Homogenization of degenerate elliptic-parabolic equations

Fabio Paronetto

Abstract

In this paper we give a result of G-convergence for a class of strongly degenerate parabolic equations in the case of periodic coefficients. The operators have the form μ(x)∂_{t} − div(a(x, t) · D) where the quadratic form associated to a(x, t) is degenerating as a Muckenhoupt weight and the coefficient μ is greater or equal to zero, possibly μ ≡ 0, that is the operator may be elliptic, parabolic or elliptic-parabolic.

1 Introduction

The problem of homogenization (see [10], [18], [32], [3]) for partial differential equations was studied since the last century (see [1] and the references therein), and in the last thirty years especially in connection with G-convergence of differential operators and Γ-convergence of functionals (see [27] and [10] as first papers dealing with this problem, and for instance [6], [38], [21]). Suppose Ω a bounded open set with Lipschitz boundary, T > 0, some functions μ : R^{n} → R and a_{ij} : R^{n} × [0, T] → R, i, j = 1, ...n, such that μ and a_{ij}(·, t) are Q-periodic functions for a. e. t ∈ [0, T] where Q = [0, 1]^{n}. Suppose the function μ ⩾ 0 and Q_{+} and Q_{0} with Lipschitz boundaries, with Q_{+} ∪ Q_{0} = Q, such that μ(x) > 0 for a. e. x ∈ Q_{+} and μ(x) = 0 for a. e. x ∈ Q_{0}. Now, if α = (α_{1}, ...α_{n}) with α_{j} ∈ Z, denote τ_{α}Q_{+} := {x ∈ R^{n} | x − α ∈ Q_{+}} and τ_{α}Q_{0} := {x ∈ R^{n} | x − α ∈ Q_{0}}, Π_{+} := ∪_{α∈Z^{n}}τ_{α}Q_{+} and Π_{0} := ∪_{α∈Z^{n}}τ_{α}Q_{0}, for h ∈ N Π_{+,h} := {x ∈ R^{n} | hx ∈ Π_{+}} and Π_{0,h} := {x ∈ R^{n} | hx ∈ Π_{0}}, and finally Ω_{+,h} = Ω ∩ Π_{+,h} and Ω_{0,h} = Ω ∩ Π_{0,h}. If μ_{h}(x) = μ(hx) we have that μ_{h} > 0 a. e. in Π_{+,h} and μ_{h} = 0 a. e. in Π_{0,h}. Then we denote by Ω_{+,h} the set Ω ∩ Q_{+,h} and by Ω_{0,h} the set Ω ∩ Q_{0,h}. Note that the equations

μ(hx)\frac{∂u}{∂t} − div(a(hx, t) · Du) = f

are parabolic in Ω_{+,h}, elliptic in Ω_{0,h}. We want to study the limit behaviour (h → +∞) of the solutions of the following problems

μ(hx)\frac{∂u}{∂t} − div(a(hx, t) · Du) = f  on Ω_{+,h} × (0, T)
u = 0  on ((∂Ω_{+,h} ∩ ∂Ω) × (0, T))
u = φ  on (Ω_{+,h} × {0})
−div(a_{h}(·, t) · Du) = f(·, t)  on Ω_{0,h} for a. e. t ∈ [0, T]
u = 0  on ∂Ω_{0,h} ∩ ∂Ω .

which can be written

\{ \begin{matrix}μ(hx)\frac{∂u}{∂t} − div(a(hx, t) · Du) = f & on Ω × (0, T) \\ u(x, t) = 0 & on ∂Ω × (0, T) \\ u(x, 0) = φ(x) & on Ω_{+,h} . \end{matrix}  (1.1)

We will make the following assumption on the matrix a: we will suppose the coefficients

a_{ij}(x, t) such that a_{ij}(·, t) Q-periodic for a. e. t ∈ [0, T]

and there exists a function λ > 0 and some constants L, M, N such that

λ(y)|ξ|^{2} ⩽ (a(y, t) · ξ, ξ) ⩽ Lλ(y)|ξ|^{2}
(a(y, t) · ξ, η) ⩽ M(a(y, t) · ξ, ξ)^{1/2}(a(y, t) · ξ, ξ)^{1/2}
|a(y, t) − a(y, s)| ⩽ Nλ(y)|t − s|

for every ξ, η ∈ R^{n}, for a. e. y ∈ R^{n} and for a. e. t, s ∈ [0, T]. On the function λ and on the function μ we will make the following assumptions:

μ ⩾ 0, λ > 0 a. e. in R^{n} ,  μ, λ Q-periodic functions ,  μ, λ, λ^{−1} ∈ L_{loc}^{1}(R^{n})

and moreover we will assume that the weights μ and λ verify the following Muckenhoupt type conditions

sup_{Q⊂R^{n}} (\frac{1}{|Q|} ∫_{Q} λdx)(\frac{1}{|Q|} ∫_{Q} λ^{−1}dx) < +∞ (A_{2})

sup_{Q⊂[0,1]^{n}} |Q|^{αr/n}(\frac{1}{|Q|} ∫_{Q} μ^{r}dx)^{1/2}(\frac{1}{|Q|} ∫_{Q} λ^{−r}dx)^{1/2} < +∞ (A_{2,2}^{αr})

(for these conditions see, for instance, [13, 28] and Section 3) for suitable constants 0 ⩽ α < 1 and r > 1 and where Q denotes a cube of R^{n} with faces parallel to the coordinate planes. The condition (A_{2,2}^{αr}) can be weakened (r = 1) requiring that μ satisfies an additional assumption (see Section 5, (H.2)). The main result is the following: denote by u_{h}(f, φ) the unique solutions to (1.1), by â the limit matrix defined as in the classical homogenization case (see, for example, [3] and Section 7) and

μ̂ := ∫−_{[0,1]^{n}}μ dy .

Theorem For every f ∈ L^{2}(0, T; L^{n}(Ω)) and φ ∈ W_{0}^{1,∞}(Ω) we have that

\begin{matrix}u_{h}(f, φ) → u(f, φ) & in L^{2}(0, T; L^{1}(Ω)) \\ a_{h} · Du_{h}(f, φ) → â · Du(f, φ) & in L^{2}(0, T; (L^{1}(Ω))^{n})−weak, \end{matrix}

where u(f, φ) is the unique solution of the following Cauchy-Dirichlet problem (the matrix â is independent of the weight μ, possibly μ ≡ 0)

μ̂\frac{∂}{∂t}u − ∑_{i,j=1}^{n}â_{ij}(t)\frac{∂^{2}}{∂x_{j}∂x_{i}} u = f on Ω × (0, T)
u(x, t) = 0  on ∂Ω × (0, T)
u(x, 0) = φ(x)  on Ω .

In the second section we give a result of completeness of a weighted Sobolev space with two different weights (inducing two different measures), μ ⩾ 0 and λ > 0, in such a way the norm will be ∥u∥^{2} = ∫_{Ω} u^{2}μ + ∫_{Ω} |Du|^{2}λ. In the section 4 we give some results about the abstract equation

Ru′ + Au = f

with R ⩾ 0 bounded operator. We give existence results for the equation Ru′ + Au = f ∈ L^{2}(0, T; L^{n}(Ω)) and a regularity result for the solution if f ∈ H^{1}(0, T; L^{n}(Ω)). Via this regularity result we prove the main theorem with a regular datum and then we approximate f ∈ L^{2}(0, T; L^{n}(Ω)) with suitable functions in H^{1}(0, T; L^{n}(Ω)). We describe the problem in Section 5, we obtain some useful compactness results in Section 6, the main theorem in Section 7. In the last section we give some examples and show that such a result is not true if homogenization also in time is considerd, i. e. a sequence of matrices

a(h^{α}x, h^{β}t) .

Notations: we will denote by C_{c}^{1}(Ω) the set of functions with compact support in Ω, by C_{per}^{1}(Q) the set of periodic functions with period the cube Q and by C_{#}^{1}(Q) the set C_{per}^{1}(Q)/R or the subset of C_{per}^{1}(Q) of the functions with mean value equal to zero. We will denote by Q a generic (open or closed) cube of R^{n} with faces parallel to the coordinate planes, by cQ the cube concentric with Q and having side length c times that of Q and and by B_{r}(x) the open ball of R^{n} centered in x with radius r. We recall that a dyadic cube is a cube of R^{n} whose vertices are in 2^{k}Z^{n} with k ∈ Z. The symbols (·, ·), |E| and p′ will indicate respectively the scalar product of R^{n}, the Lebesgue measure of the set E, and

the conjugate p′ = p/(p − 1) of p. The symbol ∫−_{E}fdx will denote the mean value of f on

E ( i. e. |E|^{−1} ∫_{E} λ(x)dx ); sometimes for a periodic function we will use the symbol ⟨f⟩ to

denote the mean value of f on its period. If f : Ω → R we will denote by \overline{f} : R^{n} → R the function extended zero out of Ω.

Acknowledgements

I wish to thank Prof. A. A. Pankov who, during is visit in Rome, was so kind to answer me many questions about one of his papers ([22]). I also want to thank G. P. Leonardi and V. Recupero for many interesting discussions, not only on the subject.

2 A result on completeness of a weighted Sobolev space

In this section we show a result (Theorem 2.3 and Corollary 2.5) on completeness of some weighted Sobolev spaces. We consider the class of admissible functions

A(Ω) = {ν : Ω → R | ν measurable and ν ⩾ 0}.

Among the class A(Ω) we consider the subclass B_{p} defined as follows: given an open set Ω and 1 < p < ∞ we say that

ν ∈ B_{p}(Ω)  if ν^{−1/(p−1)} ∈ L_{loc}^{1}(Ω) .  (2.1)

If ν ∈ B_{p}(Ω) we have that for every ω ⊂⊂ Ω

∫_{ω} |u|dx ⩽ ( ∫_{ω} |u|^{p}νdx)^{1/p}( ∫_{ω} ν^{−1/(p−1)}dx)^{(p−1)/p}.

Then if ν ∈ B_{p}(Ω) we have the inclusions

L^{p}(Ω, ν) ⊂ L_{loc}^{1}(Ω)  (2.2)

where

L^{p}(Ω, ν) = {u ∈ L_{loc}^{1}(Ω) | uν^{1/p} ∈ L^{p}(Ω)}.

The assumption that ν ∈ B_{p}(Ω) is sharp to obtain (2.2) (see [14]). Given an open set Ω, 1 < p < +∞ and n + 1 functions in A(Ω) μ, λ_{1}, ...λ_{n} such that λ_{i} ∈ B_{p}(Ω) we define

W^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) = {u ∈ L^{p}(Ω, μ) ∩ W_{loc}^{1,1}(Ω) | D_{i}u ∈ L^{p}(Ω, λ_{i}) i = 1, ...n}, (2.3)

Suppose the set {x ∈ Ω | μ(x) > 0} to have positive measure; we can endow the space W^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) with the norm

∥u∥_{W^{1,p}(Ω,μ,λ_{1},...λ_{n})}^{p} =^{def} _{∫Ω}|u|^{p}μdx + ∑_{i=1}^{n}_{∫Ω}|D_{i}u|^{p}λ_{i}dx  (2.4)

Now we want to prove that the normed space just defined is a Banach space.

Lemma 2.1 Let Ω be an open and connected subset of R^{n}. Consider a point x_{0} ∈ Ω and a sequence (u_{h})_{h} ⊂ C^{1}(Ω) such that

u_{h}(x_{0}) is a Cauchy sequence in R ,
∇u_{h} is a Cauchy sequence in (L_{loc}^{1}(Ω))^{n} .

(2.5)

Then (u_{h})_{h} is a Cauchy sequence in W_{loc}^{1,1}(Ω).

Proof - By contradiction we suppose that (u_{h})_{h} is not a Cauchy sequence in L_{loc}^{1}(Ω). Then there exists a subsequence (u_{h_{j}})_{j}, a ball B ⊂⊂ Ω and ϵ > 0 such that

∫_{B} |u_{h_{j}} − u_{h_{j+1}}| dx > ϵ  for every j.  (2.6)

Consider ω an open and connected subset of Ω such that

B ∪ {x_{0}} ⊂⊂ ω ⊂⊂ Ω

and moreover such that ω is diffeomorphic to a ball. Then it is possible to find Γ ∈ C^{1}(\overline{B} × [0, 1]; ω), such that

γ_{x}(t) := Γ(x, t) is a regular arc

γ_{x}(0) = Γ(x, 0) = x_{0}, γ_{x}(1) = Γ(x, 1) = x for every x ∈ B,

ω_{t} := {γ_{x}(t) ∈ Ω | x ∈ B} is a simply connected precompact open set ,

\frac{∂Γ}{∂t}(x, 0) = 0

(2.7)

(one can find a C^{1} diffeomorphism ψ between ω and a ball B_{0} and then take Γ(x, t) = ψ^{−1}(ψ(x_{0}) + t^{2}(ψ(x) − ψ(x_{0})))). For every j ∈ N we can write

u_{h_{j}}(x) = u_{h_{j}}(x_{0}) + ∫_{γ_{x}} (∇u_{h_{j}}(γ_{x}(t)), γ′_{x}(t))dt .  (2.8)

By plugging this expression in (2.10) we obtain

∫_{B} |u_{h_{j}}(x_{0}) − u_{h_{j+1}}(x_{0})|dx + ∫_{B} | ∫_{γ_{x}} ((∇u_{h_{j}}(γ_{x}(t)) − ∇u_{h_{j+1}}(γ_{x}(t))), γ′_{x}(t))dt|dx > ϵ

and thanks to (2.5) we deduce that definitely

∫_{B} ( ∫_{γ_{x}} |∇u_{h_{j}}(γ_{x}(t)) − ∇u_{h_{j+1}}(γ_{x}(t))||γ′_{x}(t)|dt)dx > \frac{ϵ}{2}  (2.9)

that is

\frac{ϵ}{2} < ∫_{B} ∫_{0}^{1} |(∇u_{h_{j}} − ∇u_{h_{j+1}})(γ_{x}(t))||γ′_{x}(t)|dtdx
= ∫_{0}^{1} ∫_{B} |(∇u_{h_{j}} − ∇u_{h_{j+1}})(γ_{x}(t))||γ′_{x}(t)|dxdt
⩽ max_{t∈[0,1]}|[ ∫_{B} |(∇u_{h_{j}} − ∇u_{h_{j+1}})(γ_{x}(t))||γ′_{x}(t)|dx]
= [ ∫_{B} |(∇u_{h_{j}} − ∇u_{h_{j+1}})(γ_{x}(\overline{t}))||γ′_{x}(\overline{t})|dx]
⩽ max_{x∈\overline{B}} |γ′_{x}(\overline{t})| ∫_{B} |∇u_{h_{j}}(γ_{x}(\overline{t})) − ∇u_{h_{j+1}}(γ_{x}(\overline{t}))|dy .

Since |γ′_{x}(0)| = 0 for every x ∈ B the maximum point \overline{t} will not be 0 and then ω\overline{t} will not be {x_{0}} but an open set. Consequently we can estimate the last term and obtain

\frac{ϵ}{2} < max_{x∈\overline{B}} |\frac{∂Γ}{∂t}(x, \overline{t})| max_{x∈\overline{B}} |J_{\frac{∂Γ}{∂x} (x,\overline{t})}| ∫_{ω\overline{t}} |∇u_{h_{j}}(y) − ∇u_{h_{j+1}}(y)|dy

where |J_{\frac{∂Γ}{∂x}} | is the Jacobian. Since (∇u_{h})_{h} is a Cauchy sequence in (L^{1}(ω))^{n} the last inequality is impossible. □

Remark 2.2 - The lemma also holds for a sequence

(u_{h})_{h} ⊂ C_{c}^{1}(Ω) ,  ∇u_{h} is a Cauchy sequence in (L_{loc}^{1}(Ω))^{n} ,  Ω bounded

or a sequence (Q is a cube of R^{n})

(u_{h})_{h} ⊂ C_{per}^{1}(Q) ,  ∫−_{Q}u_{h} → α ∈ R ,  ∇u_{h} is a Cauchy sequence in (L^{1}(Q))^{n} .

In fact if (u_{h})_{h} ⊂ C_{c}^{1}(Ω) we have the existence of a constant c > 0 such that

∫_{Ω} |u_{h}| ⩽ c ∫_{Ω} |∇u_{h}|.

If, by contradiction, we suppose that u_{h} is not converging in L^{1}(Ω) we have that, for ϵ > 0, there exist a ball B and a subsequence (u_{h_{j}})_{j} such that

∫_{B} |u_{h_{j+1}} − u_{h_{j}}| > ϵ .

In particular we would have

ϵ < ∫_{B} |u_{h_{j+1}} − u_{h_{j}}| ⩽ c ∫_{B} |∇u_{h_{j+1}} − ∇u_{h_{j}}| → 0 .

The same is obtained if u_{h} is periodic and its average is converging, using the inequality

∫_{Q} |u_{h} − ∫−_{Q}u_{h}| ⩽ c ∫_{Q} |∇u_{h}| .

Theorem 2.3 Consider Ω a connected open set of R^{n}, μ, λ_{1}, ...λ_{n} ∈ A(Ω) such that A = {x ∈ Ω | μ(x) > 0} has positive measure and λ_{i} ∈ B_{p}(Ω) for i = 1, ...n. Then the space of functions W^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) is complete with respect to the topology induced by the norm (2.4).

Proof - Consider a sequence of functions (u_{h})_{h} and suppose it is a Cauchy sequence with respect to the norm (2.4). Then we have that there exist n + 1 functions v_{0}, v_{1}, ...v_{n} such that

\begin{matrix}u_{h} → v_{0} in L^{p}(Ω, μ) \\ D_{i}u_{h} → v_{i} in L^{p}(Ω, λ_{i}). \end{matrix}

What is to prove is that these functions belong to L_{loc}^{1}(Ω) and that v_{i} = D_{i}v_{0}, or better that there exists a representative of v_{0} in L_{loc}^{1}(Ω) which has v_{i} as derivatives. Since L^{p}(Ω, λ_{i}) ⊂ L_{loc}^{1}(Ω) we immediatly have that v_{i} ∈ L_{loc}^{1}(Ω) for i = 1, ...n. Let us now see that v_{0} ∈ L_{loc}^{1}(Ω). By contradiction suppose that (u_{h})_{h} is not a Cauchy sequence in L_{loc}^{1}(Ω). Then there exists a subsequence (u_{h_{j}})_{j}, a ball B_{r} ⊂⊂ Ω whose radius is r and ϵ > 0 such that

∫_{B_{r}} |u_{h_{j}} − u_{h_{j+1}}| dx > 2ϵ  for every j.  (2.10)

Since |A| > 0 and |A| = lim_{δ→0^{+}} |A_{δ}| where A_{δ} = {x ∈ Ω | μ(x) > δ}, there exists δ > 0 such that |A_{δ}| > 0 and L^{p}(A_{δ}, μ) ⊂ L^{p}(A_{δ}) with continuous embedding. We can also suppose A_{δ} ⊂⊂ Ω (otherwise consider a subset A′_{δ} of A_{δ} such that A′_{δ} ⊂⊂ Ω and |A′_{δ} ∩ A| > 0) in such a way that L^{p}(A_{δ}, μ) ⊂ L^{1}(A_{δ}). Consider ω an open and connected subset of Ω such that

B_{r} ∪ A_{δ} ⊂⊂ ω ⊂⊂ Ω

and moreover such that ω is diffeomorphic to a ball. Since (u_{h})_{h} ⊂ W_{loc}^{1,1}(Ω) for every σ > 0 and for every j there exists w_{j} ∈ C^{1}(Ω) such that

∥w_{j} − u_{h_{j}}∥_{W^{1,1}(ω)} < σ .  (2.11)

Choosing σ ⩽ ϵ/2 from (2.10) one in particular has

∥w_{j} − w_{j+1}∥_{L^{1}(B_{r})} > ϵ for every j.

On the other hand since L^{p}(A_{δ}, μ) ⊂ L^{1}(A_{δ}) from (2.11) (w_{j})_{j} is a Cauchy sequence in L^{1}(A_{δ}) and in particular that there exists a set N of measure zero such that, up to subsequences,

w_{j}(x) → v_{0}(x) for every x ∈ A_{δ} \ N.

Choose x_{0} ∈ A_{δ} \ N such that

w_{j}(x_{0}) → v_{0}(x_{0}).  (2.12)

Applying the previous lemma to the sequence (w_{j})_{j} we conclude that (w_{j})_{j}, and then (u_{h})_{h}, is a Cauchy sequence in L^{1}(ω), but this fact is in contradiction with (2.10). Therefore (u_{h})_{h} admits a limit v^{∗} in L_{loc}^{1}(Ω) and since D_{i}u_{h} → v_{i} in L_{loc}^{1}(Ω) we conclude that (u_{h})_{h} is a Cauchy sequence in W_{loc}^{1,1}(Ω) and then

v_{i} = D_{i}v^{∗}.

The last thing to be verified is that v^{∗} = v_{0} in L^{p}(Ω, μ). Since u_{h} → v_{0} in L^{p}(Ω, μ), u_{h} → v_{0} in L^{p}(A_{δ}) for every δ > 0 and consequently in L_{loc}^{1}(A). On the other hand u_{h} → v^{∗} in L_{loc}^{1}(Ω) and in particular in L_{loc}^{1}(A). Then v_{0} = v^{∗} a. e. in A and v^{∗} is a representative of v_{0} in L^{p}(Ω, μ). □

Remark 2.4 - Observe that, in the hypotheses of Theorem 2.3, a Cauchy sequence (u_{h})_{h} in W^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) is a Cauchy sequence in W_{loc}^{1,1}(Ω).

This result is an extension of the result contained in [14]. In this paper the authors show the completeness of W^{1,p}(Ω, μ, λ_{1}, λ_{n}) with μ > 0 a. e. in Ω and not μ ⩾ 0. The authors in [14] show the sharpness af the assumptions λ_{i} ∈ B_{p}(Ω). As regards the other hypotheses, if we drop the assumption that Ω is connected the previous result is not true in general: it is sufficient to consider, if n = 1, Ω = (0, 1) ∪ (1, 2), μ ≡ 1 on (0, 1) and μ ≡ 0 on (1, 2), λ ≡ 1 on Ω. Then the sequence u_{n} = 1 on (0, 1) and u_{n} = n on (1, 2) is a Cauchy sequence in W^{1,p}(Ω, μ, λ) for every p, but it evidently does not admit a limit in W_{loc}^{1,1}(Ω). In the same way if we drop |{μ > 0}| > 0 and consider μ = 0 a. e. in Ω it is sufficient, for example, to consider a sequence of constants going to infinity.

To define a Sobolev space with more than one weak derivative, consider a family M of multi-indexes whose lenght is less than or equal to k ∈ N, then a family of weights Λ_{M} = {λ_{α} | α ∈ M} and define W^{k,p}(Ω, Λ_{M}) as the completion of the set of functions u ∈ W_{loc}^{k,1}(Ω) which satisfy

∥u∥_{Λ_{M}}^{p} =^{def} ∑_{|α|⩽k} _{∫Ω}|D^{α}u|^{p}λ_{α}dx < ∞  (2.13)

with respect to the norm (2.13).

Corollary 2.5 Consider a connected open set Ω, the family M of multi-indexes α with |α| = 0, ...k and a set of weights Λ_{M} = {λ_{α}|α ∈ M}. Suppose there exist some subsets of Ω, A_{α} with α ∈ M and |α| < k, with positive measure such that

λ_{α} > 0 in A_{α} if |α| < k, λ_{α} ∈ B_{p}(Ω) if |α| = k.

Then the space W^{k,p}(Ω, Λ_{M}) is complete.

Proof - By induction we suppose that for every integer value from 1 to k − 1 the thesis is true. By M_{j} we denote the set {α ∈ M : |α| > 0 and α_{j} ≠ 0} and by Λ_{j} the set of weights Λ_{M_{j}}. If we consider a sequence (u_{h})_{h} ⊂ W_{loc}^{k,1}(Ω) which is a Cauchy sequence with respect to the norm (2.13) we have that, for every j = 1, ...n, the sequence (D_{j}u_{n})_{n} converges in W^{k−1,p}(Ω, Λ_{j}). In particular (D_{j}u_{n})_{n} admits a subsequence converging in L_{loc}^{1}(Ω). Now using Lemma 2.1 we conclude. □

Now we return to spaces with only one weak derivative. To have

C_{c}^{1}(Ω) ⊂ W^{1,p}(Ω, μ, λ_{1}, ...λ_{n}),  (2.14)

so to define a space of functions which “are” zero on ∂Ω, we need an additional request on our weights. Concerning this we recall the following result, contained in [14].

Lemma 2.6 The inclusion (2.14) holds if and only if μ, λ_{1}, ...λ_{n} ∈ L_{loc}^{1}(Ω).

Now if we suppose μ, λ_{1}, ...λ_{n} to be L_{loc}^{1}(Ω) (with λ_{1}, ...λ_{n} ∈ B_{p}(Ω)), we can define the space W_{0}^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) as the completion in W^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) of C_{c}^{1}(Ω). In the same way we define W_{per}^{1,p}(Q, μ, λ_{1}, ...λ_{n}) as the completion of C_{per}^{1}(Q) where Q is a cube of R^{n} and C_{per}^{1}(Q) the subset of the Q-periodic functions of C^{1}(Q) and W_{#}^{1,p}(Ω, μ, λ_{1}, ...λ_{n}) as W_{per}^{1,p}(Ω, μ, λ_{1}, ...λ_{n})/R.

Corollary 2.7 Consider a connected and bounded open set Ω of R^{n} with Lipschitz boundary. Consider μ, λ_{1}, ...λ_{n} ∈ A(Ω) such that μ, λ_{i} ∈ L_{loc}^{1}(Ω) and λ_{i} ∈ B_{p}(Ω) for i = 1, ...n (possibly μ ≡ 0). Then the completion of C_{c}^{1}(Ω) and of C_{#}^{1}(Ω) if Ω is a cube with respect to the norm

∥u∥^{p} = _{∫Ω}|u|^{p}μdx + ∑_{i=1}^{n}_{∫Ω}|D_{i}u|^{p}λ_{i}dx

is a Banach space.

Proof - The conclusion follows from Theorem 2.3 if |{μ > 0}| > 0 and in the same way using Remark 2.2 if |{μ > 0}| = 0. □

3 Some recalls

Because of the results of the previous section with the word weight we will mean a function ν such that

ν weight if: ν ⩾ 0 a. e. in R^{n} and ν ∈ L_{loc}^{1}(R^{n}).

From now on μ and λ will be functions such that

μ, λ, λ^{−1}  weights  (3.1)

and we will define (as in (2.3) with p = 2 and λ_{i} = λ for every i = 1, ...n)

H^{1}(Ω, μ, λ) = {u ∈ W_{loc}^{1,1}(Ω) | u ∈ L^{2}(Ω, μ) and |Du| ∈ L^{2}(Ω, λ)}.

With hypotheses (3.1), by the results of the previous section, the space H^{1}(Ω, μ, λ) endowed with the topology induced by the norm

∥u∥_{H^{1}(Ω,μ,λ)} := (_{∫Ω}(u^{2}μ + |Du|^{2}λ)dx)^{1/2}  (3.2)

is a separable Hilbert space and that C_{c}^{1}(Ω) ⊂ H^{1}(Ω, μ, λ). We will denote by H_{0}^{1}(Ω, μ, λ) the closure of C_{c}^{1}(Ω) in the topology of H^{1}(Ω, μ, λ) and by H^{−1}(Ω, μ, λ) its dual space. If Q is a cube of R^{n} we denote

H_{per}^{1}(Q, μ, λ) = completion of C_{per}^{1}(Q) with respect to the norm (3.2),

and

H_{#}^{1}(Q, μ, λ) := H_{per}^{1}(Q, μ, λ)/R  (3.3)

endowed with the norm (3.2) with Ω replaced by Q and by H_{#}^{−1}(Q, μ, λ) its dual space.

Remark 3.1 - Let Ω be a bounded open set and Q_{Ω} be a cube containing Ω. Then H_{0}^{1}(Ω, μ, λ) is a subspace of H_{#}^{1}(Q_{Ω}, μ, λ).

The following definition is usually given for pairs (μ, λ) with μ, λ > 0 a. e.: we extend this to pairs of weights.

Definition 3.2 Let p, q > 1, K > 0 be constants, μ, λ two weights, λ > 0 a. e., α ∈ [0, n) and Q_{0} a cube in R^{n} (even the whole R^{n} if α = 0). We say that the pair (μ, λ) belongs to the class A_{p,q}^{α}(Q_{0}, K) if

|Q|^{\frac{α}{n}} (∫−_{Q}μdx)^{1/q}(∫−_{Q}λ^{−1/(p−1)}dx)^{(p−1)/p} ⩽ K for every cube Q ⊂ Q_{0}.  (3.4)

We denote by A_{p,q}^{α}(Q_{0}) the class ⋃_{K>0} A_{p,q}^{α}(Q_{0}, K). Moreover we denote by A_{p,q}(Q_{0}, K) the class A_{p,q}^{α}(Q_{0}, K) when α = 0 and by A_{p,q}(K) the same class when Q_{0} = R^{n}. Finally we denote by A_{p}(Q_{0}, K) the Muckenhoupt class of weights for which (3.4) holds with α = 0, p = q and μ ≡ λ, by A_{p}(K) the class A_{p}(Q_{0}, K) with Q_{0} = R^{n}, by A_{p} = ⋃_{K⩾1} A_{p}(K) and A_{∞} = ⋃_{p>1} A_{p}.

Remark 3.3 - By definition we have that A_{p,q}^{α}(Q_{0}, K) ⊂ A_{p,q}^{β}(Q_{0}, |Q_{0}|^{(β−α)/n}K) if 0 ⩽ α < β < n; A_{p}(K) ⊂ A_{q}(K) if 1 < p < q < ∞.

Remark 3.4 - Consider 0 < s < r < +∞, 1 < q ⩽ p < +∞, α ∈ [0, n/r). Then if (μ^{r}, λ^{r}) ∈ A_{2,2}^{αr}(Q_{0}, K) then (μ^{s}, λ^{s}) ∈ A_{2,2}^{αs}(Q_{0}, K′) (K′ = K^{s/r}|Q_{0}|^{\frac{r−s}{r} ( \frac{1}{q} − \frac{1}{p})}).

Remark 3.5 - Let Q_{0} be a cube, K > 0, p, q > 1, α ∈ [0, n) and consider μ, λ Q_{0}-periodic weights such that (μ, λ) ∈ A_{p,q}^{α}(Q_{0}, K). Denote with μ_{h} and λ_{h} the weights defined

μ_{h}(x) = μ(hx), λ_{h}(x) = λ(hx) for h ∈ N.

Then for every cube Q̃ there exists a constant K̃ = K̃(n, p, q, Q̃, K) (depending only on n, p, q, Q̃ and K) such that (μ_{h}, λ_{h}) ∈ A_{p,q}^{α}(Q̃, K̃) for every h ∈ N.

A_{p} weights verify the following higher sommability property (see [4, 7]): for every K ⩾ 1, for every p > 1, there exist two positive constants c = c(n, p, K) and δ = δ(n, p, K) (depending only on n, p and K) such that

(∫−_{Q}λ^{1+δ}dx)^{\frac{1}{(1+δ)}} ⩽ c (∫−_{Q}λdx) , (∫−_{Q}λ^{− \frac{1+δ}{p−1}} dx)^{\frac{1}{(1+δ)}} ⩽ c (∫−_{Q}λ^{− \frac{1}{p−1}} dx),  (3.5)

for every cube Q and λ ∈ A_{p}(K). A_{p} weights also verify the doubling property, i. e. if λ ∈ A_{p}(K) one has that for every t > 0 there exists a constant c = c(t, n, p, K) (depending only on t, n, p, K) such that

λ(tQ) ⩽ cλ(Q)  (3.6)

for every cube Q of R^{n} (see for instance [13]). The condition (3.6) implies the reverse doubling property defined as:

there exist δ, ϵ ∈ (0, 1) such that λ(δQ) ⩽ ϵλ(Q) for every cube Q of R^{n}. (3.7)

Remark 3.6 - If ν is a weight satisfying (3.7) with constants ϵ, δ ∈ (0, 1), then the weights

ν_{h}(x) := ν(hx), h ∈ N

satisfy (3.7) with the same constants ϵ, δ.

If Q_{0} is a cube and λ ∈ A_{p}(K) (see [12]) there exists a constant c = c(n, p, K, Q_{0}) (depending only on n, p, K_{1}, Q_{0}), such that

∫_{Q_{0}} |u|^{p}λdx ⩽ c ∫_{Q_{0}} |Du|^{p}λdx  (3.8)

for every u ∈ C_{0}^{∞}(Q_{0}). This result is generalized in [28] in the following way.

Theorem 3.7 Suppose Q_{0} is a proper cube of R^{n}, 1 < p ⩽ q < +∞, and u is a Lipschitz continuous function defined on Q_{0}, with either support in Q_{0} or ∫_{Q_{0}} f(x)dx = 0. Consider μ, λ > 0 a. e., weights on R^{n}. Then

[ ∫_{Q_{0}} |u(x)|^{q}μ(x)dx]^{1/q} ⩽ C(μ, λ, Q_{0})[ ∫_{Q_{0}} |Du(x)|^{p}λ(x)dx]^{1/p}

where

C(μ, λ, Q_{0}) = c(p, r) sup_{Q⊂8Q_{0}} |Q|^{1/n}(∫−_{Q}μ^{r})^{1/pr}(∫−_{Q}λ^{r(1−p′)})^{(p−1)/pr}

for any r > 1 if q = p,

C(μ, λ, Q_{0}) = c(p, q) sup_{Q⊂8Q_{0}} |Q|^{1/n−1}( ∫_{Q} μ)^{1/q}( ∫_{Q} λ^{1−p′})^{(p−1)/p}

if p < q and μ satisfies (3.7), or if p = q and both μ and λ^{1−p′} satisfy (3.7).

Remark 3.8 - It is possible to extend this theorem to pairs of weights (μ, λ) even when μ ⩾ 0, at least when p = q. Indeed consider a pair (μ, λ) with μ ⩾ 0 and λ > 0 a. e. and suppose C(μ, λ, Q_{0}) is finite. It is sufficient to consider the weight

\overline{μ} = \{ \begin{matrix}μ & where μ > 0 \\ λ & where μ = 0 \end{matrix}  (3.9)

and observe that Theorem 3.7 holds for the pair (\overline{μ}, λ) thanks to (3.5) and Remark 3.3.

If the following weighted Poincaré type inequality

∫_{Q} u^{2}μdx ⩽ c ∫_{Q} |Du|^{2}λdx for every u ∈ H_{#}^{1}(Q, μ, λ)  (3.10)

holds for a suitable positive constant c, then the norm

∥u∥_{H_{0}^{1}(Q,μ,λ)} = ∥u∥_{H_{#}^{1}(Q,μ,λ)} := ( ∫_{Q} |Du|^{2}λdx)^{1/2} u ∈ H_{#}^{1}(Q, μ, λ)  (3.11)

is equivalent on H_{#}^{1}(Q, μ, λ) (and on H_{0}^{1}(Q, μ, λ)) to the one of H^{1}(Q, μ, λ). Then, by (3.8) and Theorem 1.4 in [29], if Ω is a bounded open set with Lipschitz boundary and λ ∈ A_{2}, we have that for every μ such that the pair (μ, λ) satisfies hypotheses of Theorem 3.7 we have that

H_{0}^{1}(Ω, μ, λ) = H^{1}(Ω, λ, λ)∩W_{0}^{1,1}(Ω) ,  H_{#}^{1}(Q_{0}, μ, λ) = H^{1}(Q_{0}, λ, λ)∩W_{#}^{1,1}(Q_{0}) . (3.12)

If C(μ, λ, Q_{0}) is finite, then from Theorem 3.7 and Remark 3.8 inequality (3.10) holds. In this case we will endow H_{0}^{1}(Q_{0}, μ, λ) and H_{#}^{1}(Q_{0}, μ, λ) with the norm (3.11).

By (3.5), (3.6) and Theorem 3.7 if Ω is a bounded open set and Q_{Ω} a cube containing Ω with (μ, λ) ∈ A_{2,2}^{α}(Q_{Ω}, K_{2}) with α < 1 (see also Remark 3.3) and λ ∈ A_{2}(K_{1}) there exist σ = σ(n, K_{1}) > 0 (depending only on n, K_{1}) and c_{i} = c_{i}(n, K_{1}, K_{2}, Ω) > 0 (depending only on n, K_{1}, K_{2}, Ω), i = 1, 2, such that

c_{1} ∥λ^{−1}∥_{L^{1}(Q_{Ω})}^{−1/2}∥|Du|∥_{L^{1+σ}(Q_{Ω})} ⩽ ∥|Du|∥_{L^{2}(Q_{Ω},λ)} ⩽ c_{2} ∥λ∥_{L^{1}(Q_{Ω})}^{1/2}∥|Du|∥_{L^{(1+σ)′} (Q_{Ω})} (3.13)

for every u ∈ W_{#}^{1,(1+σ)′}(Q_{Ω}), so that

W_{0}^{1,(1+σ)′}(Ω) ⊂ H_{0}^{1}(Ω, μ, λ) ⊂ W_{0}^{1,1+σ}(Ω), H^{−1}(Ω, μ, λ) ⊂ W^{−1,1+σ}(Ω)
W_{#}^{1,(1+σ)′}(Q_{Ω}) ⊂ H_{#}^{1}(Q_{Ω}, μ, λ) ⊂ W_{#}^{1,1+σ}(Q_{Ω}), H_{#}^{−1}(Q_{Ω}, μ, λ) ⊂ W_{#}^{−1,1+σ}(Q_{Ω}).

Moreover, since H_{0}^{1}(Ω, μ, λ) continuously embeds in W^{1,1}(Ω) which continuously embeds in L^{n/(n−1)}(Ω) if n ⩾ 2 (and in L^{2}(Ω) if n = 1), there exists a positive constant c_{3} = c_{3}(n, Ω) (depending only on n and Ω) such that

∥f∥_{H^{−1}(Ω,μ,λ)} ⩽ c_{3} (_{∫Ω}λ^{−1}dx)^{1/2}∥f∥_{L^{n}(Ω)}.  (3.14)

We present here an approximation result contained in [23] which can be proved extending Theorem 2.9 and Corollary 2.10 in [23] to pairs (μ, λ) in which μ ⩾ 0 (the proofs are the same). We fix ρ a radial non-negative function in C^{∞}(R^{n}), supported in B(0, 1), such that ∫_{R^{n}} ρ = 1, and and consider (ρ_{ϵ})_{ϵ>0} a family of mollifiers, i. e. ρ_{ϵ}(x) = ϵ^{−n}ρ(x/ϵ). We define

v_{ϵ}(x, t) =^{def} ∫_{R^{n}} ρ_{ϵ}(x − y)\overline{v}(y, t)dy (x, t) ∈ R^{n} × [0, T].  (3.15)

if v ∈ L^{2}(0, T; H_{0}^{1}(Ω, μ, λ)).

Proposition 3.9 Let Ω be a bounded open set of R^{n}, T, K_{2} > 0, K_{1} ⩾ 1, α ∈ [0, 1), Q_{Ω} a cube union of dyadic cubes and containing Ω. Then there exists a positive constant c = c(n, ρ, α, K_{1}, K_{2}) (depending only on n, ρ, α, K_{1}, K_{2}) such that:

∫_{0}^{T} ∫_{Q_{Ω}} |v_{ϵ} − v|^{2}μdxdt ⩽ c ϵ^{2(1−α)}∫_{0}^{T} ∫_{Q_{Ω}} |Dv|^{2}λdxdt

(a) either for every ϵ > 0, for every pair of weights μ, λ such that λ ∈ A_{2}(K_{1}) and (μ, λ) ∈ A_{2,2}^{α}(3Q_{Ω}, K_{2}), for every v ∈ L^{2}(0, T; H_{0}^{1}(Ω, μ, λ)),

(b) or for ϵ small enough (i. e. for every ϵ ∈ (0, ϵ_{0}] for a e_{0}), for every pair of weights μ, λ such that λ ∈ A_{2}(K_{1}) and (μ, λ) ∈ A_{2,2}^{α}(9Q_{Ω}, K_{2}), for every v ∈ L^{2}(0, T; H_{#}^{1}(Q_{Ω}, μ, λ).

4 Strongly degenerate parabolic equations

In this section we give a result of existence (and uniqueness when boundary conditions are required) for the solution to an abstract strongly degenerate evolution equation

Ru′ + Au = f ,

where strongly refers to the fact that R ⩾ 0. The result about existence of such a problem are contained in [22] and now we are going to set out some results contained there. We are interested to the situation R ⩾ 0 (and A linear), even if actually in [22] a more general situation is considered, i. e. R is not necessarily non-negative. Moreover here we are also interested in problem (4.6)-(II), while in [22] only problems with initial datum is considered.

Consider a reflexive Banach space V and a Hilbert space H and identify H′ with H in such a way the embeddings

V ⊂ H ⊂ V′

are continuous and dense. Given T > 0 we define the spaces

H = L^{2}(0, T; H), V = L^{2}(0, T; V ), V′ = L^{2}(0, T; V′),  (4.1)

endowed with the standard norms

∥u∥_{H} = (∫_{0}^{T} ∥u(t)∥_{H}^{2}dt)^{1/2}, ∥u∥_{V} = (∫_{0}^{T} ∥u(t)∥_{V}^{2} dt)^{1/2}, ∥u∥_{V′} = (∫_{0}^{T} ∥u(t)∥_{V′}^{2}dt)^{1/2} .

Let R be a non-negative and linear bounded self-adjoint operator, R : H → H. We denote by H_{0} the kernel of R and by H_{+} its orthogonal space. Since R is positive, it admits a square root R^{1/2} which is a positive, linear and bounded self-adjoint operator (see, for instance, [26]). Denote by H_{+}^{1/2} the completion of H_{+} with respect to the topology induced by the norm

∥u∥_{+} := ∥R^{1/2}u∥_{H}

which satisfies

∥u∥_{+} ⩽ ∥R∥^{1/2}∥u∥_{H} .

Observe that, since R^{1/2} is self-adjoint, we have that

(Ru, v)_{H} = (R^{1/2}u, R^{1/2}v)_{H} = (u, v)_{H_{+}} .

Observe moreover that Ker R^{1/2} = Ker R. Denote by P_{+} the orthogonal projection from H onto H_{+} and by R_{+} the operator R|_{H_{+}} : H_{+} → H. Observe that

Ru = RP_{+}u and R^{1/2}u = R^{1/2}P_{+}u .

We extend the operators R, R^{1/2}, P_{+} : H → H to some operators denoted by the same letters R, R^{1/2}, P_{+} : H → H in the following way

Ru(t) := R(u(t)) , R^{1/2}u(t) := R^{1/2}(u(t)) , P_{+}u(t) := P_{+}(u(t)) .

Consider the space

W = {v ∈ V | Rv′ ∈ V′} ,  (4.2)

where Rv′ stands for (Rv)′ and the derivative is to be intended in the sense of ditributions, endowed with the norm

∥u∥_{W} = ∥u∥_{V} + ∥Ru′∥_{V′} .

The first result in [22] is the following.

Proposition 4.1 The operator P_{+} maps W into C([0, T]; H_{+}^{1/2}) and the mapping is continuous, i. e. there is a constant c > 0 such that

∥P_{+}u∥_{C([0,T];H_{+}^{1/2})} = max_{t∈[0,T]} ∥R^{1/2}u(t)∥_{H} ⩽ c ∥u∥_{W} .

Moreover

⟨Ru′, v⟩_{V′×V} + ⟨Rv′, u⟩_{V′×V} = (Ru(T), v(T))_{H} − (Ru(0), v(0))_{H} .

Thanks to this last result, given ψ ∈ H_{+}^{1/2}, we can define the space

W^{ψ} = {u ∈ W | P_{+}u(0) = ψ}  and  W^{per} = {u ∈ W | P_{+}u(0) = P_{+}u(T)}

(W^{0} will be the space with ψ = 0). Consider now a family of linear operators defined on V

A(t) : V ⟶ V′, 0 ⩽ t ⩽ T

such that

\{ \begin{matrix}t ↦ ⟨A(t)u, v⟩_{V′×V}  is measurable on [0, T] \\ ∥u∥_{V}^{2} ⩽ ⟨A(t)u, u⟩_{V′×V} ⩽ L∥u∥_{V}^{2} \\ |⟨A(t)u, v⟩_{V′×V} | ⩽ M⟨A(t)u, u⟩_{V′×V}^{1/2} ⟨A(t)v, v⟩_{V′×V}^{1/2} \end{matrix}  (4.3)

for every u, v ∈ V. Then we can construct the abstract operators

A : V ⟶ V′  where Au (t) =^{def}A(t)u(t)  0 ⩽ t ⩽ T,  (4.4)
P : W ⟶ V′,  (Pu) (t) = Ru′(t) + Au(t)  0 ⩽ t ⩽ T .  (4.5)

Consider the strongly degenerate parabolic (or elliptic-parabolic) problems

(I) \{ \begin{matrix}Ru′ + Au = f \\ P_{+}u(0) = ψ \end{matrix}  (II) \{ \begin{matrix}Ru′ + Au = f \\ P_{+}u(0) = P_{+}u(T) \end{matrix}  (4.6)

where f ∈ V′ and ψ ∈ H_{+}^{1/2}. The solution to these problems is to intended it the following way: u is a solution if, first of all,

u ∈ W and ⟨(Ru)′(t) + Au(t), v⟩_{V′×V} = ⟨f(t), v⟩_{V′×V}  (4.7)

for almost every t ∈ [0, T] and for every v ∈ V; moreover P_{+}u(0) = ψ in H_{+}^{1/2} in the first case, P_{+}u(0) = P_{+}u(T) in H_{+}^{1/2} in the second case. Define the unbounded operators L_{i} : D(L_{i}) ⊂ V → V′, i = 1, 2, where

\begin{matrix}L_{1}v = Rv′ , & D(L_{1}) = \{v ∈ W | P_{+}v(0) = 0\} , \\ L_{2}v = Rv′ , & D(L_{2}) = \{v ∈ W | P_{+}v(0) = P_{+}v(T)\} . \end{matrix}

Proposition 4.2 The operators L_{i} are densely defined closed operators. Moreover L_{i} are maximal monotone.

Proof - One may adact the corresponding proof in [37] (see Proposition 32.10 in vol. II/B). □ Theorem 4.3 For every f ∈ V′ and every ψ ∈ H_{+}^{1/2} the problem (4.6)-(I) admits a unique solution. For every f ∈ V′ the problem (4.6)-(II) admits a unique solution. Moreover

(i) P : W ⟶ V′ is linear and continuous and there exist a constant c = c(L, M) (depending only on L, M) such that

∥Pu∥_{V′} ⩽ c ∥u∥_{W} for every u ∈ W,

(ii) ⟨Pu, u⟩ ⩾ ∥u∥_{V}^{2} for every u ∈ W^{0} and for every u ∈ W^{per},

(iii) for every ψ ∈ H_{+}^{1/2} if u ∈ W^{ψ} is the solution to (4.6)-(I) there exists a constant c = c(M, L) (depending only on M, L), such that

∥u∥_{W} ⩽ c [∥Pu∥_{V′} + ∥R^{1/2}u(0)∥_{H}] ,

(iv) if u ∈ W^{per} is the solution to (4.6)-(II) there exists a constant c = c(M, L) (depending

only on M, L), such that

∥u∥_{W} ⩽ c ∥Pu∥_{V′} .

Proof - For the proof for the problem (4.6)-(I) with ψ = 0 and for the problem (4.6)-(II) one can follow, e. g., the proof of Theorem 32.D in [37] and use Proposition 4.2 since R is maximal monotone and A is bounded, coercive and strictly monotone. For the general problem (4.6)-(I) we refer to [22] (the idea is to consider the equations (R + ϵ(Id_{H} − P_{+}))u′ + Au = f with an initial datum u_{0,ϵ} such that P_{+}u_{0,ϵ} → ψ in H_{+}^{1/2} and then let ϵ go to zero). □

Remark 4.4 - The unbounded operators L^{(i)} : D(L^{(i)}) ⊂ V → V′, i = 1, 2, where

\begin{matrix}L^{(1)}v = −Rv′ , & D(L^{(1)}) = \{v ∈ W | P_{+}v(T) = 0\} , \\ L^{(2)}v = Rv′ , & D(L^{(2)}) = \{v ∈ W | P_{+}v(0) = P_{+}v(T)\} , \end{matrix}

satisfy Proposition 4.2 and then the following problems have an unique solution

\{ \begin{matrix}−Ru′ + A^{∗}u = f \\ P_{+}u(T) = 0 \end{matrix}  \{ \begin{matrix}−Ru′ + A^{∗}u = f \\ P_{+}u(0) = P_{+}u(T) . \end{matrix}

Here we present a regularity result of the solution of problem (4.6)-(I). For similar results for the classical equations see, e. g., [16], [15], [35], [2], [11]. The proof we present follows the one contained in [36]. Before we need to introduce a definition.

Definition 4.5 We say that B : [0, T] → L(X, X′), X Banach space, is regular if, for every u, v ∈ X the function

t ↦ ⟨B(t)u, v⟩_{X′×X}

is absolutely continuos on [0, T] and there exists a function b ∈ L^{1}(0, T) such that

|\frac{d}{dt}⟨B(t)u, v⟩_{X′×X} | ⩽ b(t)∥u∥_{X}∥v∥_{X} for a. e. t ∈ [0, T] .

Observe that B′(t) : X → X′ is linear for almost every t ∈ (0, T).

Remark 4.6 - If B is differentiable in the sense just defined the following fomula holds: if u ∈ W^{1,p}(0, T; X) for any p, then

(B(t)u(t))′ = B′(t)u(t) + B(t)u′(t) for a. e. t ∈ (0, T) .

In the special case B(t) = B for every t, taking v ∈ L^{p}(0, T; X) and u(t) = ∫_{0}^{t} v(s)ds, we

also derive from the formula above that

B ∫_{0}^{t} v(s)ds = ∫_{0}^{t} Bv(s)ds .  (4.8)

About the operator A : [0, T] → L(V, V′) we will make the additional assumption (besides to (4.3)):

\{ \begin{matrix}A is differentiable in the sense of Definition 4.5 for a. e. t ∈ (0, T) \\ t ↦ ⟨A′(t)u, v⟩_{V′×V}  is measurable on [0, T] \\ |⟨A′(t)u, v⟩_{V′×V} | ⩽ M′(t)∥u∥_{V} ∥v∥_{V} \end{matrix}  (4.9)

with M′ ∈ L^{2}(0, T), for almost every t ∈ (0, T) and for every u, v ∈ V (M′ would be sufficient to be in L^{1}(0, T), but we require M′ ∈ L^{2}(0, T) to prove the following result).

Theorem 4.7 Consider the problem (4.6)-(I). Assume the operator A satisfies also (4.9) and f ∈ H^{1}(0, T; V′). Given u_{0} ∈ V, ψ = P_{+}u_{0} and said u the solution of (4.6)-(I) suppose moreover that f(0) − A(0)u_{0} ∈ Im R. Then the solution u satisfies

u ∈ H^{1}(0, T; V) .

Proof - We consider the problem

\{ \begin{matrix}Ru′ + Au = f \\ P_{+}u(0) = ψ \end{matrix}

with ψ ∈ H_{+}^{1/2} (note that P_{+}u(0) = P_{+}u_{0}). From a formal point of view we have that

Ru′(0) = f(0) − A(0)u(0)

and if we derive the equation, again from a formal point of view, we have

Ru′′ + A′u + Au′ = f′ .

Then we consider the following problem

\{ \begin{matrix}Rv′ + Av = f′ − A′u \\ P_{+}v(0) = P_{+}R_{+}^{−1}(f(0) − A(0)u_{0}). \end{matrix}  (4.10)

Define the function

w(t) = u_{0} + ∫_{0}^{t} v(s)ds .

Problem (4.10) has one solution v ∈ W and in particular v ∈ L^{2}(0, T; V) so w ∈ H^{1}(0, T; V). We conclude if we show that w = u. Integrating equation in (4.10), by (4.8), we obtain

Rv(t) − Rv(0) + ∫_{0}^{t}(Av)(s)ds = f(t) − f(0) − ∫_{0}^{t}(A′u)(s)ds .

Observe that Rv(0) = RP_{+}v(0) = P_{+}Rv(0) = P_{+}(f(0)−A(0)u_{0}) = f(0)−A(0)u_{0} and that, thanks to Remark 4.6, (Rw)′ = Rv. Then equation above is, since w′ = v,

Rw′(t) = −A(0)u_{0} + f(t) − ∫_{0}^{t}(Av)(s)ds − ∫_{0}^{t}(A′u)(s)ds .

On the other hand we have

\begin{aligned}∫_{0}^{t}(Av)(s)ds = ∫_{0}^{t}(Aw′)(s)ds & = & Aw(t) − Aw(0) − ∫_{0}^{t}(A′w)(s)ds = \\ & = & Aw(t) − A(0)u_{0} − ∫_{0}^{t}(A′w)(s)ds \end{aligned}

and then, by the two last equalities, we obtain

Rw′(t) = f(t) − Aw(t) + ∫_{0}^{t} (A′w(s) − A′u(s))ds .

Subtracting to this equation Ru′ + Au = f we obtain

\{ \begin{matrix}R(w − u)′(t) + A(t)(w(t) − u(t)) = ∫_{0}^{t} (A′w(s) − A′u(s))ds \\ P_{+}w(0) = P_{+}u(0) . \end{matrix}

This means that the function w − u solves the following problem in (0, T)

\{ \begin{matrix}Ry′(t) + A(t)y(t) = h(s) \\ P_{+}y(0) = 0 \end{matrix}

where h(t) = ∫_{0}^{t} (A′w(s) − A′u(s))ds. By hypotheses on A′ we have that

∥A′(w − u)(s)∥_{V′} ⩽ M′(s) ∥w(s) − u(s)∥_{V}

for almost every s ∈ (0, T) and then

∥h(t)∥_{V′} ⩽ ∫_{0}^{t} ∥A′(w − u)(s)∥_{V′}ds ⩽ ∫_{0}^{t} M′(s)∥w(s) − u(s)∥_{V} ds .

Let T′ be the greatest value less or equal to T such that w(t) = u(t) for a. e. t ∈ [0, T′]. Then h(t) = 0 for t ∈ [0, T′]. Then by Theorem 4.3, (iii), there is constant c = c(L, M) such that

∫_{0}^{T} ∥w(t) − u(t)∥_{V}^{2} dt = ∫_{T′}^{T} ∥w(t) − u(t)∥_{V}^{2} dt
⩽ c ∫_{T′}^{T} ∥h(t)∥_{V′}^{2}dt ⩽ c ∫_{T′}^{T} ( ∫_{0}^{t} M′(s)∥w(s) − u(s)∥_{V} ds)^{2}dt
= c ∫_{T′}^{T} [ ∫_{0}^{t} |M′(s)|^{2}ds ∫_{0}^{t} ∥w(s) − u(s)∥_{V}^{2} ds]dt
⩽ c (T − T′)∥M′∥_{L^{2}(0,T)}^{2}∥w − u∥_{L^{2}(T′,T;V)}^{2}

that is 1 ⩽ c (T − T′)∥M′∥_{L^{2}(0,T)}^{2}. Since every estimate is independent of T, letting T go to T′ we obtain a contradiction and consequently T′ = T. □

Corollary 4.8 In the same hypotheses as in Theorem 4.7 the solution u of (4.6)-(I) satisfies

Ru′′ ∈ L^{2}(0, T; V′)

and moreover there exists a positive constant c depending (only ) on M, L, R^{1/2} such that

∥u∥_{V} + ∥u′∥_{V} + ∥Ru′′∥_{V′} + ∥u∥_{C([0,T];V)} ⩽ c (∥f∥_{V′} + ∥f′∥_{V′} + ∥ψ∥_{H})) .

Proof - The thesis follows from the fact that the function v by which the function w in the proof of the preceeding theorem is defined is the solution to (4.10) and from the estimates of Theorem 4.3. □

The result contained in [17] can be generalized as follows. In the sixth section we do not extend this result to moving spaces as done in [25] or in [23] (see Theorem 6.5), in general an analogous result is not true (see an example in Section 6), even if such a result will turn to be true for the solutions to the problems we will consider (see next section and Theorem 7.1).

Consider two reflexive Banach spaces B_{0}, B_{1}, B a Hilbert space such that B_{0} ⊂ B ⊂ B_{1} with continuous embeddings. Let R : B → B be a non-negative and linear bounded self-adjoint operator and P_{+} the othogonal projection from B into B_{+} = (ImR)^{⊥}.

Lemma 4.9 Suppose B_{0} ⊂ B is compact. For every η > 0 there exists c_{η} > 0 such that for every v ∈ V

∥P_{+}v∥_{B} ⩽ η∥v∥_{B_{0}} + c_{η}∥Rv∥_{B_{1}} .

Using this lemma one can prove the following theorem.

Theorem 4.10 Suppose moreover that B ⊂ B_{1} is compact. Fix two numbers p_{0}, p_{1} ∈ (1, ∞) and T > 0. The space {v ∈ L^{p_{0}}(0, T; B_{0}) | (Rv)′ ∈ L^{p_{1}}(0, T; B_{1})} compactly embeds in L^{p_{0}}(0, T; B_{+}).

Remark 4.11 - If R|_{B_{0}} : B_{0} → B_{0} it is not necessary to require B ⊂ B_{1} compact.

The corrisponing classical results is obtained when

R = Id_{B}, the identity of B

and for the classical result corresponding to Proposition 4.1 we remand to Theorem 3.1, chap. 1 in [19], to Theorem 4.3 we remand to Theorems 4.1 and Theorems 6.1, chap. 3, in [19], to Lemma 4.9 and Theorem 4.10 we remand to Lemma 5.1 and Theorem 5.1 in [17], or, to quote some more recent books, see e. g. [37], chap. 23 or [36], sections 25 - 27.

5 Hypotheses and position of the problem

We denote with Q the cube [0, 1]^{n} and consider a Q-periodic weight μ and a matrix a = [a_{ij}(y, t)]_{i,j=1}^{n} defined in R^{n} × [0, T] whose coefficients a_{ij}(·, t) are Q-periodic functions and such that for a. e. (y, t), (y, s) ∈ Q × [0, T] and for every ξ, η ∈ R^{n}

\begin{matrix}λ(y)|ξ|^{2} ⩽ ∑_{i,j=1}^{n} a_{ij}(y, t)ξ_{i}ξ_{j} ⩽ Lλ(y)|ξ|^{2} \\ | ∑_{i,j=1}^{n} a_{ij}(y, t)ξ_{j}η_{j}| ⩽ M( ∑_{i,j=1}^{n} a_{ij}(y, t)ξ_{i}ξ_{j})^{1/2}( ∑_{i,j=1}^{n} a_{ij}(y, t)η_{i}η_{j})^{1/2} \\ |a(y, t) − a(y, s)| ⩽ Nλ(y)|t − s| \end{matrix}  (5.1)

hold for some constants L, M ⩾ 1, N ⩾ 0 and λ weight in A_{2}. Observe that, even if λ were not periodic, it would be sufficient to consider, for example, λ^{∗} = a_{11}, so that λ ⩽ λ^{∗} ⩽ Lλ, and then consider L^{−1}λ^{∗} instead of λ and L^{2} instead of L. Indeed the space V_{μ,λ} and V_{μ,λ^{∗}} are the same. Therefore we will suppose

μ, a_{ij} and also λ to be Q-periodic.

Moreover, as far as the weights μ and λ are concerned, we will assume either that there exist some constants K_{1} ⩾ 1, K_{2} > 0, 0 ⩽ α < 1 and r > 1 (and of course αr < n) such that

\begin{matrix}λ ∈ A_{2}(K_{1}), \\ (μ^{r}, λ^{r}) ∈ A_{2,2}^{αr}(Q, K_{2}) \end{matrix}  (H.1)

or that there exist some constants K_{1} ⩾ 1, K_{2} > 0 such that

λ ∈ A_{2}(K_{1}),

(μ, λ) ∈ A_{2,2}^{α}(Q, K_{2}),

μ is reverse doubling, i. e. satisfies (3.7).

(H.2)

Observe that

a) both hypotheses (H.1) and (H.2) on the weight μ imply (μ, λ) ∈ A_{2,2}^{1}(Q) (see Remark 3.3 and Remark 3.4) which we require so that Theorem 3.7 holds,

b) α < 1 is needed to prove Theorem 6.6, (ii) and (iii), by Proposition 3.9,

c) moreover for every open and bounded set Ω, denoted by Q_{Ω} a cube containing Ω, there exists K̃_{2} such that (μ^{s}, λ^{s}) ∈ A_{2,2}^{s}(Q_{Ω}, K̃_{2}), s ⩾ 1, (see Remark 3.5 with h = 1).

Finally observe that if you consider the sequences of weights

μ_{h}(x) := μ(hx), λ_{h}(x) := λ(hx)

then the pairs (μ_{h}, λ_{h}) satisfy a), c) and Remark 3.6.

From now on we will denote by Ω a bounded open set with Lipschitz boundary, Q_{Ω} a cube whose vertices are points in Z^{n}, so that Q_{Ω} results to be union of dyadic cubes and μ and λ are also Q_{Ω}-periodic weights (in this way Proposition 3.9 holds). We define

Q_{+} = {x ∈ Q | μ(x) > 0} , Q_{0} = {x ∈ Q | μ(x) = 0} .

and suppose Q_{0} to be a set with Lipschitz boundary, possibly Q_{0} = Q, and Q_{+} = Q\Q_{0}, Q_{+} with Lipschitz boundary. Extending by periodicity μ (and a_{ij} and λ too) to R^{n} we denote, if α = (α_{1}, ...α_{n}) with α_{j} ∈ Z, τ_{α}Q_{+} := {x ∈ R^{n} | x−α ∈ Q_{+}} and τ_{α}Q_{0} := {x ∈ R^{n} | x−α ∈ Q_{0}}, Π_{+} := ∪_{α∈Z^{n}}τ_{α}Q_{+} and Π_{0} := ∪_{α∈Z^{n}}τ_{α}Q_{0}, Ω_{+} := Π_{+} ∩ Ω and Ω_{0} := Π_{0} ∩ Ω. For h ∈ N we define Π_{+,h} := {x ∈ R^{n} | hx ∈ Π_{+}} and Π_{0,h} := {x ∈ R^{n} | hx ∈ Π_{0}}, and finally Ω_{+,h} = Ω ∩ Π_{+,h} and Ω_{0,h} = Ω ∩ Π_{0,h}. If μ_{h}(x) = μ(hx) we have that μ_{h} > 0 a. e. in Π_{+,h} and μ_{h} = 0 a. e. in Π_{0,h}. Then we denote by Ω_{+,h} the set Ω ∩ Q_{+,h} and by Ω_{0,h} the set Ω ∩ Q_{0,h}. We want to formalize the following elliptic-parabolic problem: given f ∈ L^{2}(0, T; L^{n}(Ω)) (we will approximate f by functions in H^{1}(0, T; L^{n}(Ω))) and φ ∈ W_{0}^{1,∞}(Ω) (precisely we will consider bφ ∈ W_{0}^{1,(1+σ)′}(Ω), σ being the constant appearing in (3.13)) consider

\{ \begin{matrix}μ\frac{∂}{∂t}u − div(a · Du) = f & on Ω_{+} × (0, T) \\ u = 0 & on ((∂Ω_{+} ∩ ∂Ω) × (0, T)) \\ u = φ & on (Ω_{+} × \{0\}) \\ −div(a(·, t) · Du) = f(·, t) & on Ω_{0} for a. e. t ∈ [0, T] \\ u = 0 & on ∂Ω_{0} ∩ ∂Ω \end{matrix}  (5.2)

To write this problem in abstract form we define

\overline{μ} = \{ \begin{matrix}μ & where μ > 0 (in Ω_{+}) \\ λ & where μ = 0 (in Ω_{0}) \end{matrix}  (5.3)

and the operator

\begin{aligned}R : & L^{2}(Ω, \overline{μ}) & ⟶ & L^{2}(Ω, \overline{μ}) \\ & u & ↦ & χ_{Ω_{+}}u \end{aligned}  (5.4)

where χ_{Ω_{+}}(x) = 1 if x ∈ Ω_{+}, χ_{Ω_{+}}(x) = 0 if x ∈ Ω_{0}. In this way R is bounded, even if μ is not. Note as in this case, if H denotes L^{2}(Ω, \overline{μ}), the orthogonal space H_{+} of the kernel of R is isomorphic to L^{2}(Ω, μ), H_{+}^{1/2} = H_{+} and the orthogonal projection P_{+} : H → H_{+} coincides with R (see the previous section). Consider the space H_{0}^{1}(Ω, \overline{μ}, λ) and the triplet

H_{0}^{1}(Ω, \overline{μ}, λ) ⊂ L^{2}(Ω, \overline{μ}) ⊂ H^{−1}(Ω, \overline{μ}, λ) .

As a matter of fact (see Theorem 3.7 and Remark 3.8 and also 3.8) we have that

H_{0}^{1}(Ω, \overline{μ}, λ) = H_{0}^{1}(Ω, μ, λ) ,

and then the triplet (even if μ ≡ 0)

H_{0}^{1}(Ω, μ, λ) ⊂ L^{2}(Ω, \overline{μ}) ⊂ H^{−1}(Ω, μ, λ) .  (5.5)

Consider the family of elliptic operators defined as

A(t) = −div (a(·, t) · D) : H_{0}^{1}(Ω, \overline{μ}, λ) → H^{−1}(Ω, \overline{μ}, λ).

We denote by H_{\overline{μ},λ}, V_{μ,λ}, V′_{μ,λ} the abstrac spaces defined in (4.1) corresponding to the triplet considered in (5.5), by W_{μ,λ} the abstract space corresponding to (4.2) and by W_{μ,λ}^{0} = {u ∈ W_{μ,λ} | P_{+}u(0) = Ru(0) = 0}. Then we denote by

A_{a} : V_{μ,λ} → V′_{μ,λ}, P_{μ,a} = \frac{d}{dt} + A : W_{μ,λ} → V′_{μ,λ}

the corresponding abstract operators as done in (4.4). Note as we have, if u ∈ W_{μ,λ}(0, T; Ω) and φ ∈ C_{c}^{∞}(Ω × (0, T)), that

⟨u′, φ⟩_{μ,λ} = ∫_{0}^{T} ∫_{Ω_{+}} u(x, t)\frac{∂φ}{∂t}(x, t)μ(x)dxdt + ∫_{0}^{T} ∫_{Ω_{0}} u(x, t)\frac{∂φ}{∂t}(x, t)λ(x)dxdt ,

but

⟨Ru′, φ⟩_{μ,λ} = (R^{1/2}u, R^{1/2} \frac{∂φ}{∂t})_{H_{\overline{μ},λ}} = ∫_{0}^{T} ∫_{Ω_{+}} u(x, t)\frac{∂φ}{∂t}(x, t)μ(x)dxdt .

The abstract problem corresponding to (5.2) is then

\{ \begin{matrix}Rv′ + A_{a}v = f \\ v ∈ W_{μ,λ}^{P_{+}φ} \end{matrix}

with f ∈ V′_{μ,λ}. The analogous abstract formulation can be made for the problem (4.6)-(II).

Definition 5.1 Let L, M, ⩾ 1, N, T > 0, Ω, a, μ, λ and f as above. We call solution of the Cauchy-Dirichlet problem (5.2) a function u ∈ W_{μ,λ}^{0} such that

(P_{μ,a}u)(t) = (Ru)′(t) + (A_{a}u)(t) = f(t) for a. e. t ∈ [0, T].

Analogously we define the solution of the same equation with periodic conditions instead of Cauchy type ones.

Remark 5.2 - Given a matrix a satisfying (5.1) with μ and λ satisfying (H.1) or (H.2), by Theorem 3.7 and Theorem 4.3 we obtain that for every f ∈ Vμ,λ^{′} there exists a unique solution of the problem (5.2), or the analogous with periodic conditions. Moreover observe that A(t) = −div(a · D) is differentiable in the sense of the Definition 4.5 and

A′(t) = −div(a′ · D)

where a′ denotes the matrix whose entries are ∂_{t}a_{ij}(x, t) (the coefficients a_{ij}(x, t) for almost every x ∈ Ω are differentiable almost everywhere in t).

Now we consider the sequences

μ_{h}(x) := μ(hx), λ_{h}(x) := λ(hx), a_{ij,h}(x, t) := a_{ij}(hx, t), h ∈ N .  (5.6)

Then a_{h} satisfies (5.1) with the weight λ_{h} and with the same constants L, M. Moreover there exists a constant depending on n and K_{2}, which for simplicity we still denote with K_{2}, such that if μ and λ satisfy hypotheses (H.1) (see Remark 3.5) we have that

λ_{h} ∈ A_{2}(K_{1}), (μ_{h}^{r}, λ_{h}^{r}) ∈ A_{2,2}^{r}(Q, K_{2}),

if μ and λ satisfy hypotheses (H.2) we have that (see Remarks 3.5 and 3.6)

λ_{h} ∈ A_{2}(K_{1}), (μ_{h}, λ_{h}) ∈ A_{2,2}^{1}(Q, K_{2}), μ_{h} are equi − reverse doubling.

We choose two data f and φ in this way (see (3.13)):

f ∈ L^{2}(0, T; L^{n}(Ω)) , φ ∈ W_{0}^{1,(1+σ)′}(Ω)

spaces respectively dense in V′_{h} and H_{0}^{1}(Ω, μ_{h}, λ_{h}) for every h ∈ N. Once denoted by Ω_{+,h} the set {x ∈ Ω | μ_{h} > 0} and by Ω_{0,h} the set {x ∈ Ω | μ_{h} = 0}, we will study the asymptotic behaviour of the solutions u_{h} when h → ∞ of the following sequence of degenerate ellipticparabolic problems

\{ \begin{matrix}μ_{h} \frac{∂}{∂t}u − div(a_{h} · Du) = f & on Ω_{+,h} × (0, T) \\ u = 0 & on ((∂Ω_{+,h} ∩ ∂Ω) × (0, T)) \\ u = φ & on (Ω_{+,h} × \{0\}) \\ −div(a_{h}(·, t) · Du) = f(·, t) & on Ω_{0,h} for a. e. t ∈ [0, T] \\ u = 0 & on ∂Ω_{0,h} ∩ ∂Ω \end{matrix}  (5.7)

where the triplets are defined analogously to (5.5).

Notations: for every h ∈ N we will simply use the subindex h to denote the weighted abstract spaces related to the weights μ_{h} and λ_{h} defined in (5.6), V_{h} to denote the space H_{0}^{1}(Ω, μ_{h}, λ_{h}), V_{h} and W_{h} the correponding abstract spaces, A_{h}(t) the operators -div(a_{h}(·, t)D) : V_{h} → V′_{h}, A_{h} : V_{h} → V′_{h} the operator defined as A_{h}u(t) = A_{h}(t)u(t). Denote R_{h} (and P_{+,h} which in this case coincides with it) the operator defined by R_{h}v(x) := χ_{Ω_{+,h}}(x)v(x) and P_{h} : W_{h} → V′_{h} will be the operator defined as P_{h}u = R_{h}u′ + A_{h}u. Then the problems (5.7) written in abstract form become

find v ∈ W_{μ_{h},λ_{h}}^{φ_{h}} such that \{ \begin{matrix}R_{h}v′ + A_{h}v = f \\ P_{+,h}v = P_{+,h}φ =: φ_{h} \end{matrix}  (5.8)

We will denote

\begin{matrix}V := L^{2}(0, T; H_{0}^{1}(Ω)), V′ := L^{2}(0, T; H^{−1}(Ω)) \\ W := \{u ∈ V | u′ ∈ V′\}, ⟨f, u⟩ := ⟨f, u⟩_{V′×V} \end{matrix}

The symbols

⟨μ⟩, ⟨λ⟩, ⟨λ^{−1}⟩, ⟨a_{ij}(t)⟩

will denote respectively the mean value of μ, λ, λ^{−1}, a_{ij}(·, t) on Q.

6 Compactness type results in weighted Sobolev spaces

The results stated from now on hold with the hypotheses and the notations of the preceeding section. Therefore we will consider the constants r, K_{1}, K_{2}, α, the open set Ω and the cube Q_{Ω} fixed. We first recall the following proposition (see Corollary 4.7 in [23]) whose proof does not depend on the fact that here the weight μ can also be equal to zero in a non zero measure set, while in [23] the weight was positive almost everywhere (the only thing we need is a Poincaré inequality).

Proposition 6.1 Consider two sequences of functions v_{h}, w_{h} ∈ L^{2}(0, T; H_{#}^{1}(Q_{Ω}, μ_{h}, λ_{h})), h ∈ N. Suppose there exist a positive constant c such that

∫_{0}^{T} ∫_{Q_{Ω}} |Dv_{h}|^{2}λ_{h}dxdt ⩽ c, ∫_{0}^{T} ∫_{Q_{Ω}} |Dw_{h}|^{2}λ_{h}dxdt ⩽ c

and two function v, w (which will be in L^{2}(0, T; H^{1}(Q_{Ω}))) such that

v_{h} → v and w_{h} → w in L^{2}(0, T; L^{1}(Q_{Ω})).

Then, up to subsequences, we have that for every φ ∈ L^{∞}(Q_{Ω} × (0, T))

∫_{0}^{T} ∫_{Q_{Ω}} v_{h}w_{h}φμ_{h}dxdt → ⟨μ⟩∫_{0}^{T} ∫_{Q_{Ω}} vwφdxdt .

Remark 6.2 - In the preceeding proposition it is not possible to weaken hypotheses in the following way (even if μ_{h} > 0):

∫_{0}^{T} ∫_{Q_{Ω}} |Dv_{h}|^{2}λ_{h}dxdt ⩽ c, ∫_{0}^{T} ∫_{Q_{Ω}} w_{h}^{2}μ_{h}dxdt ⩽ c .

Call w the weak limit of w_{h} in L^{2}(0, T; L^{1}(Q_{Ω}))-weak and suppose (μ_{h})_{h} bounded in L^{r}(Q_{Ω}) with r > 1 (so we suppose μ_{h} → μ in L^{1}(Q_{Ω})-weak). One might expect to have, up to a subsequence, the following result

∫_{0}^{T} ∫_{Q_{Ω}} v_{h}w_{h}φμ_{h}dxdt → ⟨μ⟩∫_{0}^{T} ∫_{Q_{Ω}} vwφdxdt .

for every φ ∈ L^{∞}(Q_{Ω} × (0, T)), while instead we have

∫_{0}^{T} ∫_{Q_{Ω}} v_{h}w_{h}φμ_{h}dxdt → ⟨μ⟩∫_{0}^{T} ∫_{Q_{Ω}} vzφdxdt .

where z is such that w_{h}μ_{h} → z⟨μ⟩ in L^{2}(0, T; L^{1}(Q_{Ω}))-weak. Indeed if μ ∈ L_{loc}^{r}(Q_{Ω}) and u ∈ L^{2}(Ω, μ) for any r > 1 we have that

∫_{Ω} |uμ|^{p}dx ⩽ [ ∫ u^{2}μdx]^{1/2}[ ∫ μ^{p/(2−p)}dx]^{1/2}

and then there exists p ∈ (1, 2) (p = 2r/(1 + r)) such that w_{h}μ_{h} equibounded in L^{p}(Q_{Ω}). The following example shows that z may be different from w: consider w_{h}(x) = sin(hx) in (0, 2π) and μ_{h}(x) = sin(hx) + 2. Then w_{h} → 0 and μ_{h} → 2, but w_{h}μ_{h} → zμ = π.

Remark 6.3 - The space L^{2}(0, T; H_{#}^{1}(Q_{Ω}, μ_{h}, λ_{h})) contains both L^{2}(0, T; H_{0}^{1}(Ω, μ_{h}, λ_{h})) and L^{2}(0, T; H_{#}^{1}(Q, μ_{h}, λ_{h})). Moreover observe that choosing in Proposition 6.1 first w_{h} = v_{h} and φ ≡ 1, then w_{h} ≡ 1, we obtain

(i) ∫_{0}^{T} ∫_{Q_{Ω}} u_{h}^{2}μ_{h}dxdt → ⟨μ⟩∫_{0}^{T} ∫_{Q_{Ω}} u^{2}dxdt;

(ii) ∫_{0}^{T} ∫_{Q_{Ω}} u_{h}φμ_{h}dxdt → ⟨μ⟩∫_{0}^{T} ∫_{Q_{Ω}} uφdxdt for every φ ∈ L^{∞}(Q_{Ω} × (0, T)).

In the fifth section we gave an extension (Theorem 4.10) of the following classical result:

Theorem 6.4 Consider three reflexive Banach spaces B_{0}, B, B_{1} such that B_{0} ⊂ B ⊂ B_{1}. Suppose that the embedding B_{0} ↪ B is compact. Then even the embedding of {u ∈ L^{2}(0, T; B_{0}) | u′ ∈ L^{2}(0, T; B_{1})} endowed with the norm ∥u∥^{2} = ∥u∥_{L^{2}(0,T;B_{0})}^{2}+∥u′∥_{L^{2}(0,T;B_{1})}^{2} in L^{2}(0, T; B) is compact.

In [23] the following extension to moving spaces is given.

Theorem 6.5 Consider Ω an open bounded set of R^{n}, T, K_{1}, K_{2}, r positive constants with K_{1} ⩾ 1, r > 1. Consider the sequences (μ_{h})_{h} and (λ_{h})_{h} defined in (5.6) and a sequence v_{h} ∈ {w ∈ V_{h} | w′ ∈ V′_{h}} for which there exists a constant c > 0 such that

∥v_{h}∥_{V_{h}} + ∥v′_{h}∥_{V′_{h}} ⩽ c for every h ∈ N.

Then there exist p > 1, p depending (only) on n, K_{1}, and a function v ∈ L^{2}(0, T; L^{2}(Ω)) such that, up to a subsequence,

v_{h} → v in L^{2}(0, T; L^{p}(Ω)) .

In the case of strongly degenerate parabolic equations a similar compactness result does not hold in general, as the following example shows. Consider the functions

u_{1}(x) = \{ \begin{matrix}0 & if x ∈ [−2, −1] \\ x + 1 & if x ∈ [−1, 0] \\ −x + 1 & if x ∈ [0, 1] \\ 0 & if x ∈ [1, 2] \end{matrix} u_{2}(x) = \{ \begin{matrix}0 & if x ∈ [−2, −1] \\ −x − 1 & if x ∈ [−1, 0] \\ x − 1 & if x ∈ [0, 1] \\ 0 & if x ∈ [1, 2] \end{matrix}

define \overline{u} : [−1, 1] × [0, 1] → R,

\overline{u}(x, t) = \{ \begin{matrix}u_{1}(x) & if t ∈ [0, 1/2] \\ u_{2}(x) & if t ∈ (1/2, 1] \end{matrix}

and extend it by periodicity (in time). Then consider R : L^{2}(−2, 2) → L^{2}(−2, 2) defined by Rv := χ_{(−2,−1)∪(1,2)}(x)v(x). We have that \overline{u} ∈ \{v ∈ L^{2}(0, 1; H_{0}^{1}(−2, 2)) | (Rv)′ ∈ L^{2}(0, 1; H^{−1}(−2, 2))\} and the sequence u_{h}(x, t) := \overline{u}(x, ht) satisfies ∥u_{h}∥_{L^{2}(0,1;H_{0}^{1}(−2,2))} + ∥(Ru_{h})′∥_{L^{2}(0,1;H^{−1}(−2,2))} ⩽ c, but u_{h} does not strongly converge in L^{2}(0, 1; L^{1}(−2, 2)).

This example shows that is not possible to obtain a general compactness result for a sequence v_{h} ∈ W_{μ_{h},λ_{h}} with ∥v_{h}∥_{W_{μ_{h},λ_{h}}} ⩽ c, i. e. generalizing Theorem 4.10 to moving spaces to obtain a result analogous to Theorem 6.5, but this can be obtained if v_{h} are solutions of (5.8), at least when the datum f is regular enough.

Theorem 6.6 Consider Ω an open bounded set of R^{n}, T, K_{1}, K_{2}, r positive constants with K_{1} ⩾ 1, r > 1, α ∈ [0, 1). Consider the sequences (μ_{h})_{h} and (λ_{h})_{h} defined in (5.6) (satisfying or (H.1) either (H.2)) and the sequence u_{h} ∈ {w ∈ V_{h} | w′ ∈ V′_{h}} of solutions to (5.8). Then there exists a constant c > 0 such that

∥u_{h}∥_{W_{h}} = ∥u_{h}∥_{V_{h}} + ∥R_{h}u′_{h}∥_{V′_{h}} ⩽ c for every h ∈ N.

Suppose moreover that the datum f in (5.8) belongs to H^{1}(0, T; L^{n}(Ω)). Then there exists a function u ∈ H^{1}(0, T; H_{0}^{1}(Ω)) such that, up to subsequences,

u_{h} → u in C([0, T]; L^{1}(Ω)) .

and

U_{h}(t) = _{∫Ω}u_{h}^{2}(x, t)μ_{h}(x)dx → U(t) = ⟨μ⟩_{∫Ω}u^{2}(x, t)dx in C([0, T]).

Proof - The solutions are bounded, that is there exists a constant such that ∥u_{h}∥_{W_{h}} ⩽ c, from Theorem 4.3, (iii). Consider the problems (5.7)-(5.8) with the data f ∈ H^{1}(0, T; L^{n}(Ω)) and φ ∈ C_{c}^{1}(Ω). We define a sequence of functions ũ_{0,h} ∈ H_{0}^{1}(Ω_{h}, μ_{h}, λ_{h}) as the solutions to the elliptic problems

\{ \begin{matrix}A_{h}(0)u = f(0) & on Ω_{0,h} \\ u = φ & on ∂Ω_{0,h} , \end{matrix}

and than we extend them to the whole Ω in this way

u_{0,h} := \{ \begin{matrix}ũ_{0,h} & in Ω_{0,h} \\ φ & in Ω_{+,h} . \end{matrix}  (6.1)

We can look at the problems

\{ \begin{matrix}R_{h}v′ + A_{h}v = f \\ P_{+,h}v = P_{+,h}φ =: φ_{h} , \end{matrix}

in the following way, thinking the initial datum as the restriction of u_{0,h}:

\{ \begin{matrix}R_{h}v′ + A_{h}v = f \\ P_{+,h}v = P_{+,h}u_{0,h} = P_{+,h}φ =: φ_{h} . \end{matrix}

Then we have that f(0) − A_{h}(0)u_{0,h} ∈ Im R_{h} and the hypotheses of Theorem 4.7 and Corollary 4.8 are satisfied. Then in particular there exists a constant c > 0 (depending only on M, L, f, φ) such that

∥u_{h}∥_{V_{h}} + ∥u′_{h}∥_{V_{h}} ⩽ c .

In particular, by Corollary 4.8 and (3.13), we have that (u_{h})_{h} is bounded in H^{1}(0, T; W_{0}^{1,1+σ}(Ω)) and this space compactly embeds in L^{2}(0, T; L^{1}(Ω)) (by Theorem 6.4). But we have more: indeed, since (see, e. g., [11]) for a function w ∈ W^{1,p}(0, T; B) with B Banach space

w(t) − w(s) = ∫_{s}^{t} w′(τ)dτ

we have

∥u_{h}(t) − u_{h}(s)∥_{L^{1}(Ω)} ⩽ ∫_{s}^{t} ∥u′_{h}(τ)∥_{L^{1}(Ω)}dτ ⩽ |t − s|^{1/2}∥u′_{h}∥_{L^{2}(0,T;W^{1,1}(Ω))} ,

so (u_{h})_{h} is equicontinuous valued in L^{1}(Ω), and moreover the sets {u_{h}(t) | h ∈ N} are relatively compact in L^{1}(Ω) for almost every t ∈ [0, T]. Then by Lemma 1 in [30] we obtain (u_{h})_{h} relatively compact in C([0, T]; L^{1}(Ω)). Finally, u, u′ ∈ L^{2}([0, T]; H_{0}^{1}(Ω)) from Proposition 3.8, (ii), in [25] (thanks to 3.12). Now consider the functions U_{h}. Integrating the abstract equation

R_{h}u′_{h} + A_{h}u_{h} = f

in (t_{1}, t_{2}) ⊂ (0, T) we have that

U_{h}(t_{2}) − U_{h}(t_{1}) = 2 ∫_{t_{1}}^{t_{2}} [ −_{∫Ω}(a_{h}Du_{h}, Du_{h})dxdt +_{∫Ω}fu_{h}dx]dt

and then

|U_{h}(t_{2}) − U_{h}(t_{1})| ⩽ ∫_{t_{1}}^{t_{2}} [2L max_{t∈[0,T]}∥u_{h}(t)∥_{V_{h}} + max_{t∈[0,T]}∥f(t)∥_{L^{n}(Ω)} max_{t∈[0,T]}∥u_{h}(t)∥_{W^{1,1}(Ω)}]dt .

From continuity of the integral we have that (U_{h})_{h} is relatively compact in C([0, T]) and since by Remark 6.3 U_{h} → U almost everywhere in [0, T] we conclude. □

We recall this lemma, useful in the next section for homogenization and also to prove the theorem which follows, a result of compensated compactness (for some classical result about compensated compactness see, e. g., [20], [34], [33] ,[32], [31]).

Lemma 6.7 Consider two sequences w_{h} ∈ L^{2}(0, T; H_{#}^{1}(Q_{Ω}, μ_{h}, λ_{h})) and A_{h} ∈ L^{2}(0, T; (L^{2}(Q_{Ω}, λ_{h}^{−1}))^{n}), h ∈ N, for which there exist c > 0, w ∈ L^{2}(0, T; L^{1}(Q_{Ω})) and A ∈ L^{2}(0, T; (L^{1}(Q_{Ω}))^{n}) such that

∥A_{h}∥_{L^{2}(0,T;(L^{2}(Q_{Ω},λ_{h}^{−1}))^{n})} ⩽ c and A_{h} → A in L^{2}(0, T; (L^{1}(Q_{Ω}))^{n}) − weak,
∫_{0}^{T} ∫_{Q_{Ω}} |Dw_{h}|^{2}λ_{h}dxdt ⩽ c and w_{h} → w in L^{2}(0, T; L^{1}(Q_{Ω})).

Then for every φ ∈ C_{0}^{∞}(Q_{Ω} × (0, T)) we have

∫_{0}^{T} ∫_{Q_{Ω}}(A_{h}, Dφ)w_{h}dxdt → ∫_{0}^{T} ∫_{Q_{Ω}}(A, Dφ)w dxdt.

Theorem 6.8 Consider a sequence of functions v_{h} ∈ W_{h} (h = 1, 2, ...), a function v ∈ W, a constant c_{1} such that

∥v_{h}∥_{W_{h}} ⩽ c_{1} for every h, v_{h} → v in L^{2}(0, T; L^{1}(Ω)),  (6.2)

and a sequence of vectorial functions a_{h} ∈ L^{2}(0, T; (L^{2}(Ω, λ_{h}^{−1}))^{n}) (h = 1, 2, ...), a ∈ L^{2}(0, T; (L^{2}(Ω))^{n}), a constant c_{2} such that

∥a_{h}∥_{L^{2}(0,T;(L^{2}(Ω,λ_{h}^{−1}))^{n})} ⩽ c_{2}  a_{h} → a in L^{2}(0, T; (L^{1}(Ω))^{n}) − weak.  (6.3)

Moreover suppose that

u′_{h} − div (a_{h}) = f ∈ L^{2}(0, T; L^{n}(Ω)) in C_{c}^{1}(Ω × (0, T)).  (6.4)

Then

(a_{h}, Du_{h}) → (a, Du) in D′(Ω × (0, T)).

7 Homogenization

Consider a vector ξ ∈ R^{n}, the function ψ_{ξ} solution to

−div_{y}(a(y, t)D_{y}ψ_{ξ}) = −div_{y}(a(y, t) · ξ) on Q, for a. e. t ∈ (0, T)
ψ_{ξ} ∈ L^{2}(0, T; H_{#}^{1}(Q, μ, λ)) ,

the matrix and the weight defined respectively

â(t) · ξ = ⟨a(y, t) · ξ − a(y, t) · Dψ_{ξ}⟩_{y} ,  μ̂ = ⟨μ⟩  (7.1)

and the Cauchy-Dirichlet problems

\{ \begin{matrix}μ̂\frac{∂v}{∂t} − ∑_{i,j=1}^{n}â_{ij}(t)\frac{∂^{2}v}{∂x_{i}∂x_{j}} = f & on Ω × (0, T) \\ v = 0 & on ∂Ω × (0, T) \\ v = φ & on Ω × \{0\} \end{matrix}  (7.2)

with f ∈ L^{2}(0, T; L^{n}(Ω)) and φ ∈ W_{0}^{1,(1+σ)′}(Ω), σ being the positive constant appearing in (3.13).

Theorem 7.1 For every pair of weights μ and λ verifying (H.1) or (H.2) (see Section 3), and for every f ∈ L^{2}(0, T; L^{n}(Ω)) and φ ∈ W_{0}^{1,(1+σ)′}(Ω), the solutions u_{h} of the problems (5.7) (see Definition 5.1) satisfy

\begin{matrix}u_{h} → u & in L^{2}(0, T; L^{1}(Ω)) \\ a_{h} · Du_{h} → â · Du & in L^{2}(0, T; (L^{1}(Ω))^{n})−weak , \end{matrix}

where â is the matrix defined in (7.1) and u is the solution of (7.2).

Remark 7.2 - Note that if hypothesis (H.1) or (H.2) with α = 0 are satisfied it is possible to choose the datum φ ∈ L^{p}(Ω) for some p ∈ (1, ∞). This is possible estimating as in (7.3) using Theorem 4.3, (iii), since μ ∈ L^{r}(Ω) with r > 1 (if (H.1) holds it is straight forward, otherwise it follows from example d) in Section 8).

Before proving this theorem we need a result (see also [21]). Define p the polynomial p(y) = ξ_{1}y_{1} + ... + ξ_{n}y_{n} and consider ψ_{h} (h ∈ N) and ψ the solutions to the two problems

\{ \begin{matrix}[−\frac{1}{h^{α}} μ(y)\frac{∂}{∂t} − div (a^{∗}(y, t) · D)]v = −div (a^{∗}(y, t) · Dp) \\ v ∈ W_{μ,λ,#}^{T}(0, T; Q) , \end{matrix}

\{ \begin{matrix}−div (a^{∗}(y, t) · Dv) = −div (a^{∗}(y, t) · Dp) \\ v ∈ L^{2}(0, T; H_{#}^{1}(Q, μ, λ)) , \end{matrix}

where α > 0 and where W_{μ,λ,#}^{T}(0, T; Q) = {u ∈ W_{μ,λ,#}(0, T; Q) | P_{+}u(T) = 0} (see also Remark 4.4).

Lemma 7.3 The functions ψ_{h} converge to ψ in L^{2}(0, T; H_{#}^{1}(Q, μ, λ)).

Proof - It is sufficient to choose v_{h} ∈ C^{1}(0, T; H_{#}^{1}(Q, μ, λ)), v_{h}(T) = 0, bounded in W and such that ∥v_{h} − ψ∥ = ∥v_{h} − ψ∥_{L^{2}(0,T;H_{#}^{1}(Q,μ,λ))} → 0. Then

∥ψ_{h} − v_{h}∥^{2} ⩽ ⟨A^{∗}(ψ_{h} − v_{h}), ψ_{h} − v_{h}⟩
⩽ ⟨ − h^{−α}(ψ_{h} − v_{h})′, ψ_{h} − v_{h}⟩ + ⟨A^{∗}(ψ_{h} − v_{h}), ψ_{h} − v_{h}⟩
= ⟨h^{−α}v′_{h} − A^{∗}v_{h} + A^{∗}p, ψ_{h} − v_{h}⟩
= ⟨h^{−α}v′_{h} − A^{∗}v_{h} + A^{∗}ψ, ψ_{h} − v_{h}⟩
⩽ [h^{−α}∥v_{h}∥_{V′} + ∥A^{∗}(v_{h} − ψ)∥_{V′}]∥ψ_{h} − v_{h}∥

□

Proof of Theorem 7.1 - First consider problem (5.7) with a datum

f ∈ H^{1}(0, T; L^{n}(Ω))

and denote by u_{h} its solution (h ∈ N). We define by the functions ψ_{h} of the lemma above

w_{h}(x, t) := p(x) − \frac{1}{h}ψ_{h}(hx, t) .

One can verify that

−w′_{h} + A_{h}^{∗}w_{h} = 0 in D′(R^{n+1}) and w_{h} → p in L^{2}(0, T; L^{1+σ}(Q)) .

Now fix ϕ ∈ C_{c}^{∞}(Ω×(0, T)) and subtract ⟨−w′_{h} +A_{h}^{∗}w_{h}, ϕu_{h}⟩ to ⟨u′_{h} +A_{h}u_{h}, ϕw_{h}⟩ to obtain

⟨u′_{h} + A_{h}u_{h}, ϕw_{h}⟩ − ⟨−w′_{h} + A_{h}^{∗}w_{h}, ϕu_{h}⟩ = ⟨f, ϕw_{h}⟩ .

Obviously

∫_{0}^{T}_{∫Ω}fϕw_{h}dxdt → ∫_{0}^{T}_{∫Ω}fϕpdxdt .

Since w_{h} are bounded in V = L^{2}(H_{#}^{1}), by Proposition 6.1 and Theorem 6.6 we have that the terms involving the temporal derivative, up to a subsequence, satisfy

−∫_{0}^{T}_{∫Ω}u_{h}w_{h} \frac{∂ϕ}{∂t}μ_{h}dxdt → −⟨μ⟩∫_{0}^{T}_{∫Ω}up\frac{∂ϕ}{∂t}dxdt .

Since a_{h} · Du_{h} and a_{h}^{t} · Dw_{h} are bounded in L^{2}(0, T; L^{2}(Ω, λ_{h}^{−1})), thanks to (3.13) we can suppose the existence of two vectors η, θ ∈ R^{n} such that, up to a subsequence,

a_{h} · Du_{h} → η, a_{h}^{t} · Dw_{h} → θ in L^{2}(0, T; L^{1+σ}(Ω)^{n}) .

Now using Lemma 6.7, taking the limit we obtain

∫_{0}^{T}_{∫Ω}[ − ⟨μ⟩up\frac{∂ϕ}{∂t}dxdt + (ξ, Dϕ)pdxdt − (η, Dϕ)u]dxdt = ∫_{0}^{T}_{∫Ω}fpϕ dxdt

and

u′ − div ξ = f .

Now multiplying this last equality by ϕp and proceeding as in the classical homogenization (see, e. g., [3]) and using Lemma 7.3 and the results of the previous section one obtain that the limit function u ∈ L^{2}(0, T; H_{0}^{1}(Ω)) with u′ ∈ L^{2}(0, T; H^{−1}(Ω)) and satisfies the limit equation

μ̂\frac{∂u}{∂t} − ∑_{i,j=1}^{n}â_{ij}(t)\frac{∂^{2}u}{∂x_{i}∂x_{j}} = f

if f ∈ H^{1}(0, T; L^{n}(Ω)). Now we prove that the initial datum is φ if f ∈ H^{1}(0, T; L^{n}(Ω)) and φ ∈ W_{0}^{1,(1+σ)′}(Ω). Now we define a sequence of functions ũ_{0,h} ∈ H_{0}^{1}(Ω_{h}, μ_{h}, λ_{h}) as the solutions to the elliptic problems

\{ \begin{matrix}A_{h}(0)u = f(0) & on Ω_{0,h} \\ u = φ & on ∂Ω_{0,h} , \end{matrix}

and consider u_{0,h} as in (6.1). We have that ∥ũ_{0,h}∥_{H_{0}^{1}(Ω_{h},μ_{h},λ_{h})} ⩽ c with c depending (only) on L, M, f, φ. Then the sequence ∥u_{0,h}∥_{H_{0}^{1}(Ω,μ_{h},λ_{h})} turns to be bounded, then (u_{0,h})_{h} turns to be weakly compact in W^{1,1}(Ω) and then, up to a subsequence, strongly convergent to a function u_{0} in L^{1}(Ω). Therefore we gave that, up to a subsequence, u_{0,h} strongly conveges to a function u_{0} in L^{1}(Ω). Moreover we have, if χ_{h}(x) = 1 for x ∈ Ω_{+,h} and χ_{h}(x) = 0 for x ∈ Ω_{0,h},

u_{0,h} = χ_{h}u_{0,h} + (1 − χ_{h})u_{0,h} = χ_{h}φ + (1 − χ_{h})u_{0,h} → \frac{|Q_{+}|}{|Q|}φ + \frac{|Q_{0}|}{|Q|}u_{0}

where the convergence is in L^{1}(Ω)-weak. Then u_{0} = |Q_{+}|φ + |Q_{0}|u_{0} implies that

u_{0} = φ ,

i. e. the initial datum for the limit problem is the function φ. Then we have proved the theorem for f ∈ H^{1}(0, T; L^{n}(Ω)). Now we show the convergence of the solutions and of the coerresponding momenta even for f ∈ L^{2}(0, T; L^{n}(Ω)). Suppose f ∈ L^{2}(0, T; L^{n}(Ω)) and, for ϵ > 0, fix g ∈ H^{1}(0, T; L^{n}(Ω)) such that ∥f−g∥_{L^{2}(0,T;L^{n}(Ω))} < ϵ. We will denote by u_{h}(f) and u_{h}(g) the solutions to (5.7) and by u(f) and u(g) the solutions to (7.2) with respectively data f, φ and g, φ. By Theorem 4.3, point (ii), we have that

∥u_{h}(f) − u_{h}(g)∥_{V_{h}} ⩽ ∥f − g∥_{V′_{h}} .

By (3.14) and (3.13) we have that

∥u_{h}(f) − u_{h}(g)∥_{L^{2}(0,T;L^{1}(Ω))} ⩽ c ∥f − g∥_{L^{2}(0,T;L^{n}(Ω))} ⩽ c ϵ

and analogously ∥u(f) − u(g)∥_{L^{2}(0,T;L^{1}(Ω))} ⩽ c ϵ. Then

\begin{aligned}∥u_{h}(f) − u(f)∥_{L^{2}(0,T;L^{1}(Ω))} & ⩽ & ∥u_{h}(f) − u_{h}(g)∥_{L^{2}(0,T;L^{1}(Ω))} + \\ & & +∥u_{h}(g) − u(g)∥_{L^{2}(0,T;L^{1}(Ω))} + \\ & & +∥u(g) − u(f)∥_{L^{2}(0,T;L^{1}(Ω))} . \end{aligned}  (7.3)

Taking the limit and since for g ∈ H^{1}(0, T; L^{n}(Ω)) we have u_{h}(g) → u(g) in L^{2}(0, T; L^{1}(Ω)), we conclude lim sup_{h} ∥u_{h}(f)−u(f)∥_{L^{2}(0,T;L^{1}(Ω))} ⩽ 2 c ϵ. As regards the momenta for a fixed Φ ∈ (C_{c}^{∞}(Ω × (0, T)))^{n} we write

(a_{h} · Du_{h}(f), Φ) − (a · Du(f), Φ) = [(a_{h} · Du_{h}(f), Φ) − (a_{h} · Du_{h}(g), Φ)]+
+[(a_{h} · Du_{h}(g), Φ) − (a · Du(g), Φ)] + [(a · Du(g), Φ) − (a · Du(f), Φ)] .

As before the central term is going to zero since g ∈ X. We estimate, for example, the first term, the last being similar. By (5.1) we have

|∫_{0}^{T}_{∫Ω}(a_{h} · Du_{h}(g), Φ) − (a_{h} · Du_{h}(f), Φ)| ⩽ ML∥|DΦ|∥_{∞}[T_{∫Ω}λ_{h}dx]^{1/2}∥u_{h}(g) − u_{h}(f)∥_{V_{h}}

and this quantity can be estimated as before to find that it is O(ϵ). Now take a sequence (g_{h})_{h} ⊂ H^{1}(0, T; L^{n}(Ω)) such that

g_{h} → f in L^{2}(0, T; L^{n}(Ω))  and  g_{h}(0) compact in L^{n}(Ω) .

Arguing as before we obtain that the functions

u_{0,h} := \{ \begin{matrix}ũ_{0,h} & in Ω_{0,h} \\ φ & in Ω_{+,h} . \end{matrix} where \{ \begin{matrix}A_{h}(0)ũ_{0,h} = g_{h}(0) & on Ω_{0,h} \\ u = φ & on ∂Ω_{0,h} , \end{matrix}

converge to φ. □

8 Some examples and comparison with the classical result

In this last section we want to observe how the limit matrix â(t) in (7.1) is independent of the weight μ, that a similar result is not true if we consider homogenization also in time (i. e. a_{h}(x, t) = a(h^{α}x, h^{β}t)) and we want to show some possible choices of μ. By â(t) we will denote the matrix defined in (7.1).

a) If we consider μ ≡ 1 we have the classical (degenerate) parabolic problems (see for instance [25] and [23])

\frac{∂u}{∂t} − div(a_{h} · Du) = f  on Ω × (0, T)
u = 0  on ∂Ω × (0, T)
u = φ  on Ω × {0} .

Then the solutions converge in L^{2}(0, T; L^{1}(Ω)) to the solution of

\frac{∂u}{∂t} − ∑_{i,j=1}^{n}â_{ij}(t)\frac{∂^{2}u}{∂x_{i}∂x_{j}} = f  on Ω × (0, T)
u = 0  on ∂Ω × (0, T)
u = φ  on Ω × {0} .

b) If we consider μ ≡ 0 we have the classical (degenerate) elliptic problems (see for instance [8] and [9])

−div(a(·, t) · Du) = f(·, t) on Ω for a. e. t ∈ [0, T]
u = 0  on ∂Ω

In this case the solutions converge in L^{2}(0, T; L^{1}(Ω)) to the solution of

− ∑_{i,j=1}^{n}â_{ij}(t)\frac{∂^{2}u}{∂x_{i}∂x_{j}} = f(·, t) on Ω for a. e. t ∈ [0, T]
u = 0  on ∂Ω

Before showing other possible choices we want to stress that the limit matrix is the same for the parabolic and for the elliptic case. Consider the classical case (not degenerate), i. e. a sequence of parabolic operators like

\frac{∂}{∂t} − div(a · D)  (8.1)

with

λ_{0}|ξ|^{2} ⩽ (a(x, t) · ξ, ξ) ⩽ Λ_{0}|ξ|^{2} , for a. e. (x, t) ∈ Ω × (0, T) ,

where λ_{0}, Λ_{0} ∈ R, 0 < λ_{0} < Λ_{0}, the situation consider by F. Colombini and S. Spagnolo in [5]. This corrisponds to consider λ(x) = λ_{0} for every x ∈ Ω and μ ≡ 1. In [5] the authors obtain a result of compactness with respect to G-convergence for a sequence of parabolic operators like (8.1) defined by matrices satisfying the following condition

lim_{τ→0} sup_{h} ∫_{a}^{b} ∫_{ω} |a_{h}(x, t + τ) − a_{h}(x, t)|dxdt = 0  (H)

for every (a, b) ⊂⊂ (0, T) and ω ⊂⊂ Ω. F. Colombini and S. Spagnolo prove that (see [5] for the definitions and the result) given a sequence (a_{h})_{h} satisfying hypothesis (H) the sequence of parabolic operators P_{h} = \frac{∂}{∂t} − div(a_{h}(x, t)·D) G-converges to P = \frac{∂}{∂t} −div(a(x, t)·D) if and only if the family of sequences of elliptic operators A_{h}(t) = −div(a_{h}(·, t) · D) G-converges to A(t) = −div(a(·, t) · D) for almost every t ∈ (0, T). Moreover they give an example of a sequence (a_{h}(x, t))_{h} which does not satisfy (H) such that the parabolic G-limit is \frac{∂}{∂t} − div(a_{1}(x, t) · D), the elliptic G-limit is −div(a_{2}(·, t) · D) for almost every t ∈ (0, T), but a_{1} ≠ a_{2}. Observe that even if a(x, t) ∈ C^{1}(\overline{Ω}×[0, T]) and we define a_{h}(x, t) = a(h^{α}x, h^{β}t) with β > 0, the sequence (a_{h})_{h} does not satisfy condition (H). This clarifies that the result of the previous section, it is not possible to obtain a result of G-convergence for a general sequence

μ_{h} \frac{∂}{∂t} − div(a_{h} · D)

with the limit matrix independent of the coefficients μ_{h}. In the homogenization case it is possible only for homogenization in space, i. e. a_{h}(x, t) = a(hx, t). In this sense the result of the previous section is sharp.

c) Interesting are also examples in which the problem is mixed. Take, for example, the simplest situation, that is

μ = \{ \begin{matrix}0 & in Q_{0} \\ 1 & in Q_{+} \end{matrix}

where Q_{+} and Q_{0} are open sets with Lipschitz boundaries such that Q_{+} ∪Q_{0} = [0, 1]^{n}. In this case the solution of the mixed problems

μ_{h} \frac{∂u}{∂t} − div(a_{h} · Du) = f  on Ω_{+,h} × (0, T)
u = 0  on ((∂Ω_{+,h} ∩ ∂Ω) × (0, T))
u = φ  on (Ω_{+,h} × {0})
−div(a(·, t) · Du) = f(·, t) on Ω_{0,h} for a. e. t ∈ [0, T]
u = 0  on ∂Ω_{0,h} ∩ ∂Ω

converge in L^{2}(0, T; L^{1}(Ω)) to the solution of

|Q_{+}|\frac{∂u}{∂t} − ∑_{i,j=1}^{n}â_{ij}(t)\frac{∂^{2}u}{∂x_{i}∂x_{j}} = f  on Ω × (0, T)
u = 0  on ∂Ω × (0, T)
u = φ  on Ω × {0} .

d) A general interesting sistuation is the following: if α = 0 (H.1) or (H.2) are equivalent to the existence of a constant c > 0 (see [23], section 6) such that

0 ⩽ μ ⩽ c λ

(see also [24] for μ = λ).

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F. Paronetto Dipartimento di Matematica “Ennio De Giorgi”, Via per Arnesano 73100 Lecce, Italy

E-mail: fabio.paronetto@unile.it