. Now verify thatrbelongs to F(C1, C₂, µ₀). Clearlyr0 andr∞C₁. To verify thatrsatisfies also (iii), (iv), (v), consider a countable set Z, dense in H₀¹(Ω), and for every u, v ∈ Z define the functions F_h^u,v(t) =

Ωu(x)v(x)r_h(x, t)dx. Since (r_h)_h ⊂ F(C₁, C₂, µ₀) the sequence (F_h^u,v)_h turns out to be equicontinuous and equibounded. Then there is a function, denoted by F^u,v, and a subsequence F_h^u,v

j such that F_h^u,v

j →F^u,v uniformly in [0, T]. Since Z is countable we can find a sequenceh_jk such thatF_h^u,v

jk →F^u,v uniformly in [0, T] for everyu, v∈Z (and in fact for everyu, v∈H₀¹(Ω)). This in particular implies that

T 0

F_h^u,v

jk(t)η(t)dt→ ^T

0

F^u,v(t)η(t)dt for everyη ∈L¹(Ω). Sincer_hj →rin L^∞(Ω×(0, T))-weak∗

T 0

F_h^u,v

jk(t)η(t)dt=

T

0 Ω

u(x)v(x)η(t)r_hjk(x, t)dxdt→ ^T

0 Ω

u(x)v(x)η(t)r(x, t)dxdt

and thenF^u,v(t) =

Ω

u(x)v(x)r(x, t)dx.

Notice thatF^u,v∈W^1,∞: then for every η∈C_c¹(0, T) we have

T 0

d dt

F_h^u,v

jk(t)

η(t)dt=− ^T

0

F_h^u,v

jk(t)η(t)dt→ ^T

0

d dt

F^u,v(t) η(t)dt

and then we derive

−C₂u_H0¹_(Ω)v_H0¹_(Ω) ^T

0

η(t)dt ^T

0

d dt

F^u,v(t)

η(t)dtC₂ u_H0¹_(Ω)v_H¹_0(Ω) ^T

0

η(t)dt from which

d dt

F^u,v(t)C₂ u_H0¹_(Ω)v_H¹_0(Ω). Analogously we derive that _dt^d

Ωu²(x)r(x, t)dxµ₀u²_H1 0(Ω).

Givenr∈ F(C₁, C₂, µ₀) we introduce the families of operators R: [0, T]−→ L(L²(Ω)), R(t)u:=r(·, t)u(·) R: [0, T]−→ L(H₀¹(Ω), H⁻¹(Ω)), R(t)u, v:= d

dt _Ωu(x)v(x)r(x, t)dx R:H −→ H Ru(t) :=R(t)u(t)

R:V −→ V Ru, v_V×V:=

T 0

R(t)u(t), v(t)_H−1(Ω)×H₀¹(Ω)dt

(8)

and define the following Banach space W =

u∈ V (Ru)∈ V

, u_W :=u_V+(Ru)_V (9)

where (Ru) denotes the derivative in the distributional sense ofRuwith respect to the variablet.

An approximation result we will need later is the following.

Lemma 2.2. Consider R defined in (8). Then for every u∈ W and σ >0 there exist v ∈C¹([0, T];H₀¹(Ω)) and S ∈ C²([0, T];L(H₀¹(Ω), H⁻¹(Ω))) (defined analogously to R by a s∈ F(C₁, C₂, µ₀), ^∂s_∂t ∈ F(C2, K,−K) whereK=K(C₁, σ))such that

u−v_W < σ , (Ru)−(Sv)_V < σ.

Moreover S can be chosen in such a way that S(0) = 0.

Proof. Fixu∈ W=WRandσ >0. From Proposition 2.4 in [8] we derive the existence ofv∈C¹([0, T];H₀¹(Ω)) such that

u−vW < σ/2. Consider a family of mollifiers (ρ)_>0 and, after defining

R(t) :=¯

R(t) ift∈[0, T] 0 ift∈[0, T], consider

R(t) =

R

R(τ)ρ¯ (t−τ)dτ

and the corresponding Ru(t) := R(t)u(t). Note that R ∈ W^1,∞(0, T;L(H₀¹(Ω), H⁻¹(Ω))) and R → R in L^∞(0, T;L(H₀¹(Ω), H⁻¹(Ω)))∩W^1,q(0, T;L(H₀¹(Ω), H⁻¹(Ω))) for every q <+∞.

Clearly R(t)u, u_H⁻¹_(Ω)×H0¹_(Ω)=

RR¯(τ)ρ(t−τ)dτ u, uµ₀u²_H1 0(Ω).

Observe that, sincev∈C¹([0, T];H₀¹(Ω)),v ∈ WR andv∈ WR for every >0. Then, since (Ru)−(Rv)V (Ru)−(Rv)V+(Rv)−(Rv)V,

to prove the result it is sufficient to estimate(Rv)−(Rv)_V. SinceR→RinL^∞(0, T;L(H₀¹(Ω), H⁻¹(Ω))) we have that

Rv→ Rv inH,

and therefore it is sufficient to analyse the quantity Rv− Rv. We consider a function φ ∈ V and by the

generalized H¨older’s inequality we have for everyp >2 Rv− Rv, φ_V×V= ^T

0

R(t)v(t)−R(t)v(t), φ(t)_H−1×H₀¹dt

T

0

R(t)−R(t)^2p/(p−2)_L(H1

0,H⁻¹)dt^p−2_2p T 0

v(t)^p_H1

0(Ω)dt¹_p T 0

φ(t)²_H1 0(Ω)dt¹₂

.

Taking the supremum with respect toφ,φ_H¹_0(Ω)= 1, we obtain that

Rv− RvV R−R_L^2p/(p−2)_(0,T;L(H¹_0(Ω),H⁻¹_(Ω)))v_L^p_(0,T;H0¹_(Ω))

for a p >2 and sinceR→R inL^q(0, T;L(H₀¹(Ω), H⁻¹(Ω))) for everyq <+∞we conclude choosingS =R forsufficiently small.

Moreover we can choose S in such a way S(0) = 0. To do this it is sufficient to consider the function η(t) =δ⁻¹t in [0, δ] andη(t) = 1 in [δ, T], and choose

S(t) = ^t

0

η(τ)R(τ)dτ+R(0).

It is sufficient to estimate(Rv)−(Sv)V. First we estimate Rv− Sv in H:

T 0

R(t)v(t)−S(t)v(t)²_L2(Ω)dtR²_L∞(0,T;L(H¹₀(Ω),H⁻¹(Ω)))v²_Vδ². Similarly we can obtain Rv− SvV R_L^∞_(0,T;L(H0¹_(Ω),H⁻¹_(Ω)))vV√

δ. Since δ is arbitrary we are

done.

For the following result see Proposition 2.6 in [8].

Theorem 2.3. For every u, v∈ W the following holds:

d

dt(Ru(t), v(t))_L2(Ω)= Ru(t), v(t)_H⁻¹_(Ω)×H0¹_(Ω)

+ Ru(t), v(t)_H−1(Ω)×H₀¹(Ω)+ Rv(t), u(t)_H−1(Ω)×H₀¹(Ω). (10) Moreover the functiont→(R(t)u(t), u(t))_L2(Ω) is continuous and there is a constantc=c(T,R) (depending only on T,RC₁)such that

max[0,T]|(R(t)u(t), u(t))_L²_(Ω)|c u²_W. (11) Remark 2.4. Observe that, if R(t)v(x) = r(x, t)v(x) for every v ∈ L²(Ω) and for a r ∈ F(C₁, C₂, µ₀), by Theorem 2.3 we deduce that ifu∈ W then

u(t)∈L²(Ω^r₊(t), r(·, t)) for everyt∈[0, T],

where Ω^r₊(t) is defined in (7) and L²(Ω^r₊(t), r(·, t)) denotes the completion of L²(Ω^r₊(t)) with respect to the normv²:=

Ωv²(x)r(x, t)dx. Observe that L²(Ω)⊂L²(Ω^r₊(t), r(·, t)) and

Ω^r₊(t)

v²(x)r(x, t)dxC₁

Ω

v²(x)dx

for everyv ∈L²(Ω) and everyt∈[0, T].

We recall the definition of the class M_Ω×(0,T)(λ₀,Λ₀, M), with λ₀ Λ₀ and M positive real numbers, given in (1), characterised by

a= [a_ij(x, t)]ⁿ_i,j=1 such that λ₀|ξ|²

a(x, t)·ξ, ξ

Λ₀|ξ|² a(x, t)·ξ, ηM

a(x, t)·ξ, ξ_1/2

a(x, t)·η, η_1/2

(12)

for every ξ, η ∈ Rⁿ, for a.e. (x, t) ∈ Ω×(0, T). By M_Ω×(0,T)(λ₀,Λ₀, M, N), N non-negative constant, we denote the subclass ofM_Ω×(0,T)(λ₀,Λ₀, M) satisfying the further assumption

a(x, t)−a(x, s)N|t−s| (13)

fora.e. x∈Ω and every s, t∈[0, T]. For simplicity we define the family of operators A: [0, T]→ L(H₀¹(Ω), H⁻¹(Ω)) A(t)u, v_H−1(Ω)×H₀¹(Ω)=

Ω

a(x, t)·Du(x), Dv(x) dx A:V → V Au(t) =A(t)u(t).

(14)

Observe that under assumption (13) if we choose a ∈ M_Ω×(0,T)(λ₀,Λ₀, M, N) we can define A : [0, T] → L(H₀¹(Ω), H⁻¹(Ω))

A(t)u, v

H⁻¹×H¹₀ =

Ω

a(x, t)·Du(x), Dv(x)

dx wherea_ij(x, t) =∂a_ij

∂t (x, t)

which by (13) turns out to be bounded. We recall now an existence result contained in [8] (Th. 3.8). Before we give the definition of solution.

Definition 2.5. Givena ∈ M_Ω×(0,T)(λ₀,Λ₀, M),r ∈ F(C1, C₂, µ₀) withµ₀>−2λ0, f ∈ V, ϕ∈L²(Ω^r₊(0), r(·,0)) (see Rem. 2.4 for the definition of this space) a functionu∈ W is a solution of

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂

∂t(ru)−div(a·Du) =f on Ω×(0, T)

u= 0 in∂Ω×(0, T)

u(x,0) =ϕ in Ω^r₊(0)

(15)

if

(Ru)(t), v_H⁻¹_(Ω)×H0¹_(Ω)+ Au(t), v_H⁻¹_(Ω)×H¹_0(Ω)= f(t), v_H⁻¹_(Ω)×H0¹_(Ω)

Ω

u(x,0)−ϕ(x)₂

r(x,0) dx= 0

for every v ∈H₀¹(Ω) and fora.e. t∈ [0, T] and the initial datum makes sense in L²(Ω^r₊(0), r(·,0)) thanks to Theorem 2.3. Ifr≡0 the initial condition has no meaning and in this case a solution is a functionu∈ V such that

Au(t), v_H−1(Ω)×H₀¹(Ω)= f(t), v_H−1(Ω)×H¹₀(Ω)

for everyv ∈H₀¹(Ω) and fora.e. t∈[0, T].

Theorem 2.6. Consider a∈ M_Ω×(0,T)(λ₀,Λ₀, M),r∈ F(C₁, C₂, µ₀) withµ₀ >−2λ₀. For everyf ∈ V and ϕ∈L²(Ω^r₊(0), r(·,0))problem(15)has a unique solutionu∈ Wand there exists a constantc=c(µ₀, λ₀,Λ₀, C₂)

(depending only onµ₀,λ₀,Λ₀,C₂)such that u_W c

f_V+ϕ(·)r^1/2(·,0)_L²_(Ω)

. (16)

Statement of the problem- Fixf ∈ V andϕ∈L²(Ω) and consider (a_h)_h⊂ M_Ω×(0,T)(λ₀,Λ₀, M, N), (r_h)_h⊂ F(C₁, C₂, µ₀)

µ₀>−2λ0.

(17)

Assumptionµ₀>−2λ₀is required to have existence to problems (18) (see Th. 2.6).

Then we consider the sequence of elliptic-parabolic problems

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂

∂t(r_hu)−div(a_h·Du) =f on Ω×(0, T)

u= 0 in ∂Ω×(0, T)

u(x,0) =ϕ in Ω_h,+(0)

(18)

where Ω_h,+(t) := Ω^r₊^h(t). We consider ϕ ∈ L²(Ω) so that problem (18) makes sense for every h ∈ N, since L²(Ω) is dense inL²(Ω_h,+(0), r_h(·,0)) for everyh∈N(see Rem. 2.4).

We want to study the asymptotic behaviour of the solutions u_h when h→+∞and characterise the limit problem.

The main difficulty is the lack of compactness of the solutions in L²(0, T;L²(Ω)) which is natural in the classical case,i.e. whenr_h≡1.

In this framework the natural compactness result reads as follows in the theorem below (see Th. 2.14 and Th. 2.18 in [8]). Before we define

W_h=

u∈ V (r_hu)∈ V

. (19)

Theorem 2.7. Consider a sequence(u_h)_h such that u_h∈ Wh andu_hWh c for a positive constant c. Then (u_h)_h is relatively compact

(i) in L²(0, T;L²(Ω))if r_h→r inL^∞(Ω×(0, T))-weakly ∗and r >0almost everywhere;

(ii) in L²(Q₊;r), the completion ofC_c(Q₊) with respect to the norm ur^1/2_L²_(Q+) where Q₊ ={(x, t)∈ Ω×(0, T)|r(x, t)>0}, if r_h→r inL^∞(Ω×(0, T)) (strongly).

We by-pass the problem of the lack of compactness in L²(0, T;L²(Ω)) via the following regularity result (see Th. 3.11 and Cor. 3.13 in [8]).

Theorem 2.8. Consider the problem (15). Assume that, besides to assumptions of Theorem 2.6, ∂r/∂t ∈ F(K1, K₂), i.e. satisfies (i)−(iv)of(6)with constantsK₁, K₂, andasatisfies(13). Suppose moreover thatf ∈ H¹(0, T;H⁻¹(Ω)) and that there existsu₀∈H₀¹(Ω)such thatu₀=ϕinΩ^r₊(0)andf(0) + div (a(x,0)Du₀(x))−

∂r∂t(x,0) =r(x,0)v(x) for somev∈L²(Ω^r₊(0)). Then the solution u satisfies u∈H¹(0, T;H₀¹(Ω)) and u_H1(0,T;H₀¹(Ω)) c for a positive constantc depending (only) on µ₀, λ₀,Λ₀, C₂, N, K₂,f_H¹_(0,T;H⁻¹_(Ω)), r^1/2(·,0)v_L²_(Ω),r^1/2(·,0)u_0L²_(Ω).

3. Preliminary compactness results

In this section we will suppose more regularity on the sequence (r_h)_h than we will require to state the main theorem (see Th. 4.5). Precisely in this section we require (see (12), (13) and (6))

(a_h)_h⊂ M_Ω×(0,T)(λ₀,Λ₀, M, N), (r_h),⊂ F(C1, C₂, µ₀), ∂r_h

∂t

h⊂ F(K1, K₂, ν₀) (20) for some constantsK₁, K₂, ν₀. Consider the problems

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂

∂t(r_hu)−div(a_h·Du) =g on Ω×(0, T)

u= 0 in ∂Ω×(0, T)

u=ψ_h in Ω_h,+(0)× {0}

(21)

whereg∈H¹(0, T;H⁻¹(Ω))⊂C([0, T];H⁻¹(Ω)) andψ_h is the solution to E_hw:=−div(a_h(x,0)·Dw(x)) +∂r_h

∂t (x,0)w(x) =g(x,0) in Ω

w= 0 in ∂Ω, (22)

where the linear, elliptic and bounded operatorE_h : H₀¹(Ω) →H⁻¹(Ω) can be considered because a_h andr_h are continuous in the variablet.

Before stating the main result we recall the following lemma. Denote by c_P the constant appearing in the Poincar´e’s inequality

Ω

u²(x)dxc_P

Ω

|Du|²(x)dx, u∈H₀¹(Ω). (23)

Lemma 3.1. Considera, a₁, a₂...∈ MΩ(λ₀,Λ₀, M)and supposea_h −→^EG a(see Def.1.1). Consider a sequence of functions (b_h)_h ⊂L^∞(Ω), b_h γ for every h∈N where −λ₀/c_P < γ 0 and suppose b_h →b inL^∞(Ω)- weak∗. Then for every f ∈H⁻¹(Ω) it results that

w_h→w inL²(Ω)

a_h·Dw_h→a·Dw inL²(Ω)ⁿ-weak, wherew_h andw denote respectively the solutions

−div(a_h·Dv) +b_hv=f in Ω

v= 0 in ∂Ω

−div(a·Dv) +bv=f inΩ

v= 0 in∂Ω.

Proof. Since−λ₀/c_P < γ0 we have that

Ω

b_hv²dx

Ω

γv²dxc_Pγ

Ω

|Dv|²dx

and then the elliptic operators v → −div(a_h·Dv) +b_hv are equicoercive. Since the solutions are compact in L²(Ω) we have, up to a subsequence, that −b_hw_h converges to−bw weakly in L²(Ω) and then strongly in H⁻¹(Ω). Then we obtain the thesis observing thatw_h solve the problems

−div(a_h·Dv) =f_h:=f−b_hw_h in Ω

v= 0 in∂Ω.

Remark 3.2. As a consequence we have that, if a_h(·,0) −→^EG a(·,0) and (r_h)_h ⊂ F(C1, C₂, µ₀), (^∂r_∂t^h)_h ⊂ F(K1, K₂, ν₀) for some K₁, K₂, ν₀ and |^∂r_∂t^h|(x,0) α for every h ∈ N with 0 α < λ₀/c_P are such that r_h→rinL^∞(Ω×(0, T))-weak∗(see also Rem. 2.1) equation (22) admits a unique solution since

Ω

u²(x)∂r_h

∂t (x,0)dx−α

Ω

u²(x)dx >−λ₀

Ω

|Du|²(x)dx.

The solutionsψ_h of (22) satisfy

ψ_h→ψ in L²(Ω), a_h·Dψ_h→a·Dψ in L²(Ω)ⁿ-weak, whereψ is the solution of

Ew:=−div(a(x,0)·Dw(x)) +∂r

∂t(x,0)w(x) =g(x,0) in Ω

w= 0 in ∂Ω.

Now we state the first compactness result.

Lemma 3.3. Consider the problems (21) with the data g∈H¹(0, T;H⁻¹(Ω)) andψ_h =E⁻¹_h g(0) and denote by u_h∈ W_h the corresponding solutions. Then the sequence

(u_h)_h is bounded inH¹(0, T;H₀¹(Ω)).

As a consequence we obtain that

(u_h)_h is relatively compact inC([0, T];L²(Ω)), the sequence t→

Ω

u²_h(x, t)r_h(x, t)dx is relatively compact in C([0, T]).

Proof. Sinceg(0) + div(a_h(·,0)·Dψ_h)−∂r_h

∂t (·,0)ψ_h= 0, hypotheses of Theorem 2.8 are satisfied. Then we have that the solutions satisfy the estimations

u_hH¹_(0,T;H0¹_(Ω))c, u_hC([0,T];H¹_0(Ω))c.

We deduce that (u_h)_h is equibounded inC([0, T];H₀¹(Ω)) and therefore the sets{u_h(t)|h∈N} are relatively compact inL²(Ω) for everyt∈[0, T]. Moreover

u_h(t)−u_h(s) = ^t

s u_h(τ)dτ and then

u_h(t)−u_h(s)_H¹_0(Ω) ^t

s u_h(τ)_H0¹_(Ω)dτ |t−s|^1/2u_hV,

so (u_h)_h is also equicontinuous valued in H₀¹(Ω) (and in particular in L²(Ω)). Then by Lemma 1 in [12] we obtain (u_h)_hrelatively compact in C([0, T];L²(Ω)).

To prove the second statement denote for simplicity by (u_h)_h a subsequence converging inC([0, T];L²(Ω)) and calluthe limit inC([0, T];L²(Ω)). Consider the sequence (r_h)_h: by Remark 2.1 we have the existence of

r∈ F(C₁, C₂, µ₀) such that, up to a subsequence,r_h→rin L^∞(Ω×(0, T))-weak∗. Then, fort, s∈[0, T],

Ω

u²_h(x, t)r_h(x, t)dx−

Ω

u²_h(x, s)r_h(x, s)dx

Ω

u²_h(x, t)r_h(x, t)dx−

Ω

u²_h(x, t)r_h(x, s)dx +

Ω

u²_h(x, t)r_h(x, s)dx−

Ω

u²_h(x, s)r_h(x, s)dx. By (6) and the continuity ofu_h: [0, T]→L²(Ω) we infer thatt→

Ωu²_h(x, t)r_h(x, t)dxis continuous. Moreover

Ω

u²_h(x, t)r_h(x, t)dx−

Ω

u²(x, t)r(x, t)dx

Ω

u²_h(x, t)r_h(x, t)dx−

Ω

u²(x, t)r_h(x, t)dx +

Ω

u²(x, t)r_h(x, t)dx−

Ω

u²(x, t)r(x, t)dx r_h∞

Ω

|u²_h(x, t)−u²(x, t)|dx+

Ω

u²(x, t)

r_h(x, t)−r(x, t) dx

by which we obtain the thesis.

Theorem 3.4. Consider a sequence of functions v_h ∈ Wh (h= 1,2, ...), a function v∈ W, a constant c₁ such that

v_hWh c₁ for everyh, v_h→v in C([0, T];L²(Ω)), (24) and a sequence of vectorial functions m_h, m∈L²(0, T; (L²(Ω))ⁿ) (h= 1,2, ...), a constantc₂ such that

m_hL²_(0,T;(L²(Ω))ⁿ)c₂ m_h→m inL²(0, T; (L²(Ω))ⁿ)-weak. (25) Moreover suppose that

(r_hv_h)−div (m_h) =f ∈ V inC_c¹(Ω×(0, T)). (26)

Then

m_h, Dv_h

→ m, Dv

in D(Ω×(0, T)).

Proof. Fix a functionϕ∈C_c^∞(Ω×(0, T)) and multiply equation (26) byv_hϕ. We obtain

T 0 Ω

(m_h, Dv_h)ϕdxdt= f −(r_hv_h), v_hϕ − ^T

0 Ω

(m_h, Dϕ)v_hdxdt. (27)

Clearly f, v_hϕ → f, vϕas h→ ∞ and

T 0 Ω

(m_h, Dϕ)v_hdxdt → ^T

0 Ω

(m, Dϕ)vdxdt. By (10) and since ϕhas compact support in Ω×(0, T) we have that

−2 (r_hv_h), v_hϕ=

T

0 Ω

v_h²(x, t)r_h(x, t)∂ϕ

∂t(x, t)dxdt− ^T

0 Ω

v²_h(x, t)ϕ∂r_h

∂t (x, t)dxdt which converges to

T

0 Ω

v²(x, t)r(x, t)∂ϕ

∂t(x, t)dxdt− ^T

0 Ω

v²(x, t)ϕ∂r

∂t(x, t)dxdt=−2 (rv), vϕ.

Then T 0 Ω

(m_h, Dv_h)ϕdxdt→ f −(rv), vϕ − ^T

0 Ω

(m, Dϕ)vdxdt and since, multiplying (26) byφ∈C_c^∞(Ω×(0, T)) and taking the limit, we also have

(rv)−div (m) =f inV,

we obtain the thesis.

4. The definition of G -convergence and the compactness result

In this section we give the main result, a compactness result with respect to G-convergence defined below (see Def. 4.3) for a sequence of operators (see (12) and (6))

P_hu= ∂

∂t(r_hu)−div(a_h·Du),

(a_h)_h⊂ M_Ω×(0,T)(λ₀,Λ₀, M, N), (r_h)_h⊂ F(C1, C₂, µ₀).

(28)

The statement of this result is divided in two theorems. In the first one (Th. 4.1) we suppose the regularity required in the previous section, i.e. (20), and prove a partial result: the existence of a limit operator in divergence form. The second result (Th. 4.5) is a kind of uniqueness result: with less assumptions on the coefficients,i.e. satisfying the assumptions in (28), we will prove that the matrix defining the limit operator is independent of the sequence (r_h).

Theorem 4.1. Consider a sequence(a_h)_h⊂ M_Ω×(0,T)(λ₀,Λ₀, M, N)and(r_h)_h⊂ F(C1, C₂, µ₀)with(^∂r_∂t^h)_h⊂ F(K₁, K₂, ν₀) and ^∂r_∂t^h(x,0) = 0 for every h∈N. There exist a matrix a∈ M_Ω×(0,T)(λ₀, M²Λ₀, M

Λ₀/λ₀) and a function r∈ F(C₁, C₂, µ₀)such that for every f ∈ V andϕ∈L²(Ω) the solutions u_h of problems (18), h∈N, satisfy, up to a subsequence,

u_h→u in L²(0, T;L²(Ω)) and a_h·Du_h→a·Du inL²(0, T, L²(Ω)ⁿ)-weak whereuis the solution of ⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

∂(ru)

∂t −div(a·Du) =f inΩ×(0, T)

u= 0 in∂Ω×(0, T)

u=ϕ inΩ^r₊(0)× {0}.

(29)

Remark 4.2. The result is true also if in (18) we consider a sequence of data (f_h)_h ⊂ V, (f_h)_h relatively compact inV, and (ϕ_h)_h⊂L²(Ω),ϕ_h relatively compact inL²(Ω).

Proof. First, by Remark 2.1, from (r_h)_hwe can extract a subsequence, still denoted by (r_h)_h, such thatr_h→r in L^∞(Ω×(0, T)-weak∗ andr∈ F(C1, C₂, µ₀). Denote byRthe operator defined by

R:L²(0, T;L²(Ω))→L²(0, T;L²(Ω)), (Ru)(x, t) =r(x, t)u(x, t). (30) Analogously (to short) we denote byR_h the operators associated tor_h and byA_hthe operators associated to a_h as defined in (14). FixX a countable and dense subset of H¹(0, T;H⁻¹(Ω)) and Y ={y ∈H₀¹(Ω) | y = E⁻¹g(0) forg ∈ X} where E is the operator defined in Remark 3.2 (the EG-limit, up to a subsequence, of {a_h(·,0)}_h). Then considerg∈X andψ=E⁻¹g(0) (and denote byu_h(g, ψ) the solutions to the problems (21) withψ_h=E_h⁻¹

Eψ

where E_hare the operators in (22).