THE EVOLUTION DAM PROBLEM FOR NONLINEAR DARCY’S LAW AND DIRICHLET BOUNDARY CONDITIONS *

A. Lyaghfouri

Abstract:In this paper, we study a time dependent dam problem modeling a non- linear fluid flow through a homogeneous or nonhomogeneous porous medium governed by a nonlinear Darcy’s law. We prove existence and uniqueness of a weak solution.

Introduction

The dam problem consists of finding the flow region and the pressure of fluid flow through a porous medium Ω under gravity. The free boundary represents the region separating the wet and the dry part of the porous medium. Assuming the flow governed by a linear Darcy’s law and taking Dirichlet boundary conditions on some part of the boundary, this problem has been widely studied from several points of view both for the stationary and the evolutionary case.

For the stationary case, the first results are due to C. Baiocchi ([5], [6]) who solved the case of rectangular dams by introducing the so called Baiocchi’s trans- formation, which leads him to consider problems of variational inequalities. Then he established existence and uniqueness results. Although this method is not adaptable for the general case, many authors ([7], [8], [16], [23]), used same techniques to treat questions related to heterogeneous or three dimensional rect- angular dams.

Few years after, the steady problem has been studied in the general case by H.W. Alt ([2], [3], [4]), H. Br´ezis, D. Kinderlehrer and G. Stampacchia [9], J. Carrillo and M. Chipot [15]. An existence theorem has been proved and the uniqueness of the solution established up to a certain class of disturbing functions.

The regularity of the free boundary was also investigated.

Received: May 21, 1997; Revised: June 11, 1997.

AMS Mathematical Subject Classification: 35R35, 76S05.

* Supported by the European Science Foundation Scientific Programme on the Mathematical Treatment of Free Boundary Problems.

The evolution dam problem has been solved first by A. Torelli ([24], [25]) in the case of a rectangular domain. He used a similar transformation to the Baiocchi’s one, which allowed him to reduce the problem to a quasi-variational inequality problem. He obtained, in this way, results of existence, uniqueness and regularity of his solution. Unfortunately, this method is not adapted for the general case.

Later, it was G. Gilardi [19] who proved an existence theorem for a weak
formulation of the evolution dam problem when the porous medium is assumed
to be a general Lipschitz bounded domain of R^{n}. In [18] E. Dibenedetto and
A. Friedman proved an existence theorem in a general way by an other method,
both for compressible or incompressible flow. Moreover they proved uniqueness
for rectangular dams. The question of uniqueness of the solution, in its generality,
remains open until solved by J. Carrillo [14].

In this study, we consider an incompressible fluid flow governed by a gener- alized nonlinear Darcy’s law relating the velocity v of the fluid to its pressure p by:

v=−A(x,∇(p+x_{n}))

where A is a function defined in Ω×R^{n} and x = (x_{1}, ..., x_{n}) denotes points in
R^{n}.

The prime example of nonlinear Darcy’s laws for a homogeneous porous
medium (see [17]) corresponds to theq-Laplacian: A(x, ξ) =|ξ|^{q−2}ξ. For hetero-
geneous media we have the following measurable perturbations of theq-Laplacian:

A(x, ξ) =|a(x)·ξ|^{q−2}a(x)·ξ, wherea(x) is a measurable positive definite matrix
representing the permeability of the medium at x. Note that when q = 2, we
rediscover the well known linear Darcy Law.

In addition, we would like to consider a model of Dirichlet boundary condi- tions. The paper is organized as follows: In section 1, we begin by transforming the problem usually stated in terms of the pressure function into a problem for the hydrostatic headu=p+xn. The dry part is described by a bounded func- tion g. Then we give a weaker formulation to our problem. In section 2 and 3 we prove an existence theorem by means of regularization and by using the Tychonoff fixed point theorem. In section 4, we prove some properties of the solutions. In particular for any solution (u, g),u is bounded and A-subharmonic and g is continuous in time variable. In section 5, we assume that q ≤ n+ 1, A(x, ξ) =A(ξ) andA(e)·ν ≤0 on the bottom of the dam. Then from section 4, we derive a monotonicity property for g. Making use of this result, we prove a comparison theorem and with the help of the continuity of g we deduce the uniqueness of the solution.

1 – Statement of the problem

The dam is a bounded locally Lipschitz domain Ω inR^{n}(n≥2) (see Figure 1).

The boundary Γ of Ω is divided in two parts: an impervious part Γ_{1}and a pervious
one Γ_{2} which is assumed to be nonempty and relatively open in Γ. Let now T
be a positive number, Q = Ω×(0, T) and ϕ a nonnegative Lipschitz function
defined inQ. We define:

Σ_{1} = Γ_{1}×(0, T), Σ_{2} = Γ_{2}×(0, T), Σ_{3} = (Γ_{2}×(0, T))∩ {ϕ >0}

and Σ_{4}= (Γ_{2}×(0, T))∩ {ϕ= 0}.

Fig. 1

We shall be interested with the problem of finding the pressure p and the
saturationχ of the fluid. For convenience, we set: ψ=ϕ+x_{n},u =p+x_{n} and
g = 1−χ. Starting from the nonlinear Darcy’s law, the mass conservation law
and taking a Dirichlet boundary condition on Σ_{2}, the flow is governed by the
following equations:

(1.1)

i) u≥x_{n}, 0≤g≤1, g(u−x_{n}) = 0 in Q
ii) div^{³}A(x,∇u)−gA(x, e)^{´}+gt= 0 in Q
iii) u=ψ on Σ_{2}

iv) g(·,0) =g_{0} in Ω

v) ^{³}A(x,∇u)−gA(x, e)^{´}·ν= 0 on Σ_{1}
vi) ^{³}A(x,∇u)−gA(x, e)^{´}·ν≤0 on Σ_{4}

whereeis the vertical unit vector ofR^{n}, i.e.e= (0,1) with 0∈R^{n−1},ν denoting
the outward unit normal to∂Ω, g0 is a given function satisfying 0≤g0 ≤1 and
A: Ω×R^{n} −→ R^{n} is a mapping that satisfies the following assumptions with
some constantsq >1 and 0< α≤β <∞:

(1.2)

the function x 7−→ A(x, ξ) is measurable∀ξ ∈R^{n}, and
the function ξ 7−→ A(x, ξ) is continuous for a.e x∈Ω,
for allξ ∈R^{n} and a.e. x∈Ω

A(x, ξ)·ξ ≥α|ξ|^{q} ,
(1.3)

|A(x, ξ)| ≤β|ξ|^{q−1} ,
(1.4)

for allξ, ζ ∈R^{n} such thatξ6=ζ and a.e. x∈Ω

³A(x, ξ)− A(x, ζ)^{´}·(ξ−ζ)>0 ,
(1.5)

∃r >1 : div(A(x, e))∈L^{r}(Ω).
(1.6)

The condition (1.1) i) means that we look for a nonnegative pressure p and g(·, t) characterises the wet region Ω(t) at time t. (1.1) iii) means that the trace pressure at the bottoms of fluid reservoirs is equal to the one of the fluid and equal to the atmospheric one when the boundary of Ω is in contact with the air.

(1.1) iv) is an initial data. (1.1) v) and (1.1) vi) are due to the fact that the flux
of fluid vanishes on Σ_{1} (since Γ_{1} is impervious) and is nonnegative on Σ_{4} where
the fluid is free to exit from our porous medium.

From the strong formulation (1.1), we are led to consider the following weak formulation:

(P)

Find (u, g)∈L^{q}(0, T, W^{1,q}(Ω))×L^{∞}(Q) such that :
i) u≥x_{n}, 0≤g≤1, g(u−x_{n}) = 0 a.e. in Q;
ii) u=ψ on Σ_{2} ;

iii) Z

Q

³A(x,∇u)−gA(x, e)^{´}· ∇ξ+g ξ_{t}dx dt+
Z

Ω

g_{0}(x)ξ(x,0)dx≤0

∀ξ∈W^{1,q}(Q), ξ= 0 on Σ3, ξ≥0 on Σ4, ξ(x, T) = 0 a.e. in Ω .

2 – A regularized problem

We first introduce the following approximated problem:

(Pε)

Find uε∈W^{1,q}(Q) such that : uε=ψ on Σ2 ,
Z

QA(x,∇uε)·∇ξ+ε|uεt|^{q−2}u_{εt}·ξt+G_{ε}(u_{ε})^{³}ξ_{t}− A(x, e)·∇ξ^{´}dx dt=

= Z

ΩGε(uε(x, T))ξ(x, T)dx− Z

Ωg0(x)ξ(x,0)dx

∀ξ∈W^{1,q}(Q), ξ= 0 on Σ_{2} ,
whereG_{ε}: L^{q}(Q) (resp.L^{q}(Ω)) −→ L^{∞}(Q) (resp.L^{∞}(Ω)) is defined by

(2.1) G_{ε}(v) =

0 if v−x_{n}≥ε

1−(v−x_{n})/ε if 0≤v−x_{n}≤ε

1 if v−x_{n}≤0.

Then we have:

Theorem 2.1. Assume thatϕis a nonnegative Lipschitz continuous function
and thatAsatisfies (1.2)–(1.5). Then, there exists a solutionu_{ε} of (P_{ε}).

Proof: It will be done in three steps:

Step 1: We define

V=^{n}v∈W^{1,q}(Q) / v= 0 on Σ_{2}^{o} and K=^{n}v∈W^{1,q}(Q) / v=ψon Σ_{2}^{o}.
For u∈K, we consider the map:

A(u) : W^{1,q}(Q)−→R, ξ7−→ hA(u), ξi=
Z

QA(x,∇u)·∇ξ+ε|ut|^{q−2}utξtdx dt .
Then the operatorA defined byA: u∈K 7−→A(u), satisfies:

Lemma 2.2. If we denote by(W^{1,q}(Q))^{0}the dual space ofW^{1,q}(Q), we have
i) For everyu∈K,A(u)∈(W^{1,q}(Q))^{0};

ii) A is continuous from K into(W^{1,q}(Q))^{0};
iii) A is monotone and coercive.

Proof: (see [20] for example).

Step 2: Forv∈W^{1,q}(Q), we consider the map: f_{v}: W^{1,q}(Q)−→R,
ξ 7−→

Z

Q

G_{ε}(v)^{³}A(x, e)·∇ξ−ξt

´dx dt+ Z

Ω

G_{ε}(v(x, T))ξ(x, T)−g_{0}(x)ξ(x,0)dx .
It is clear thatf_{v} ∈(W^{1,q}(Q))^{0}. Using Lemma 2.2, we deduce (see [20]) that for
everyv∈W^{1,q}(Q) there exists a unique uε solution of the variational problem
(2.2) u_{ε} ∈K , hAuε, ξi=hfv, ξi ∀ξ∈V .

Step 3: Now, let us consider the map F_{ε} defined by: F_{ε}: W^{1,q}(Q) −→ K,
v7−→u_{ε}. Let us denote by B(0, R(ε)) the closed ball inW^{1,q}(Q) of center 0 and
radiusR(ε). Then we have:

Lemma 2.3.

i) ∃R(ε)>0/ F_{ε}(B(0, R(ε))) ⊂ B(0, R(ε));

ii) F_{ε}: B(0, R(ε))−→B(0, R(ε))is weakly continuous.

Proof: i) Note that u_{ε}−ψ is a suitable test function to (2.2), so:

(2.3)

Z

QA(x,∇uε)· ∇uε+ε|uεt|^{q}dx dt=

= Z

QA(x,∇uε)· ∇ψ+ε|uεt|^{q−2}u_{εt}·ψ_{t}dx dt
+

Z

Q

G_{ε}(v)A(x, e)· ∇(uε−ψ)dx dt

− Z

QG_{ε}(v) (u_{ε}−ψ)_{t}dx dt−
Z

Ωg_{0}(x) (u_{ε}−ψ)(x,0)dx
+

Z

ΩGε(v(x, T)) (uε−ψ)(x, T)dx .

Using (1.3), (1.4), (2.1), (2.3) and H¨older’s inequality, we get for some con-
stantsc_{i}

min(α, ε) Z

Q|∇uε|^{q}+|uεt|^{q}dx dt≤c1|uε|^{q−1}_{1,q} +c2|uε|1,q+c3 .

By Poincar´e’s Inequality, this leads for some constantsc^{0}_{i}to|uε|^{q}_{1,q}≤c^{0}_{1}|uε|^{q−1}_{1,q} +
c^{0}_{2}|uε|1,q+c^{0}_{3} from which we deduce that: |uε|1,q ≤R(ε) where R(ε) is some con-
stant depending onε. So we have: F_{ε}(B(0, R(ε)))⊂B(0, R(ε)). More precisely,
we have proved that: F_{ε}(W^{1,q}(Q))⊂B(0, R(ε)).

ii) Let (v_{i})_{i∈I} be a generalized sequence in C =B(0, R(ε)) which converges
tov in C weakly.

Set u^{i}_{ε} = F_{ε}(v_{i}) and u_{ε} = F_{ε}(v). We want to prove that (u^{i}_{ε})_{i∈I} converges
to uε weakly in C. Since C is compact with respect to the weak topology, it
is enough to show that (u^{i}_{ε})_{i∈I} has u_{ε} as the unique limit point for the weak
topology inC. So letube a weak limit point for (u^{i}_{ε})i∈I inC. Using the compact
imbedding: W^{1,q}(Q),→L^{q}(Q), one can construct a sequence (u^{i}_{ε}^{k})_{k∈}_{N} such that
{ik/ k∈N} ⊂ I, u^{i}_{ε}^{k} * u weakly in W^{1,q}(Q) and u^{i}_{ε}^{k} → u strongly inL^{q}(Q).

Choose u^{i}_{ε}^{k} −u_{ε} as a suitable test function for (2.2) written for u^{i}_{ε}^{k} and u_{ε}.
Subtract the equations, so that

(2.4) Z

Q

³A(x,∇u^{i}_{ε}^{k})− A(x,∇uε)^{´}· ∇(u^{i}_{ε}^{k} −u_{ε}) +

+ε^{³}|u^{i}_{εt}^{k}|^{q−2}u^{i}_{εt}^{k} − |uεt|^{q−2}u_{εt}^{´}·(u^{i}_{ε}^{k}−u_{ε})_{t}dx dt=

= Z

Q

³Gε(vik)−Gε(v)^{´ ³}A(x, e)· ∇(u^{i}_{ε}^{k}−uε)−(u^{i}_{ε}^{k}−uε)t

´dx dt

+ Z

Ω

³G_{ε}(v_{i}_{k}(x, T))−G_{ε}(v(x, T))^{´}(u^{i}_{ε}^{k}−u_{ε})(x, T)dx .

Now we have by (1.4), (2.1), H¨older’s inequality and the fact that|u^{i}_{ε}^{k}−uε|1,q ≤
2R(ε):

(2.5)

¯

¯

¯

¯ Z

Q

³G_{ε}(v_{i}_{k})−G_{ε}(v)^{´ ³}A(x, e)· ∇(u^{i}_{ε}^{k}−u_{ε})−(u^{i}_{ε}^{k}−u_{ε})_{t}^{´}dx dt

¯

¯

¯

¯≤

≤c1(ε)· |vik−v|q . (2.6)

¯

¯

¯

¯ Z

Ω

³G_{ε}(v_{i}_{k}(x, T))−G_{ε}(v(x, T))^{´}(u^{i}_{ε}^{k}−u_{ε})(x, T)dx

¯

¯

¯

¯≤

≤c2(ε)·^{¯}^{¯}_{¯}(vik−v) (·, T)^{¯}^{¯}_{¯}

q .
Now due to (1.5), (2.4)–(2.6) and the compact imbeddings: W^{1,q}(Q),→L^{q}(Q)
andW^{1−}^{1}^{q}^{,q}(Ω× {T}),→L^{q}(Ω× {T}), we get:

(2.7)

k→+∞lim Z

Q

³A(x,∇u^{i}_{ε}^{k})− A(x,∇uε)^{´}· ∇(u^{i}_{ε}^{k} −uε)dx dt= 0,

k→+∞lim Z

Q

³|u^{i}_{εt}^{k}|^{q−2}u^{i}_{εt}^{k} − |uεt|^{q−2}u_{εt}^{´}·(u^{i}_{ε}^{k}−u_{ε})_{t}dx dt= 0 .

Using (2.7), we deduce that (see [11]) there exists a subsequence of (u^{i}_{ε}^{k}) also
denoted by (u^{i}_{ε}^{k}) such that ∇u^{i}_{ε}^{k} → ∇uε and u^{i}_{εt}^{k} → u_{εt} a.e. in Q. Taking into

account the boundedness of (u^{i}_{ε}^{k}) in W^{1,q}(Q), we get: ∇u^{i}_{ε}^{k} * ∇uε weakly in
L^{q}(Q) andu^{i}_{εt}^{k} * uεt weakly inL^{q}(Q). So we have uε =u and uε is the unique
weak limit point of (u^{i}_{ε}) in C. Thus u^{i}_{ε} = F_{ε}(v^{i}) * u_{ε} = F_{ε}(v) weakly in C.

Hence the continuity ofFε holds.

At this step, applying the Tychonoff fixed point theorem on B(0, R(ε)) (see [22]), we derive thatFε has a fixed point. Thus (Pε) has at least one solution.

Let us now show that our sequence (u_{ε}) is uniformly bounded inL^{∞}(Q). More
precisely we have:

Proposition 2.4. Let u_{ε} be a solution of (P_{ε}) and let ε_{0} > 0. Then we
have for anyε ∈ (0, ε_{0}) and H such that H ≥max(ε_{0}+ max{xn,(x^{0}, x_{n})∈Ω},
max{ψ(x, t),(x, t)∈Σ_{2}})

(2.9) x_{n}≤u_{ε}≤H a.e. in Q .

Proof: i) Since (u_{ε}−H)^{+} is a suitable test function for (P_{ε}), we have by
(2.1) and the choice ofH

(2.10)

Z

QA(x,∇uε)· ∇(uε−H)^{+}+ε|uεt|^{q−2}u_{εt}(u_{ε}−H)^{+}_{t} dx dt=

= Z

QGε(uε)^{³}A(x, e)· ∇(uε−H)^{+}−(uε−H)^{+}_{t} ^{´}dx dt

− Z

Ω

g_{0}(x) (u_{ε}−H)^{+}(x,0)dx+
Z

Ω

G_{ε}(u_{ε}(x, T)) (u_{ε}−H)^{+}(x, T)dx

=− Z

Ωg_{0}(x) (u_{ε}−H)^{+}(x,0)dx≤0 .
Then we deduce from (1.3) and (2.10)

Z

Qα|∇(uε−H)^{+}|^{q}+ε|(uε−H)^{+}_{t} |^{q}dx dt≤0,
which leads to|∇(uε−H)^{+}|=|(uε−H)^{+}_{t}|= 0 a.e. in Q. Thus (u_{ε}−H)^{+}= 0
andu_{ε}≤H a.e. in Q.

ii) We denote by (·)^{−} the negative part of a function. Thenξ= (uε−xn)^{−} is
a test function for (P_{ε}) and one has by taking into account (2.1):

(2.11) Z

QA(x,∇uε)· ∇(uε−x_{n})^{−}+ε|uεt|^{q−2}u_{εt}·(u_{ε}−x_{n})^{−}_{t} dx dt=

= Z

Ω(u_{ε}−x_{n})^{−}(x, T)dx+
Z

QA(x, e)· ∇(u_{ε}−x_{n})^{−}dx dt

− Z

Ω

g_{0}(x) (u_{ε}−x_{n})^{−}(x,0)dx−
Z

Q

(u_{ε}−x_{n})^{−}_{t} dx dt .

Integrating by part the last term of (2.11), we obtain (2.12)

Z

[uε≤xn]

³A(x,∇uε)−A(x,∇xn)^{´}·(∇uε−∇xn)+ε|(uε−xn)_{t}|^{q}dx dt ≤ 0.

Using (1.5) and (2.12) we conclude that uε≥xna.e. in Q.

Now we give an a priori estimate for ∇uε and u_{εt}.

Proposition 2.5. Under assumptions of Proposition 2.4, we have for any
ε∈(0, ε_{0}):

(2.13)

Z

Q

³α|∇uε|^{q}+ε|uεt|^{q}^{´}dx dt≤C ,

whereC is a constant independent ofε.

Proof: Using the fact that u_{ε}−ψ is a suitable test function, we get
(2.14)

Z

QA(x,∇uε)· ∇uε+ε|uεt|^{q}dx dt=

= Z

QA(x,∇uε)· ∇ψ dx dt+ Z

Q

ε|uεt|^{q−2}u_{εt}·ψ_{t}dx dt
+

Z

QGε(uε)A(x, e)· ∇(uε−ψ)dx dt− Z

QGε(uε) (uε−ψ)tdx dt +

Z

Ω

G_{ε}(u_{ε}(x, T)) (u_{ε}−ψ)(x, T)dx−
Z

Ω

g_{0}(x) (u_{ε}−ψ)(x,0)dx .

First let us set: E_{ε}(y) =
Z _{y}

0

(1−H_{ε}(s))ds and H_{ε}(s) = 1 ∧ s^{+}

ε . We have
0≤(1−H_{ε}(y))y≤E_{ε}(y)≤y ∀y≥0 and using (2.9), we get

(2.15)

Z

Q−Gε(u_{ε}) (u_{ε}−ψ)_{t}dx dt=

= Z

Q−G_{ε}(u_{ε}) (u_{ε}−x_{n})_{t}dx dt+
Z

QG_{ε}(u_{ε})ϕ_{t}dx dt

=− Z

Q

∂

∂tE_{ε}(u_{ε}−x_{n})dx dt+
Z

Q

G_{ε}(u_{ε})ϕ_{t}dx dt

= Z

Ω

·

E_{ε}^{³}u_{ε}(x,0)−x_{n}^{´}−E_{ε}^{³}u_{ε}(x, T)−x_{n}^{´}

¸ dx+

Z

Q

G_{ε}(u_{ε})ϕ_{t}dx dt≤C .

Next, by (2.9) the last two terms of (2.14) are bounded. So using (1.3), (2.14), (2.15) and H¨older’s inequality, we derive for some constant C > 0:

0 ≤ U_{ε} ≤ C(1 +U_{ε}^{1/q} + U_{ε}^{1/q}^{0}) where U_{ε} =
Z

Q

(α|∇uε|^{q} +ε|uεt|^{q})dx dt.

Hence we get (2.13) sinceq, q^{0} >1.

In the following proposition, we show that u_{ε} satisfies an inequality similar to
(P) iii).

Proposition 2.6. Letu_{ε} be a solution of(P_{ε}). Then we have:

Z

QA(x,∇uε)·∇ξ+ε|uεt|^{q−2}u_{εt}·ξ_{t}+G_{ε}(u_{ε})^{³}ξ_{t}− A(x, e)·∇ξ^{´}dx dt+

(2.16) +

Z

Ω

g_{0}(x)ξ(x,0)dx≤0

∀ξ∈W^{1,q}(Q), ξ= 0 on Σ_{3}, ξ≥0 on Σ_{4}, ξ(x, T) = 0 a.e. x∈Ω.
Proof: Letξ as in (2.16). For anyδ >0, (^{p}_{δ}^{ε} ∧ξ), where p_{ε} =u_{ε}−x_{n}, is a
test function for (P_{ε}). So we can write

Z

QA(x,∇uε)·∇

µpε

δ ∧ξ

¶

− A(x, e)·∇

µpε

δ ∧ξ

¶

+ε|uεt|^{q−2}uεt·
µpε

δ ∧ξ

¶

t

dx dt+ +

Z

Q

H_{ε}(u_{ε}−x_{n})A(x, e)·∇

µp_{ε}
δ ∧ξ

¶

dx dt − Z

Q

H_{ε}(u_{ε}−x_{n})
µp_{ε}

δ ∧ξ

¶

t

dx dt=

= Z

Ω

³1−g_{0}(x)^{´}·
µp_{ε}

δ ∧ξ

¶

(x,0)dx .

The first integral in the left side of this equality can be written as Z

[uε−xn<δξ]

³A(x,∇uε)− A(x,∇xn)^{´}· ∇u_{ε}−x_{n}

δ +ε

δ|uεt|^{q}dx dt+
+

Z

[uε−xn≥δξ]

³A(x,∇uε)− A(x, e)^{´}· ∇ξ+ε|uεt|^{q−2}u_{εt}ξ_{t}dx dt .
Using (1.5) and the fact that 1−g0(x)≥0 a.e. x∈Ω, we obtain

(2.17) Z

[uε−xn≥δξ]

³A(x,∇uε)− A(x, e)^{´}· ∇ξ+ε|uεt|^{q−2}u_{εt}ξ_{t}dx dt+
+

Z

QH_{ε}(u_{ε}−x_{n})A(x, e)· ∇
µp_{ε}

δ ∧ξ

¶

dx dt− Z

QH_{ε}(u_{ε}−x_{n})
µp_{ε}

δ ∧ξ

¶

t

dx dt≤

≤ Z

Ω(1−g0(x))ξ(x,0)dx .

Let us show that:

δ→0lim Z

QHε(uε−xn)A(x, e)· ∇

µuε−xn

δ ∧ξ

¶

dx dt= (2.18)

= Z

QHε(uε−xn)A(x, e)· ∇ξ dx dt ,

δ→0lim Z

QH_{ε}(u_{ε}−x_{n})

µuε−xn

δ ∧ξ

¶

t

dx dt= Z

QH_{ε}(u_{ε}−x_{n})ξ_{t}dx dt .
(2.19)

Indeed, taking in account (1.6) we can use the divergence formula to get:

Z

QHε(uε−xn)A(x, e)· ∇

µuε−xn

δ ∧ξ

¶

dx dt=

=− Z

Q

div^{³}H_{ε}(u_{ε}−x_{n})A(x, e)^{´}·

µu_{ε}−x_{n}
δ ∧ξ

¶ dx dt

+ Z

∂Q

H_{ε}(u_{ε}−x_{n})A(x, e)·ν·

µu_{ε}−x_{n}
δ ∧ξ

¶

dσ(x, t) .

Since ξ∧ u_{ε}−xn

δ −→ ξ a.e. on [u_{ε} > x_{n}] when δ goes to 0, we obtain by the
Lebesgue theorem,

δ→0lim Z

QH_{ε}(u_{ε}−x_{n})A(x, e)· ∇

µu_{ε}−x_{n}
δ ∧ξ

¶

dx dt =

= − Z

Q

div^{³}H_{ε}(u_{ε}−xn)A(x, e)^{´}·ξ dx dt+
Z

∂Q

H_{ε}(u_{ε}−xn)A(x, e)·ν·ξ dσ(x, t) =

= Z

Q

H_{ε}(u_{ε}−x_{n})A(x, e)· ∇ξ dx dt ,
which proves (2.18). Similarly we establish (2.19).

So combining (2.18)–(2.19) and lettingδ →0 in (2.17) we obtain (2.16) since Z

Ω

ξ(x,0)dx=− Z

Q

ξ_{t}dx dt.

3 – Existence of a solution

Theorem 3.1. Assume thatϕis a nonnegative Lipschitz continuous function and thatAsatisfies (1.2)–(1.6). Then there exists a solution (u, g) of (P).

The proof will consist in passing to the limit, when ε goes to 0, in (P_{ε}).

To do this we shall need some lemmas.

First from definition (2.1) of G_{ε} and estimates (2.9), (2.13) and (1.4), we de-
duce the existence of a subsequenceε_{k} ofεand functions: u∈L^{q}(0, T, W^{1,q}(Ω)),
g∈L^{q}^{0}(Q), A_{0}∈L^{q}^{0}(Q) such that:

uεk * u weakly in L^{q}(0, T, W^{1,q}(Ω)),
(3.1)

A(x,∇uεk)* A_{0} weakly in L^{q}^{0}(Q) ,
(3.2)

G_{ε}_{k}(u_{ε}_{k})* g weakly in L^{q}^{0}(Q) .
(3.3)

Then we can prove:

Lemma 3.2. Let u, g be defined by (3.1) and (3.3) respectively. Then we have:

i) u=ψ on Σ_{2}, u≥x_{n} a.e. inQ;

ii) 0≤g≤1 a.e. inQ.

Proof: We consider the setK_{1}={v∈L^{q}(0, T, W^{1,q}(Ω))/ v≥x_{n}a.e. in Q,
v = ψ on Σ2}. K1 is closed and convex in L^{q}(0, T, W^{1,q}(Ω)), then it is weakly
closed. Since u_{ε}_{k} ∈ K_{1}, u ∈ K_{1} and i) holds. In the same way, we prove that
g∈K2 = {v∈L^{q}^{0}(Q) / 0≤v≤1 a.e. in Q} and ii) holds.

Lemma 3.3. Let u, g be defined by (3.1) and (3.3) respectively. Then we have:

(3.4) g(u−x_{n}) = 0 a.e. in Q .

Proof: First, note that (3.4) is not an obvious result as it is in the stationary
case (see [17]), since we do not know, a priori, whetheru_{ε} converges strongly to
u in L^{q}(Q) because the imbedding L^{q}(0, T, W^{1,q}(Ω)) ,→ L^{q}(Q) is not compact.

To overcome this difficulty, we are going to prove a strong convergence of the
sequence (G_{ε}_{k}(u_{ε}_{k}))_{k} in a suitable space.

Next, we have for θ∈D(Q),θ≥0 andp_{ε}_{k} =u_{ε}_{k} −x_{n}
0 ≤

Z

QG_{ε}_{k}(u_{ε}_{k}) (u_{ε}_{k} −x_{n})θ dx dt =

= Z

Q∩[0≤p_{εk}≤εk]

(1−H_{ε}_{k}(p_{ε}_{k}))p_{ε}_{k}θ dx dt ≤ ε_{k}· |θ|∞· |Q|.
So

(3.5) lim

k→+∞

Z

Q

G_{ε}_{k}(u_{ε}_{k}) (u_{ε}_{k}−x_{n})θ dx dt= 0 .

To get (3.4), it suffices to prove that:

(3.6) lim

k→+∞

Z

Q

G_{ε}_{k}(u_{ε}_{k}) (u_{ε}_{k}−x_{n})θ dx dt=
Z

Q

g(u−x_{n})θ dx dt .

For this purpose we introduce the function: w_{ε}_{k} = ε_{k}|uεkt|^{q−2}u_{ε}_{k}_{t}+G_{ε}_{k}(u_{ε}_{k}).

From (2.13), sinceq^{0} >1, we have:

(3.7) ε_{k}|uεkt|^{q−2}u_{ε}_{k}_{t}−→0 strongly in L^{q}^{0}(Q).
Then from (3.3) and (3.7) we deduce that

(3.8) w_{ε}_{k} * g weakly in L^{q}^{0}(Q) .
We are going to prove that:

(3.9) w_{ε}_{k}_{t}* g_{t} weakly in L^{q}^{0}(0, T, W^{−1,q}^{0}(Ω)).

So, let us show thatg_{t}∈L^{q}^{0}(0, T, W^{−1,q}^{0}(Ω)). Let ξ∈D(0, T, W_{0}^{1,q}(Ω)). Since ξ
is a test function for (Pεk), we obtain, after lettingk→+∞in (Pεk), and taking
into account (3.2), (3.3) and (3.7)

Z

Qg ξtdx dt=− Z

Q

³A0−gA(x, e)^{´}· ∇ξ dx dt

from which we deduce (see [10]) thatg_{t}=−div(A_{0}−gA(x, e))∈L^{q}^{0}(0, T,W^{−1,q}^{0}(Ω)).

Now we have in the distributional sense:

w_{ε}_{k}_{t}=−div^{³}A(x,∇u_{ε}_{k})−G_{ε}_{k}(u_{ε}_{k})A(x, e)^{´} and w_{ε}_{k}_{t}*div^{³}gA(x, e)−A_{0}^{´}=g_{t}
weakly in L^{q}^{0}(0, T, W^{−1,q}^{0}(Ω)).
So (3.9) holds.

At this stage let us introduce the spaceW defined by:W={v∈L^{q}^{0}(0, T, L^{q}^{0}(Ω))/

v_{t}∈L^{q}^{0}(0, T, W^{−1,q}^{0}(Ω))}which is a Banach space for the norme: kvk_{L}q0

(0,T,L^{q}^{0}(Ω))+
kv_{t}k_{L}q0

(0,T,W^{−1,q}^{0}(Ω)). Since L^{q}^{0}(Ω) and W^{−1,q}^{0}(Ω) are reflexifs, the imbedding
L^{q}^{0}(Ω) ,→ W^{−1,q}^{0}(Ω) being continuous and compact (see [1]), we deduce that
(see [20]), the imbedding

(3.10) W ,→L^{q}^{0}(0, T, W^{−1,q}^{0}(Ω)) is compact.

Now, the sequencew_{ε}_{k} ∈W and it is bounded in W by (3.8) and (3.9). So up to
a subsequence still denoted byεk, we have wεk * wweakly inW. But it is easy
to see thatw=g. We then deduce from (3.7) and (3.10)

(3.11) G_{ε}_{k}(u_{ε}_{k})→g strongly in L^{q}^{0}(0, T, W^{−1,q}^{0}(Ω)).

Finally, to conclude, it suffices to remark that: (u_{ε}_{k}−xn)θ *(u−xn)θ weakly in
L^{q}(0,T,W_{0}^{1,q}(Ω)). Then by (3.11), we get (3.6). Consequently

Z

Qg(u−xn)θ dxdt= 0

∀θ∈D(Q), θ≥0 and sinceg(u−x_{n})≥0 a.e. in Q, we get (3.4).

Remark 3.4. We have (see [20]) the imbedding W ,→C^{0}([0, T], W^{−1,q}^{0}(Ω))
theng∈C^{0}([0, T], W^{−1,q}^{0}(Ω)). In section 4, we shall improve this regularity and
prove thatg∈C^{0}([0, T], L^{p}(Ω)), ∀p∈[1,+∞[.

Lemma 3.5. Letu,A_{0}andgbe defined by (3.1), (3.2) and (3.3) respectively.

Then we have:

(3.12) Z

Q

³A_{0}−gA(x, e)^{´}· ∇(u−ψ)ξ dx dt=
Z

Q

g(ϕ ξ)_{t}dx dt ∀ξ∈D(0, T) .

Proof: Let ζ be a smooth function such that d(suppζ,Σ2) > 0 and
suppζ ⊂R^{n}×(τ_{0}^{0}, T −τ_{0}^{0}) for T > τ_{0}^{0} >0. Then there exists τ_{0} > 0 such that:

∀τ ∈ (−τ0, τ0), (x, t) 7−→ ζ(x, t−τ) is a test function for (Pεk). So we get, for
allτ ∈(−τ0, τ_{0}), after letting kgo to +∞

(3.13)

Z

Q

³A_{0}(x, t)−g(x, t)A(x, e)^{´}· ∇ζ(x, t−τ)dx dt−

− ∂

∂τ

³Z

Q

g(x, t+τ)ζ(x, t)dx dt^{´}= 0
since

− Z

Qg(x, t)ζ_{t}(x, t−τ)dx dt= ∂

∂τ

³Z

Qg(x, t)ζ(x, t−τ)dx dt^{´}

= ∂

∂τ

³Z

Qg(x, t+τ)ζ(x, t)dx dt^{´}.

Now it is easy to see that (3.13) still holds for functions ζ inL^{q}(0, T, W^{1,q}(Ω))
such thatζ= 0 on Σ_{2} and ζ = 0 on Ω×((0, τ_{0})∪(T −τ_{0}, T)). So if we consider
ξ∈D(τ_{0}, T−τ_{0}),ξ ≥0 and set: ζ = (u−ψ)ξ, we have ∀τ ∈(−τ_{0}, τ_{0})

(3.14) Z

Q

µ³

A_{0}(x, t)−g(x, t)A(x, e)^{´}·∇^{³}(u−ψ)ξ^{´}−g(ϕ ξ)_{t}

¶

(x, t−τ)dx dt=

= ∂G

∂τ (τ) withG(τ) =

Z

Q

g(x, t+τ) ((u−x_{n})ξ)(x, t)dx dt.

From (3.14) we know that G ∈ C^{1}(−τ0, τ_{0}). Moreover by Lemmas 3.2 and
3.3, we getG(τ)≥0 =G(0) ∀τ ∈(−τ0, τ0). So 0 is an absolute minimum forG
in (−τ0, τ_{0}) and

(3.15) ∂G

∂τ(0) = 0.

Combining (3.14) and (3.15) we get (3.12) for all ξ∈D(τ0, T−τ0),ξ ≥0.

Thanks to Lemma 3.5, we are going to prove a result which allows us to pass
to the limit in (P_{ε}_{k}).

Lemma 3.6. The sequence(u_{ε}_{k}) (resp.A(x,∇uεk)) converges strongly tou
(resp.A(x,∇u)) in L^{q}(0, T, W^{1,q}(Ω))(resp. L^{q}^{0}(Q)).

To prove Lemma 3.6, we need a lemma:

Lemma 3.7. Letu andA_{0} defined by (3.1) and (3.2) respectively. Then we
have

(3.16) Z

QA(x,∇u)·∇ξ dx dt= Z

QA0(x, t)·∇ξ dx dt ∀ξ ∈L^{q}(0, T, W^{1,q}(Ω)).
Proof: Letθ∈D(0, T),θ≥0. Choose ξ= (u_{ε}_{k}−ψ)θas a test function for
(P_{ε}_{k}) and write (3.12) for ξ =θ. Subtract the equations, we obtain:

(3.17)

Z

QθA(x,∇uεk)· ∇uεkdx dt=

= Z

Q

θ A_{0}· ∇u dx dt+
Z

Q

³A(x,∇uεk)−A_{0}^{´}· ∇ψ θ dx dt

− Z

Qε_{k}|u_{ε}_{k}_{t}|^{q−2}u_{ε}_{k}_{t}^{³}(u_{ε}_{k}−ψ)θ^{´}

tdx dt +

Z

QA(x, e)·^{³}Gεk(uεk)∇(uεk−ψ)−g∇(u−ψ)^{´}θ dx dt

− Z

Q

G_{ε}_{k}(u_{ε}_{k})^{³}(u_{ε}_{k} −ψ)θ^{´}

tdx dt− Z

Q

g(ϕ θ)_{t}dx dt .

By (3.2) we have:

(3.18) lim

k→+∞

Z

Q

³A(x,∇uεk)−A_{0}^{´}· ∇ψ θ dx dt= 0 .

Remark that Z

Q

ε_{k}|uεkt|^{q−2}u_{ε}_{k}_{t}·^{³}(u_{ε}_{k} −ψ)θ^{´}

tdx dt=

= Z

Qε_{k}|uεkt|^{q}θ dx dt+

Z

Qε_{k}|uεkt|^{q−2}uεktuεkθtdx dt−

Z

Qε_{k}|uεkt|^{q−2}uεkt·(ψ θ)tdx dt.

The first term of the above equality is nonnegative. Moreover using (3.7) and the
fact thatu_{ε}_{k} is uniformly bounded, we get:

(3.19) lim−

Z

Qε_{k}|uεkt|^{q−2}uεkt·^{³}(uεk−ψ)θ^{´}

tdx dt≤0. From (3.4), we deduce:

(3.20) Z

QA(x, e)·^{³}G_{ε}_{k}(u_{ε}_{k})∇(u_{ε}_{k} −ψ)−g∇(u−ψ)^{´}θ dx dt=

= Z

Q

θ G_{ε}_{k}(u_{ε}_{k})A(x, e)· ∇(uεk−x_{n})dx dt

− Z

Q

³Gεk(uεk)−g^{´}A(x, e)· ∇(ϕ θ)dx dt .
By (3.3) the last term of (3.20) goes to 0. Applying the divergence formula, we
get:

(3.21)

¯

¯

¯

¯ Z

Qθ Gεk(uεk)A(x, e)· ∇(uεk −xn)dx dt

¯

¯

¯

¯=

=

¯

¯

¯

¯ Z

Q

θA(x, e)· ∇

µZ _{min(u}_{εk}_{−x}_{n}_{,ε}_{k}_{)}

0

³1−H_{ε}_{k}(s)^{´}ds

¶ dx dt

¯

¯

¯

¯

≤εk

µZ

Qθ|div(A(x, e))|dx dt+ Z

∂Ω×(0,T)θ|A(x, e)·ν|dσ(x, t)

¶ .

We obtain from (3.20)–(3.21):

(3.22) lim

k→+∞

Z

QA(x, e)·^{³}G_{ε}_{k}(u_{ε}_{k})∇(uεk−ψ)−g∇(u−ψ)^{´}θ dx dt= 0 .
The last two terms of the right hand side of (3.17) can be written:

(3.23) − Z

Q

G_{ε}_{k}(u_{ε}_{k})^{³}(u_{ε}_{k} −ψ)θ^{´}

tdx dt− Z

Q

g(ϕ θ)_{t}dx dt=

=− Z

QG_{ε}_{k}(u_{ε}_{k}) (u_{ε}_{k}−x_{n})θ_{t}dx dt−
Z

QG_{ε}_{k}(u_{ε}_{k}) (u_{ε}_{k}−x_{n})_{t}θ dx dt
+

Z

Q

³G_{ε}_{k}(u_{ε}_{k})−g^{´}(ϕ θ)_{t}dx dt .

Arguing as in (3.5), the first term of the right side of (3.23) converges to 0.

Integrating by parts, we can see as in (3.21) that the second term goes also to 0.

Using (3.3), we get (3.24) lim

k→+∞− Z

Q

G_{ε}_{k}(u_{ε}_{k})^{³}(u_{ε}_{k} −ψ)θ^{´}

tdx dt− Z

Q

g(ϕ θ)_{t}dx dt= 0 .
Combining (3.17), (3.18), (3.19), (3.22) and (3.24) we conclude that:

(3.25) lim Z

Q

θA(x,∇uεk)· ∇uεkdx dt ≤ Z

Q

θ A_{0}(x, t)· ∇u dx dt

∀θ∈D(0, T), θ≥0 .
Let now v∈L^{q}(0, T, W^{1,q}(Ω)) and θ∈D(0, T) such that θ≥0. Using (1.5)
we have:

Z

Qθ^{³}A(x,∇u_{ε}_{k})− A(x,∇v)^{´}·(∇u_{ε}_{k}− ∇v)dx dt≥0 ∀k∈N
which can be written for allk∈N:

(3.26) Z

Q

θA(x,∇uεk)· ∇uεkdx dt − Z

Q

θA(x,∇uεk)· ∇v dx dt −

− Z

Q

θA(x,∇v)· ∇(uεk −v)dx dt ≥ 0 . Passing to the limit sup in (3.26) and taking into account (3.1)–(3.2) and (3.25), we get:

(3.27)

Z

Qθ^{³}A_{0}(x, t)− A(x,∇v)^{´}· ∇(u−v)dx dt≥0 .

If we choosev=u±λ ξ withξ ∈L^{q}(0, T, W^{1,q}(Ω)) andλ∈[0,1] in (3.27) we
obtain, after lettingλgo to 0 and taking into account (1.2) and (1.4)

Z

Qθ^{³}A_{0}(x, t)− A(x,∇u)^{´}· ∇ξ dx dt= 0 ∀θ∈D(0, T), θ≥0,

∀ξ ∈L^{q}(0, T, W^{1,q}(Ω))
and by density we get (3.16).

Proof of Lemma 3.6: Taking ξ=θ uin (3.16) with θ∈D(0, T), we get (3.28)

Z

Q

θA(x,∇u)· ∇u dx dt= Z

Q

θ A_{0}(x, t)· ∇u dx dt .

Using (3.25) and (3.28) we obtain:

(3.29) lim Z

Q

θA(x,∇uεk)· ∇uεkdx dt ≤ Z

Q

θA(x,∇u)· ∇u dx dt . Combining (3.1)–(3.2) and (3.28)–(3.29), one can prove easily:

(3.30) lim

Z

Q

θ^{³}A(x,∇uεk)− A(x,∇u)^{´}· ∇(uεk−u)dx dt≤0 .

Now, since ∇uεk * ∇u weakly in L^{q}(Q), we conclude by (3.30) because A
satisfies the Browder’s property (S_{+}) (see [12]), that ∇uεk → ∇u strongly in
L^{q}(Q). Moreover the mapping L^{q}(Q) → L^{q}^{0}(Q), v 7→ A(·, v) being continuous,
we deduce thatA(x,∇u_{ε}_{k}) converges strongly toA(x,∇u) inL^{q}^{0}(Q). Now by the
Poincar´e Inequality one can see thatu_{ε}_{k}converges strongly inL^{q}(0, T, W^{1,q}(Ω)).

Proof of Theorem 3.1: It is clear that (P) i) and (P) ii) follow from
Lemma 3.2 and Lemma 3.3. Let ξ ∈ W^{1,q}(Q), ξ = 0 on Σ_{3}, ξ ≥ 0 on Σ_{4} and
ξ(x, T) = 0 a.e. in Ω. Lettingkgo to +∞in (2.16) written forξ and using (3.3),
(3.7) and Lemma 3.6, we get (P) iii). This achieves the proof of Theorem 3.1.

Remark 3.8. Note that the Lemma 3.7 is sufficient for the proof of Theorem 3.1, however the result of Lemma 3.6 is more precise.

4 – Some properties

Let us first prove a technical lemma which generalizes Lemma 3.5.

Lemma 4.1.Let(u, g)be a solution of (P), letv∈W^{1,q}(Q)andF∈W_{loc}^{1,∞}(R^{2}),
such that:

i) F(u−xn, v)∈L^{q}(0, T, W^{1,q}(Ω));

ii) F(ψ−xn, v)∈W^{1,q}(Q);

iii) F(z_{1}, z_{2})≥0 for a.e. (z_{1}, z_{2})∈R^{2};
iv) either ∂F

∂z_{1}(z_{1}, z_{2})≥0 a.e.(z_{1}, z_{2})∈R^{2}, or ∂F

∂z_{1}(z_{1}, z_{2})≤0 a.e.(z_{1}, z_{2})∈R^{2}.
Then we have ∀ξ∈D(Ω×(0, T)):

(4.1) Z

Q

³A(x,∇u)−gA(x, e)^{´}· ∇^{³}F(u−x_{n}, v)ξ^{´}+g^{³}F(0, v)ξ^{´}

tdx dt=

= Z

Q

³A(x,∇u)−gA(x, e)^{´}· ∇^{³}F(ψ−x_{n}, v)ξ^{´}+g^{³}F(ψ−x_{n}, v)ξ^{´}

tdx dt .

Particularly, ifF(ψ−x_{n}, v)ξ = 0on Σ_{2}, then
(4.2)

Z

Q

³A(x,∇u)−gA(x, e)^{´}· ∇^{³}F(u−x_{n}, v)ξ^{´}+g^{³}F(0, v)ξ^{´}

tdx dt= 0 .
Proof:Arguing as in the proof of Lemma 3.5, we get forξ∈D(R^{n}×(τ_{0}, T−τ0)),
ξ≥0,τ0>0 and ζ = (F(u−xn, v)−F(ψ−xn, v))ξ

(4.3) Z

Q

³A(x,∇u(x, t))−g(x, t)A(x, e)^{´}· ∇^{³}F(u−x_{n}, v)ξ^{´}(x, t−τ) +
+g(x, t)^{³}F(0, v)ξ^{´}

t(x, t−τ)dx dt −

− Z

Q

³A(x,∇u(x, t))−g(x, t)A(x, e)^{´}· ∇^{³}F(ψ−xn, v)ξ^{´}(x, t−τ)−

−g(x, t)^{³}F(ψ−xn, v)ξ)t(x, t−τ)dx dt =

= ∂

∂τG(τ) withG(τ) =

Z

Q

g(x, t+τ) ((F(u−xn, v)−F(0, v))ξ)(x, t)dx dt. Since the integrals
on the left hand side of (4.3) are continuous functions on τ, we deduce that
G∈C^{1}(−τ0, τ_{0}). Using the monotonicity of F and (3.4), we can see that 0 is an
extremum forGin (−τ0, τ_{0}) and

(4.4) ∂G

∂τ(0) = 0. From (4.3) and (4.4) we deduce the Lemma.

From Lemma 4.1, we have:

Corollary 4.2. Let(u, g) be a solution of (P). Then:

Z

QA(x,∇u)· ∇ µ

min

µ(u−x_{n}−k)^{+}

ε ,1

¶ ξ

¶

dx dt= 0

∀ε >0, ∀k≥0, ∀ξ∈D(R^{n}×(0, T)) such that ξ≥0, ξ= 0onΣ3 .

Proof: It suffices to chooseF(z1, z2) = min(^{(z}^{1}^{−k)}_{ε} ^{+},1) in Lemma 4.1 and to
take in account (3.4).

Corollary 4.3. Let(u, g) be a solution of (P). Then we haveu∈L^{∞}(Q).

Proof: Since u≥x_{n} a.e. inQ, it suffices to prove thatu is bounded above.

So letH be a constant such that:

H≥max µ

max^{n}xn, (x^{0}, xn)∈Ω^{o}, max^{n}ψ(x, t), (x, t)∈Σ2

o¶ .

Letξ be a nonnegative function in D(0, T). Then if one apply Lemma 4.1 with
F(z_{1}, z_{2}) = (z_{1}−z_{2})^{+} andv=H−x_{n}, we get by taking in account (3.4) and the
choice ofH:

(4.5)

Z

Q

ξ(t)A^{³}x,∇(u−H)^{´}· ∇(u−H)^{+}dx dt= 0 .

Since (u−H)^{+}= 0 on Σ2, we deduce from (1.3) and (4.5) that u≤H a.e. in Q.

Theorem 4.4. Let (u, g) be a solution of (P). Then we have in the distri- butional sense:

(4.6) div^{³}A(x,∇u)−gA(x, e)^{´}+g_{t}= 0 .
Moreover, ifdiv(A(x, e))≥0in D^{0}(Ω), we have:

(4.7) div^{³}gA(x, e)^{´}−gt= div^{³}A(x,∇u)^{´}≥0 .

Proof: i) Taking ±ξ ∈D(Q) as a test function for (P), we get (4.6).

ii) Letξ ∈D(Q),ξ≥0, then from Corollary 4.2, we have forε >0 andk= 0:

(4.8)

Z

QA(x,∇u)· ∇ µ

min

µu−xn

ε ,1

¶ ξ

¶

dx dt= 0 . Note thatξ = 0 on∂Q, so

(4.9)

Z

QA(x, e)· ∇ Ãµ

1−min

µu−x_{n}
ε ,1

¶¶

ξ

!

dx dt≤0 . Adding (4.8) and (4.9), we get:

1 ε Z

Q∩[u−xn<ε]ξ^{³}A(x,∇u)− A(x,∇xn)^{´}(∇u− ∇xn)dx dt+
+

Z

Qmin

µu−xn

ε ,1

¶³

A(x,∇u)− A(x,∇x_{n})^{´}· ∇ξ dx dt ≤

≤ − Z

QA(x, e)· ∇ξ dx dt .