QUASI–LINEAR DEGENERATE ELLIPTIC PROBLEMS WITH L1 DATA

´sir and Olivier Guibe ´ Dominique Blanchard(1), Franc ¸ ois De Laboratoire de Math´ematiques Rapha¨el Salem UMR CNRS 6085, Site Colbert Universit´e de Rouen F-76821 Mont Saint Aignan cedex E-mail : [email protected] [email protected] [email protected] Abstract. We prove existence and uniqueness of a solution for a class of quasi–linear problems with L1 data. The diffusion matrix A(x, u) is allowed to degenerate with respect to the unknown u. We obtain uniqueness of the solution under a weak assumption on A(x, u) that permits to consider highly oscillating or/and increasing coefficients (with respect to u).

1. Introduction In this paper we study a class of possibly degenerate elliptic problems of the type (1.1)

− div(A(x, u)Du) = f

(1.2)

u=0

in Ω, on ∂Ω,

where Ω is a bounded domain of RN (N ≥ 1), f ∈ L1 (Ω) and A(x, s) : Ω × R → RN ×N is a Carath´eodory function with values in the space of matrices on R. We assume that there exists a continuous function α : R → R+ such that A(x, s) ≥ α(s)I for any s ∈ R and almost any x in Ω, so that equation (1.1) may degenerate on the subset {x ∈ Ω ; α(u(x)) = 0} of Ω. Moreover the function α(s) may vanish when |s| → +∞ and equation (1.1) may not be uniformly elliptic for large values of |s|. With respect to the “extensive” literature devoted to degenerate elliptic problems that ranges from “porous medium” equations (see e.g. the references in [16] and [17]) to problems with degeneracy at infinity (see e.g. [1], [6], [11] and [15] among the most recent papers), the originality of the present paper, as far as existence results are concerned, is to consider a diffusion matrix A(x, s) (which is even not assumed to be symmetric). In particular, this precludes any change of unknown in equation (1.1) as in the estimates that follows from (1.1) in order to maintain both coercivity and continuity of the diffusion matrix. Moreover we just assume that the data f belongs to L1 (Ω). In the case where the matrix A(x, s) is uniformly coercive (i.e. if α(s) ≥ α0 > 0 ∀s ∈ R), one can prove existence of a solution of (1.1)–(1.2), without any growth condition on Date: May 10, 2011. Key words and phrases. existence, uniqueness, degenerate elliptic problems, integrable data. (1)Laboratoire Jacques-Louis Lions, Universit´ e Paris VI, Boˆıte courrier 187, 75252 Paris cedex 05

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´ ´ D. BLANCHARD, F. DESIR, O. GUIBE

A(x, s) with respect to s using the notion of renormalized solution (see e.g. [14] and [18]). Denoting by TK (K ≥ 0) the truncation at height K, the uniform coercivity of A(x, s) leads
N to the estimate DTK (u) ∈ L2 (Ω) . As a consequence the field A(x, u)DTK (u) belongs
N to L2 (Ω) whatever the growth of A(x, s) may be. This is enough to give a sense to a renormalized solution of (1.1)–(1.2). The same arguments hold true if α(s) is strictly positive on R but may tend to 0 as |s| R +∞ R0 tends to infinity provided that 0 α(s) ds = −∞ α(s) ds = +∞ which is a condition that insures that a solution u is finite almost everywhere in Ω. As an example, one can prove existence solution of (1.1)–(1.2) for α(t) = 1/(1 + |t|)m for m ≤ 1 (and A(x, s) ≥ α(s)I). In the case where the function α may vanish on R, one cannot expect that DTK (u) belongs


2 (Ω) N but one only has, for any K ≥ 0, DT 2 (Ω) N where α to L α e (u) ∈ L e(s) = K
Rs e(u) in (1.1)). This is enough 0 α(t) dt (this could be seen by plugging the test function TK α to define De α(u) almost everywhere in Ω (see [4]) and then to define Du almost everywhere α e(u) almost everywhere on the subset {x ∈ Ω ; α(u)(x) 6= in Ω through setting Du = Dα(u) 0} = Ω0 and e.g. Du = 0 almost everywhere on Ω \ Ω0 . By contrast with the coercive

N case, the condition DTK α e(u) ∈ L2 (Ω) is not sufficient, in general, to give a sense to a renormalized formulation of (1.1) since there is no reason for A(x, u)DTK (u) to belong
N to L2 (Ω) . An assumption that combines both the structure of the matrix A(x, s) and the growth of its coefficients with respect to s is needed. We introduce in this paper such a condition, namely for any K ≥ 0 there exists a constant AK such that |A(x, s)| ≤ AK and |A(x, s)ξ|2 ≤ AK A(x, s)ξ · ξ for any |s| ≤ K and any ξ ∈ RN , almost everywhere in Ω (see (2.5) and (2.6) in Section 2). Assumption (2.5) is classical. Remark that (2.6) is not stricto sensu a condition on the growth of A(x, s) with respect to s : in the usual symmetric case it is always satisfied (whatever the growth of A(x, s) with respect to s may be!). Indeed if A(x, s) is a symmetric matrix then |A(x, s)ξ|2 ≤ |A(x, s)||A1/2 (x, s)ξ|2 ≤ AK A(x, s)ξ · ξ where A1/2 (x, s) denotes the unique symmetric matrix such that A1/2 (x, s)A1/2 (x, s) = A(x, s). As a consequence of Theorem 2.3, we obtain an existence result in the degenerate and symmetric case without any growth assumption on A(x, s) (with respect to s). In the general case (i.e. for non symmetric matrices) and loosely speaking, Assumption 2.6 is concerned with the growth of the antisymmetric part of A(x, s) with respect to the symmetric part of A(x, s) (see the comments on the assumptions in Subsection 2.1). We now turn to the uniqueness problem for equation of the type (1.1). In the non degenerate case and as far as variational solutions are concerned (i.e. for f ∈ H −1 (Ω)), the most significant results in this direction can be found in [2], [9] [12], [13], where, in short, the authors assume that the matrix A(x, s) is Lipschitz continuous with respect to s or at least exhibit a strongly controlled modulus of continuity (in [9] an extra term of the form div(φ(x, u)) is involved in (1.1) with φ(x, s) Lipschitz continuous with respect to s). For merely integrable data f (which is the case in the present paper), for non degenerate problems and if a 0–order term is involved in (1.1), namely if the operator is of the form λu − div(A(x, u)Du) with λ > 0, then one can prove uniqueness of the renormalized solution under a local Lipschitz continuity assumption on A(x, s) with respect to s adapting e.g. the techniques developed in [3], [18] and [20] or [5] for parabolic nonlinear version of (1.1)–(1.2). In the degenerate case (i.e. when α may vanish) a 0–order term λu, with λ > 0, permits to obtain the uniqueness of the field α e(u) which, in turn, implies the uniqueness of u. Indeed such results still hold true for a 0-order term β(u) where β is a strictly monotone function.

QUASI–LINEAR DEGENERATE ELLIPTIC PROBLEMS WITH L1 DATA

3

With respect to the results mentioned above, a natural question arises: does the uniqueness of u (in the non degenerate case) or of α e(u) (in the degenerate case) still hold true under local Lipschitz continuity of A(x, s) with respect to s for Problems of the type (1.1)–(1.2) (i.e. without any 0-order term) ? We do not answer this question in the present paper so that the problem still remains open. Actually we prove an uniqueness result under a fairly technical assumption A(x, s) (see (3.2)) which is a global condition on R but which allows both strong growth of A(x, s) and on its modulus of continuity with respect to s. As an

example we obtain uniqueness of the field α e(u) for A(x, s) = 1 + b(x) exp(s) sin2 exp(s2 ) B(x) where b is a non negative function belonging to L∞ (Ω) and B is a coercive and symmetric matrix lying in (L∞ (Ω))N ×N . Similar conditions have been introduced independently in the recent paper [19] to prove uniqueness of the solution in the very close framework of entropy solution when the function α(s) > 0 for any s ∈ R. Our assumption (3.2) and the one used in [19], even if similar, are not equivalent. Actually in the coercive case assumption stated in [19] forces both A(x, s) and its modulus of continuity to grow at most like exp(c|s|) while it is not the case with (3.5) (see the example above). The paper is organized as follows. In Section 2 we detail the assumptions that hold true in the whole paper and we give the definition of a renormalized solution of (1.1)–(1.2). Then we prove existence of a such a solution (Theorem 2.3). This section is completed by a few estimates on renormalized solutions. Section 3 is concerned with the uniqueness of a renormalized solution of (1.1)–(1.2) under the additional assumptions mentioned above. 2. Existence result This section is organized as follows. In subsection 2.1 we detail the assumptions on A(x, s). In subsection 2.2 we prove an existence result (Theorem 2.3) and in 2.3 we derive a few estimates on renormalized solutions. 2.1. Assumptions and notations. In the whole paper we assume that (2.1)

f ∈ L1 (Ω);

A(x, s) : Ω × R → RN ×N is a Carath´eodory function such that there exists a non negative continuous function α : R → R+ with Z 0 Z +∞ (2.2) α(s) ds = α(s) ds = +∞, −∞

0

(2.3)

A(x, s)ξ · ξ ≥ α(s)|ξ|2 ,

∀s ∈ R, ∀ξ ∈ RN , almost everywhere in Ω;

(2.4)

α(s) = 0 implies that A(x, s) = 0 almost everywhere in Ω.

Moreover, we assume that for any K ≥ 0, there exists a constant AK such that (2.5) A(x, s) ≤ AK almost everywhere in Ω, ∀s ∈ R such that |s| ≤ K, and (2.6)

A(x, s)ξ 2 ≤ AK A(x, s)ξ · ξ

almost everywhere in Ω, ∀s ∈ R such that |s| ≤ K.

A few comments on assumptions (2.3), (2.4), (2.5) and (2.6) follow. Conditions (2.3) and (2.4) say that the matrix A(x, s) degenerates if and only if α(s) = 0. As mentioned in the introduction, assumption (2.6) is not a condition that concerns only the growth of A(x, s)

´ ´ D. BLANCHARD, F. DESIR, O. GUIBE

4

with respect to s but rather a coupling hypothesis between the structure of A(x, s) and the growth of its coefficients: (2.6) is satisfied as soon as A(x, s) is symmetric (see again the introduction). As far as the non symmetric case is concerned, (2.6) is also always satisfied if A(x, s) does not degenerate (i.e. if α(s) > 0 for any s ∈ R). It only remains to examine the case where A(x, s) is non symmetric and vanishes. Conditions (2.6) appears to be a constraint on the growth of the antisymmetric part of A(x, s) with respect to that of the symmetric part. Since one can always assume that A(x, s) = Λ(x, s) + C(x, s) where Λ(x, s) is a diagonal matrix with Λ(x, s)ξ · ξ ≥ α(s)|ξ|2 and where C(x, s) is a antisymmetric matrix, assumption (2.6) may be rewritten as 2 Λ (x, s)|ξ|2 − C2 (x, s)ξ · ξ ≤ AK Λ(x, s)ξ · ξ a.e. in Ω, ∀|s| ≤ K. The following notations will be used throughout the paper. For any positive real number K, we denote by TK (r) the truncation function at height K, TK (r) = min(−K, max(r, K)). For any integer n ≥ 1, let us define the bounded positive functions (2.7)

θn (s) =


1 T2n (s) − Tn (s) , n

hn (s) = 1 − θn (s) .

For any positive real number a and any measurable function v on Ω, we denote by {|v| < a} (resp. {|v| ≤ a}) the measurable subset of Ω defined by {x ∈ Ω ; |v(x)| < a} (resp. {x ∈ Ω ; |v(x)| ≤ a}). Moreover 1lE will be used to shorten the notation 1l{x∈E} for the Rr characteristic function of the subset E. At last we set α e(r) = 0 α(s) ds which is a C 1 non decreasing function on R. 2.2. Definition of a renormalized solution to (1.1)–(1.2). As mentioned in the introduction, we use the framework of renormalized solution to solve (1.1)–(1.2). In order to give the definition of such a solution in our setting a few preliminaries are necessary. Let us first recall that since f belongs to L1 (Ω), even formal a priori estimates are obtained by using bounded test functions in (1.1). As usual the simplest one is TK (u) 2 and, using (2.3), it leads to α(u) DTK (u) ∈ L1 (Ω) for any K ≥ 0. Then (and still formally) one can plug TK (e α(u)) as a test function and obtain α(u)Du · DTK (e α(u)) ∈ L1 (Ω) for any K ≥ 0. Indeed, as usual when degenerate equations are involved, none of the two estimates allows to show that u belongs to a Sobolev space (or even a Lebesgue space) because α(u) vanishes on the subset {x ∈ Ω α(u(x)) = 0}. By contrast, the second estimate that may be formally rewritten TK (e α(u)) ∈ H01 (Ω) for any K ≥ 0, allows us to give a sense to De α(u) almost everywhere in Ω as shown in [4]. Therefore we are in a position to define the field Du on Ω through the formula ( Dα e(u) if α(u)(x) 6= 0, α(u) (x) Du(x) = (2.8) 0 if α(u)(x) = 0. This is the definition of Du that will be used throughout the paper (see the comments after Definition 2.1). A renormalized solution of (1.1)–(1.2) is proposed below.

QUASI–LINEAR DEGENERATE ELLIPTIC PROBLEMS WITH L1 DATA

5

Definition 2.1. A measurable function u : Ω −→ R (u is finite almost everywhere in Ω) is a renormalized solution of (1.1)–(1.2) if
(2.9) TK α e(u) ∈ H01 (Ω), for any K ≥ 0, (2.10) (2.11)

1l{|eα(u)| 0) and assume that there exists w ∈ C 1 (R) such that Z t w(z) dz (A.1) A(x, s) − A(x, t) ≤ s

(A.2)

w≥0

(A.3)

|w0 | ≤ C2 w1+η ,

where C2 > 0 and η > 0. Then the matrix A(x, s) verifies condition (3.2). Sketch of proof. The proof relies on the construction of the function ϕ using standard real analysis. We leave some details to the reader. Since α(s) = α0 we can assume that α0 = 1 and then α e(t) = t, ∀t ∈ R. Step 1. For any n ≥ 2, let us define the function gn by Z t n
0 0 (A.4) ∀t ≥ 0, gn (t) = 1 + |w (z)| + |w (−z)| dz + w(0) + 1 . 0

From (A.1)–(A.3) it follows that g ∈ C 1 (R+ ) and 
 1 ≤ g 0 (t) ≤ (C2 + 1)n gn (t) 1+η/n , n ∀t ≥ 0, (A.5)
1+1/(n−1)  1 ≤ gn (t) ≤ gn0 (t) , Z t (A.6) ∀t, s ∈ R, A(x, s) − A(x, t) ≤ gn (|z|) dz . s

Let us emphasize that we can choose n such that η/n and 1/(n − 1) are small enough. Step 2. Let µ > 0 and from Step 1 let ψ ∈ C 1 (R+ ) such that (
1+µ 1 ≤ ψ 0 (t) ≤ M1 ψ(t) , ∀t ≥ 0, (A.7)
1+µ 1 ≤ ψ(t) ≤ ψ 0 (t) , where M1 > 0 and (A.8)

∀t, s ∈ R,

Z t A(x, s) − A(x, t) ≤ ψ(|z|) dz . s


3 e = ψ(z) dz, ∀t ≥ 0 and ϕ(t) = ψ(|t|) e + 1) − 1 sign(t), ∀t ∈ R. Let us define ψ(t) 0 We now claim that if µ is chosen small enough then A(x, s) verifies (3.2) with the function ϕ. Rt



´ ´ D. BLANCHARD, F. DESIR, O. GUIBE

26

e The regularity of ψ implies that ϕ ∈ C 1 (R) ∩ We first derive some properties on ϕ and ψ. and
2 e ∀t ∈ R, ϕ0 (t) = 3ψ(|t|) ψ(|t|) + 1 ≥ 1,

e e + 1) sign(t). ∀t 6= 0, ϕ00 (t) = 3 ψ(|t|) + 1 2ψ 2 (t) + ψ 0 (t) ψ(t)

C 2 (R∗ )

From (A.7) and since ψ is an increasing function it follows that ∀t > 0,  Z t
1+µ

2 e e ψ 0 (z) dz + ψ(0) + 3 ψ(t) 1 ≤ ϕ00 (t) ≤ 6 ψ(|t|) + 1 ψ(t) M1 ψ(|t|) +1 (A.9) 0 Z t  
1+µ
µ e ψ(z) dz + ψ(0) + 3M1 ϕ0 (t) ≤ 6 ψ(t) + 1 ψ(t) M1 ψ (t) 0


1+µ


e + 1 ψ(t) e + ψ(0) + 3M1 ϕ0 (t) 1+µ ≤ 6 ψ(t) M1 ψ(t)
1+µ ≤ M2 ϕ0 (t) , where M2 is a positive constant depending on M1 and ψ(0). Moreover inequalities (A.7) give also that for any t > 0
e +1 , (A.10) ψ(t)1−µ ≤ M3 ψ(t) 2
e ≤ M4 ψ(t) 1+µ+µ , (A.11) ψ(t) where M3 and M4 are positive constants. Let 0 < K ≤ K0 (K0 will be chosen later) and r, s be two real numbers such that ϕ(s) − ϕ(r) ≤ K. To prove condition (3.2) let us write ϕ0 (r) − ϕ0 (s) A(x, s) A(x, r) 1 (A.12) ϕ0 (s) − ϕ0 (r) ≤ ϕ0 (s) A(x, s) − A(x, r)| + A(x, r) ϕ0 (r)ϕ0 (s) . We first deal with the case rs > 0. Without loss of generality we assume that 0 < s < r (the case r < s < 0 is obtained by symmetry). From (A.8) and (A.12) we have R Z r sr ϕ00 (z) dz A(x, s) A(x, r) 1 ψ(z) dz + A(x, r) 0 ϕ0 (s) − ϕ0 (r) ≤ ϕ0 (r) ϕ (r)ϕ0 (s) s Gathering (A.7)–(A.11) together with standard analysis arguments and a calculus lead to Z Z r A(x, s) A(x, r) 1 M2 ϕ0 (r)µ r 0 0 ϕ (z) dz ϕ (z) dz + |A(x, r)| 0 ϕ0 (s) − ϕ0 (r) ≤ 2 s e ϕ (r)ϕ0 (s) s 3ϕ0 (r)(1 + ψ(s))
K M2 K e ∞ (Ω) + ψ(r) ≤ + kA(x, 0)k L ϕ0 (r)1−µ ϕ0 (s) 3ϕ0 (r)(1 + ϕ(s))2/3 M5 K M6 K ≤ 0 +
1−2µ 2/3 e ϕ (r)(1 + ϕ(r) + ϕ(s)) ψ(r)1−µ 1 + ψ(r) ϕ0 (s) ≤ ≤

ϕ0 (r)(1

M5 K + + ϕ(r) + ϕ(s))2/3

ϕ0 (r)(1

M5 K + + ϕ(r) + ϕ(s))2/3

M7 K 2µ+µ2

2−2µ− e 1+µ+µ2 ϕ0 (s) (1 + ψ(r)) M8 K

(1 + ϕ(r) + ϕ(s))

1 3

2−2µ−

2µ+µ2 1+µ+µ2




. ϕ0 (s)

QUASI–LINEAR DEGENERATE ELLIPTIC PROBLEMS WITH L1 DATA

27

We are now in a position to choose µ > 0 such that 2µ + µ2
2 1 def 1 2 − 2µ − < . 0 such that M2 K0 < 1/2. Then we have Z r
µ
µ 1 0 0 (A.14) 0 ≤ ϕ (r) − ϕ (s) ≤ ϕ00 (z) dz ≤ M2 ϕ0 (r) K ≤ M2 K0 ϕ0 (r) ≤ ϕ0 (r). 2 s At last, inequalities (A.13) and (A.14) yield that A(x, s) A(x, r) M9 K (A.15) ϕ0 (s) − ϕ0 (r) ≤ (ϕ0 (r))1/2 (ϕ0 (r))1/2 (1 + ϕ(r) + ϕ(s))δ , where M9 is a positive constant independent of t, s and K, that is (3.2) when rs > 0 When rs ≤ 0, since ϕ0 ≥ 1 we have |r| ≤ K ≤ K0 and |s| ≤ K ≤ K0 . Since ϕ0 ∈ C 1 (R∗ ) ∩ C(R), (A.8) and (A.12) give that there exists M10 > 0 (independent of r, s and K) such that A(x, s) A(x, r) M10 K ≤ (A.16) −
, ϕ0 (s) ϕ0 (r) ϕ0 (s)1/2 ϕ0 (r)1/2 1 + |ϕ(s)| + |ϕ(r)| δ that is (3.2) for rs ≤ 0. The proof of Lemma A.1 is complete.



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