Definition and existence
of renormalized solutions
of elliptic equations
with general measure data
Gianni DAL MASO1
François MURAT2
Luigi ORSINA3
Alain PRIGNET4
Abstract. We introduce a new definition of solution for the nonlinear monotone
elliptic problem
−div (a(x, ∇u)) = µ in Ω,
u=0
on ∂Ω,
where µ is a Radon measure with bounded variation on Ω. We prove the existence
of such a solution, a stability result and partial uniqueness results.
Introduction
Consider the elliptic problem
−div (a(x, ∇u)) = µ in Ω,
u = 0 on ∂Ω,
(1)
where Ω is a bounded, open subset of RN , N ≥ 2, u 7→ −div (a(x, ∇u)) is a
0
monotone operator from W01,p (Ω) (p > 1) into W −1,p (Ω), and µ is a Radon
measure with bounded variation on Ω.
1
SISSA, V. Beirut 4, 34013 Trieste, Italy
Laboratoire d’Analyse Numérique, Université Paris VI, Tour 55-65, 4 Place Jussieu,
75252 Paris cedex 05, France
3
Dipartimento di Matematica, Università di Roma I, P.le A. Moro 2, 00185 Roma, Italy
4
École Normale Supérieure, 46 Allée d’Italie, 69364 Lyon cedex 07, France
2
1
When dealing with this problem, one of the main difficulties is actually
to introduce a notion of solution, and then of course to prove the existence
and uniqueness of such a solution.
In the linear case, the notion of “solution by transposition” was introduced by Stampacchia [11], who proved the existence and uniqueness of such
a solution. In the nonlinear case, the existence of a solution in the sense of
distributions was proved by Boccardo & Gallouët [2], but this solution is not
unique in general, as shown by a counterexample for the linear case due to
Serrin [10] (see [9] for a detailed version). Later, in the case were µ belongs to
the smaller class L1 (Ω), three definitions of solutions of (1) were introduced
independently: the notion of “entropy solution” [1], the notion of “solution
obtained as limit of approximations” [4], and the notion of “renormalized
solution” [8]. For each of these definitions (which can actually be proved
to be equivalent) existence, uniqueness, and continuity of the solution with
respect to the data were proved.
The present Note announces our paper [5], in which we generalize these
definitions to the case where µ is a general Radon measure with bounded
variation, and in which we prove the existence of such a solution, a stability
result, partial uniqueness results, and some other properties.
Preliminaries
Let p and p0 be real numbers, with 1 < p ≤ N , and p1 + p10 = 1. Let
a : Ω × RN → RN be a Carathéodory function (that is, a(·, ξ) is measurable
on Ω for every ξ in RN , and a(x, ·) is continuous on RN for almost every x
in Ω) which satisfies the following hypotheses
a(x, ξ) · ξ ≥ α|ξ|p − a0 (x),
(2)
|a(x, ξ)| ≤ β [b(x) + |ξ|]p−1 ,
(3)
(a(x, ξ) − a(x, ξ 0 )) · (ξ − ξ 0 ) > 0 ,
(4)
for almost every x in Ω and for every ξ and ξ 0 in RN with ξ 6= ξ 0 , where
α > 0, β > 0, a0 ∈ L1 (Ω), and b ∈ Lp (Ω), with b ≥ 0, are given.
Thanks to hypotheses (2), (3), and (4), u 7→ −div (a(x, ∇u)) is a coercive,
continuous, bounded, and strictly monotone operator defined on W01,p (Ω)
2
0
0
with values in its dual space W −1,p (Ω); therefore, for every µ in W −1,p (Ω)
there exists a unique solution of (1) which belongs to W01,p (Ω) and solves the
equation in the sense of distributions, that is
u ∈ W01,p (Ω),
div (a(x, ∇u)) = µ in D0 (Ω),
(5)
(see, for example, [7]).
We define Mb (Ω) as the set of Radon measures with bounded total vari0
ation on Ω. If p > N , the space Mb (Ω) is embedded in W −1,p (Ω): this
ensures the existence and uniqueness of a solution of (5) in that case, and
explains the restriction p ≤ N that we have imposed above.
We define M0 (Ω) as the set of the measures of Mb (Ω) which are absolutely continuous with respect to the capacity of W01,p (Ω) (which we denote
by capp (E, Ω) for any subset E of Ω); in other words, µ belongs to M0 (Ω)
if and only if µ belongs to Mb (Ω) and µ(E) = 0 for every Borel set E such
that capp (E, Ω) = 0.
We recall the following result of decomposition of measures (see [3] and [6]).
Proposition 1 For any measure µ in Mb (Ω), there exists a function f in
0
L1 (Ω), a vector g in (Lp (Ω))N , two disjoint Borel sets E + and E − of Ω, and
two non-negative measures λ+ and λ− in Mb (Ω), such that λ+ is concentrated on E + , λ− is concentrated on E − , capp (E + , Ω) = capp (E − , Ω) = 0,
and
µ = f − div (g) + λ+ − λ− .
Moreover, µ0 = f − div (g) belongs to M0 (Ω) and the decomposition of µ
into µ = µ0 + λ+ − λ− is unique.
Given a positive real number k, we define the truncation Tk : R → R by
Tk (s) = max(−k, min(k, s)) for every s in R.
Definition 2 Let u : Ω → R be a measurable function which is finite almost
everywhere on Ω. Suppose that, for every k > 0, Tk (u) belongs to W01,p (Ω).
Then there exists (see [1], [8]) a unique measurable vector-valued function
v : Ω → RN such that
∇Tk (u) = v χ{|u|≤k}
almost everywhere in Ω,
for every k > 0.
This function v is called the gradient of u and is denoted by ∇u.
3
We will use Definition 2 throughout this paper. We explicitly remark
that the gradient defined in this way is not in general the gradient in the
usual sense of distributions, since it is possible that u or ∇u are not locally
integrable; nevertheless, ∇u coincides with the usual gradient in the sense of
1,1
distributions if the function u belongs to Wloc
(Ω).
Definition of renormalized solutions
We are now in a position to define the notion of renormalized solution.
Definition 3 Let µ be a measure in Mb (Ω), which is decomposed into µ =
µ0 + λ+ − λ− as in Proposition 1. A function u defined on Ω is a renormalized
solution of problem (1) if the following two statements hold:
(i) the function u is measurable, finite almost everywhere, such that Tk (u)
belongs to W01,p (Ω) for every k > 0, and such that the function ∇u
introduced in Definition 2 satisfies
|∇u|p−1 belongs to Lq (Ω), for every q <
N
;
N −1
(6)
(ii) if w belongs to W01,p (Ω) ∩ L∞ (Ω) and if there exist k > 0 and w+∞ ,
w−∞ which belong to W 1,r (Ω) ∩ L∞ (Ω), for some r > N , such that
(
w = w+∞ almost everywhere on the set {x : u(x) > k},
w = w−∞ almost everywhere on the set {x : u(x) < −k},
(7)
then
Z
a(x, ∇u) · ∇w dx =
Ω
Z
Ω
w dµ0 +
Z
w+∞ dλ+ −
Ω
Z
w−∞ dλ− .
(8)
Ω
Other definitions, which turn out to be equivalent to Definition 3, are
given in [5].
Observe that every term in (8) has a meaning. Indeed, the integral in the
left hand side can be written as
Z
{u<−k}
a(x, ∇u)·∇w dx +
Z
a(x, ∇u)·∇w dx +
{|u|≤k}
Z
{u>k}
4
a(x, ∇u)·∇w dx,
where all three terms are well defined: actually, (6) and (3) imply that
a(x, ∇u) belongs to (Lq (Ω))N , for every q < NN−1 , so that the first and the
third term are finite in view of (7); moreover, as a(x, ∇u) = a(x, ∇Tk (u))
almost everywhere on the set {x : |u(x)| ≤ k}, the second term is well defined, since w and Tk (u) belong to W01,p (Ω) and, consequently, a(x, ∇Tk (u))
0
belongs to (Lp (Ω))N by (3). Similarly, each term in the right hand side of
(8) is well defined since w belongs to L1 (Ω, µ0 ), because W01,p (Ω) ∩ L∞ (Ω) is
embedded in L1 (Ω, µ0 ) for every µ0 of M0 (Ω), and since w+∞ and w−∞ are
continuous and bounded on Ω.
We remark that it is possible to choose any function of Cc1 (Ω) as test
function in (8), which implies that u is a solution of (1) in the sense of
distributions. But there are much more test functions which are admissible
in (8), such as w = Tj (u − ϕ), with j > 0 and ϕ in W01,p (Ω) ∩ L∞ (Ω), so
that u is an entropy solution of (1) in a sense similar to the sense of [1], or
w = h(u) ϕ, with h ∈ W 1,∞ (R), supp (h0 ) compact in R, and ϕ in Cc1 (Ω),
which implies that u is a renormalized solution of (1) in a sense similar to
the sense of [8].
Note finally that if u is a renormalized solution of (1), the function
u is not defined at every point x ∈ Ω, but only capp -quasi everywhere.
Nevertheless, equation (8) strongly suggests that u = +∞ on the set E +
where λ+ is concentrated, and that u = −∞ on the set E − where λ−
is concentrated, even if these statements have no precise meaning since
capp (E + , Ω) = capp (E − , Ω) = 0.
Existence and stability results
One of our main results is the following result of existence.
Theorem 4 For any measure µ in Mb (Ω), there exists at least a renormalized solution of (1).
In order to prove Theorem 4, we first decompose µ into µ = f − div (g) +
λ − λ− as in Proposition 1, and we approximate µ by a sequence µε =
⊕
∞
fε −div (gε )+λ⊕
ε −λε , where fε , gε , λε , and λε are functions of C (Ω) which
satisfy hypotheses (11) to (14) below (it is easy to obtain such approximations
+
5
by regularizing f, g, λ+ , and λ− ). We then observe that the unique solution
of the approximate problem
uε ∈ W01,p (Ω),
−div (a(x, ∇uε )) = fε − div (gε ) + λ⊕
ε − λε
in D0 (Ω), (9)
is actually a renormalized solution of (1) with right hand side µε = fε −
div (gε ) + λ⊕
ε − λε .
The existence result of Theorem 4 is then a consequence of the following
stability result, which states the strong convergence of the truncations of uε .
Theorem 5 Let fε , gε , λ⊕
ε , and λε be sequences which satisfy
0
N
fε ∈ L1 (Ω), gε ∈ (Lp (Ω)) , λ⊕
ε ∈ Mb (Ω), λε ∈ Mb (Ω),
(10)
fε converges to f weakly in L1 (Ω),
(11)
p0
gε converges to g strongly in (L (Ω))N ,
(12)
+
λ⊕
ε is non negative and converges to λ tightly in Mb (Ω),
(13)
−
λ
ε is non negative and converges to λ tightly in Mb (Ω),
(14)
and let uε be a renormalized solution of (1) with right hand side µε = fε −
div (gε ) + λ⊕
ε − λε . Then there exists a subsequence, still denoted by ε, and
a function u, which is a renormalized solution of (1) with right hand side
µ = f − div (g) + λ+ − λ− , such that
Tk (uε ) → Tk (u) strongly in W01,p (Ω),
for every k > 0.
Let us emphasize the fact that in hypotheses (13) and (14) the two se
quences λ⊕
ε and λε are assumed to be non-negative: if these hypotheses do
not hold, the result of Theorem 5 can fail. In contrast, λ⊕
ε and λε are not
assumed to be the positive and negative part of a given measure λε ; for this
+
−
reason we used notation λ⊕
ε and λε and not notation λε andλε .
Further properties and uniqueness
The next result states the link between the measures λ+ and λ− and the
density of energy of a renormalized solution u on the set where u is very
large.
6
Theorem 6 Let µ be a measure in Mb (Ω), which is decomposed into µ =
µ0 + λ+ − λ− as in Proposition 1. Let u be a renormalized solution of (1).
For every ϕ in C 0 (Ω) we have:
Z
1 Z
lim
a(x, ∇u) · ∇u ϕ dx = ϕ dλ+ ,
n→+∞ n {n≤u<2n}
Ω
Z
1 Z
a(x, ∇u) · ∇u ϕ dx = ϕ dλ− .
lim
n→+∞ n {−2n<u≤−n}
Ω
We conclude this Note with two results concerning uniqueness.
Theorem 7 Further to (2), (3), and (4), assume that a is strongly monotone,
and that it is locally Lipschitz continuous, if p ≥ 2, or Hölder continuous,
if p ≤ 2. Let u and û be two renormalized solutions of (1), with the same
measure µ of Mb (Ω) as right hand side. If one of the following two conditions
holds
1 Z
|∇Tn (u − û)|p dx = 0 or u − û ∈ L∞ (Ω),
lim
n→+∞ n Ω
then u = û.
Both conditions in Theorem 7 are unfortunately restrictive, but provide
uniqueness results in classes of “comparable” solutions. Other restrictive
conditions given in [5] allow one to recover the uniqueness results proved in
[1] and [8] when µ belongs to L1 (Ω).
References
[1] P. Benilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre,
J.L. Vazquez, An L1 theory of existence and uniqueness of nonlinear
elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995),
241–273.
[2] L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right hand
side measures, Comm. Partial Differential Equations, 17 (1992), 641–
655.
[3] L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann.
Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539–551.
7
[4] A. Dall’Aglio, Approximated solutions of equations with L1 data. Application to the H-convergence of parabolic quasi-linear equations, Ann.
Mat. Pura Appl., 170 (1996), 207–240.
[5] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions
of elliptic equations with general measure data, in preparation.
[6] M. Fukushima, K. Sato, S. Taniguchi, On the closable part of preDirichlet forms and the fine support of the underlying measures, Osaka
J. Math., 28 (1991), 517–535.
[7] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites
non linéaires, Dunod et Gauthier-Villars, Paris (1969).
[8] P.-L. Lions, F. Murat, Solutions renormalisées d’équations elliptiques
non linéaires, to appear.
[9] A. Prignet, Remarks on existence and uniqueness of solutions of elliptic
problems with right-hand side measures, Rend. Mat., 15 (1995), 321–
337.
[10] J. Serrin, Pathological solutions of elliptic differential equations, Ann.
Scuola Norm. Sup. Pisa Cl. Sci, 18 (1964), 385–387.
[11] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques
du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble),
15 (1965), 189–258.
8
Définition et existence
de solutions renormalisées
d’équations elliptiques
dont le second membre est une mesure
Résumé. Nous introduisons une nouvelle définition de solution pour le problème
elliptique non linéaire monotone
−div (a(x, ∇u) = µ dans Ω,
u=0
sur ∂Ω,
où µ est une mesure de Radon sur Ω dont la variation totale est bornée. Nous montrons l’existence d’une solution satisfaisant cette nouvelle définition, un résultat
de stabilité et des résultats partiels d’unicité.
Version française abrégée
Considérons le problème elliptique
−div (a(x, ∇u)) = µ dans Ω,
u = 0 sur ∂Ω,
(1)
où Ω est un ouvert borné de RN , N ≥ 2, u 7→ −div (a(x, ∇u)) est un
0
opérateur monotone de W01,p (Ω) (1 < p ≤ N ) dans W −1,p (Ω), et µ est une
mesure de de Radon sur Ω dont la variation totale est bornée.
Des notions de solution du problème (1), qui permettent de démontrer
l’existence et l’unicité d’une solution, ont été données récemment dans [1], [4]
et [8] quand µ est une fonction de L1 (Ω). Dans cette Note, nous annonçons
les résultats de notre article [5], dans lequel nous introduisons une notion
de solution de (1) qui généralise ces définitions dans le cas où µ est une
mesure de Radon générale dont la variation est bornée. Nous y démontrons
l’existence d’une solution satisfaisant cette nouvelle définition, un résultat de
stabilité, des résultats d’unicité et quelques autres propriétés.
Soit Mb (Ω) l’ensemble des mesures de Radon sur Ω dont la variation
totale est bornée, et soit M0 (Ω) l’ensemble des mesures de Mb (Ω) qui sont
9
absolument continues par rapport à la capacité de W01,p (Ω) ; en d’autres
termes, µ appartient à M0 (Ω) si et seulement si µ appartient à Mb (Ω) et
µ(E) = 0 pour tout borélien E tel que capp (E, Ω) = 0.
Rappelons un résultat de décomposition des mesures (voir [3] et [6]).
Proposition 1 Étant donnée une mesure µ de Mb (Ω), il existe f ∈ L1 (Ω),
0
g ∈ (Lp (Ω))N , deux boréliens disjoints E + et E − de Ω et deux mesures
positives λ+ et λ− de Mb (Ω), avec λ+ concentrée sur E + , λ− concentrée sur
E − et capp (E + , Ω) = capp (E − , Ω) = 0, tels que
µ = f − div (g) + λ+ − λ− .
De plus, µ0 = f − div (g) appartient à M0 (Ω), et la décomposition µ =
µ0 + λ+ − λ− est unique.
Soit Tk (s) = max(−k, min(k, s)) pour s réel, et k > 0. Nous pouvons
maintenant introduire notre définition de solution renormalisée du problème
(1).
Définition 3 Soit µ une mesure de Mb (Ω), qui est décomposée en µ =
µ0 + λ+ − λ− grâce à la Proposition 1. Une fonction u définie sur Ω est une
solution renormalisée du problème (1) si les deux assertions suivantes sont
vérifiées :
(i) la fonction u est mesurable, presque partout finie, telle que Tk (u) appartienne à W01,p (Ω) pour tout k > 0, et telle que ∇u, que nous
définissons ici comme égal à ∇Tk (u) presque partout sur l’ensemble
{x : |u(x)| ≤ k}, vérifie |∇u|p−1 ∈ Lq (Ω) pour tout q < NN−1 ;
(ii) pour toute fonction w ∈ W01,p (Ω) ∩ L∞ (Ω), telle qu’il existe k > 0 et
w+∞ et w−∞ qui appartiennent à W 1,r (Ω) pour un certain r > N , avec
w = w+∞ presque partout sur l’ensemble {x : u(x) > k}, et w = w−∞
presque partout sur l’ensemble {x : u(x) < −k}, on a
Z
Ω
a(x, ∇u) · ∇w dx =
Z
Ω
w dµ0 +
10
Z
Ω
w+∞ dλ+ −
Z
Ω
w−∞ dλ− .
Nos résultats principaux sont, outre des résultats partiels d’unicité (voir
Théorème 7 de la version anglaise), un théorème d’existence et un théorème
de stabilité.
Théorème 4 Pour toute mesure µ de Mb (Ω), il existe au moins une solution
renormalisée de (1).
Théorème 5 Soient fε , gε , λ⊕
ε et λε des suites qui vérifient (10) à (14),
et soit uε une solution renormalisée de (1) pour le second membre µε =
fε − div (gε ) + λ⊕
ε − λε . Alors il existe une sous suite de ε et une solution
renormalisée u de (1) pour le second membre µ = f − div (g) + λ+ − λ− , telles
que
Tk (uε ) → Tk (u) fortement dans W01,p (Ω),
11
pour chaque k > 0.