Definition 4.1. Viscosity sub/super/solution for the non-local eikonal equation)
2. Assumptions and preliminary results
Givenpandp two real numbers, withp >1 and 1p+p1 = 1, leta: Ω×IRN →IRN be a Carath´eodory function such that for almost everyxin Ω and for everyξ, η in IRN (ξ=η):
|a(x, ξ)| ≤C0[k0(x) +|ξ|p−1], (2.1) a(x, ξ)ξ≥C1|ξ|p−k1(x), (2.2) (a(x, ξ)−a(x, η))(ξ−η)>0, (2.3)
a(x,0) = 0, (2.4)
whereC0 andC1 are two positive real constants,k0 is a nonnegative function in Lp(Ω) andk1 is a nonnegative function in L1(Ω). Thanks to hypotheses (2.1)–
(2.4) the operatorA:u→ −div(a(x,∇u)) mapsW01,p(Ω) into its dualW−1,p(Ω) and for every F ∈ W−1,p(Ω) there exists a unique function u ∈ W01,p(Ω) such
that
A(u) =F in Ω
u= 0 on∂Ω, (2.5)
in the weak sense (see, e.g., [25]).
Let Mb(Ω) the space of Radon measures à on Ω whose total variation |à| is bounded on Ω. As usual we identifyMb(Ω) with the dual of the Banach space C0(Ω) of continuous functions that are zero on the boundary, so that the duality isà, u=
Ωu dà, for everyuin C0(Ω) and the norm isàMb(Ω)=|à|(Ω).
In order to include in our analysis also the case of thin obstacles, it is con- venient to introduce the notion of p-capacity. Given a compact set K ⊆ Ω, its p-capacity with respect to Ω is given by
Cp(K) = inf
Ω
|∇z|2dx:z∈C0∞(Ω), z≥χK
,
whereχK is the characteristic function ofK. This definition can be extended to any open subsetAof Ω in the following way:
Cp(A) = sup{Cp(K) :Kcompact,K⊆A}. Finally, it is possible to define thep-capacity of any set B⊆Ω as:
Cp(B) = inf{Cp(A) :Aopen,B⊆A}.
A property holdsCp-quasi everywhere (abbreviated asCp-q.e.) when it holds up to sets ofp-capacity zero.
A functionv : Ω→IR isCp-quasi continuous (resp.Cp-quasi upper semicon- tinuous) if, for everyε >0 there exists a setE such that Cp(E)< ε andv|Ω\E is continuous (resp. upper semicontinuous) in Ω\E. We recall also that ifuandvare Cp-quasi continuous andu≤v a.e. in Ω then alsou≤v Cp-q.e. in Ω. A function u∈ W01,p(Ω) always has a Cp-quasi continuous representative, which is uniquely defined (and finite) up to a set ofp-capacity zero. In the sequel we shall always
Obstacle Problems for Monotone Operators with Measure Data 295 identifyuwith itsCp-quasi continuous representative, so that the pointwise values ofuare defined Cp-quasi everywhere.
In the sequelMb,0p (Ω) will be the special subspace ofMb(Ω) of all measures which vanish on all sets ofp-capacity zero. Moreover, we denote the positive cones ofMb(Ω) andMb,0p (Ω) byMb+(Ω) andMb,0p,+(Ω), respectively.
It is well known that, ifàbelongs toW−1,p(Ω)∩Mb(Ω), thenàis inMb,0p (Ω), everyuinW01,p(Ω)∩L∞(Ω) is summable with respect toàand
à, u=
Ω
u dà,
whereã,ãdenotes the duality pairing betweenW−1,p(Ω) andW01,p(Ω), while in the right hand sideudenotes theCp-quasi continuous representative and, conse- quently, the pointwise values ofuare definedà-almost everywhere.
A self-contained presentation of all these notions can be found, for instance, in [19].
Let us recall here the following result, that is the analogous of the Lebesgue decomposition theorem and can be proved in the same way (see Lemma 2.1 in [18]).
Proposition 2.1. For every measure à∈Mb(Ω) there exists a unique pair of mea- sures(àa, às), withàa ∈Mb,0p (Ω)andàs concentrated on a set ofp-capacity zero, andà=àa+às. If àis nonnegative, so are àa andàs.
Let us fix a functionψ: Ω→IR, and the corresponding convex set Kψ(Ω) :={z Cp−quasi continuous in Ω : z≥ψ Cp−q.e. in Ω}. For everyj >0 we define the truncation functionTj: IR→IR by
Tj(t) =
t if|t| ≤j jsign(t) if|t|> j.
Let us consider the space T01,p(Ω) of all functions u: Ω → IR which are almost everywhere finite and such that Tj(u) ∈ W01,p(Ω) for every j > 0. It is easy to see that every functionu∈T01,p(Ω) has aCp-quasi continuous representative with values in IR, that will always be identified with the function u. Moreover, for every u ∈ T01,p(Ω) there exists a measurable function Φ : Ω → IRN such that
∇Tj(u) = Φχ{|u|≤j} a.e. in Ω (see Lemma 2.1 in [2]). This function Φ, which is unique up to almost everywhere equivalence, will be denoted by ∇u. Note that
∇ucoincides with the distributional gradient ofuwheneveru∈T01,p(Ω)∩L1loc(Ω) and∇u∈L1loc(Ω,IRN).
In order to study the elliptic problem A(u) =à in Ω,
u= 0 on∂Ω (2.6)
whenàis a bounded Radon measure, we cannot use the variational formulation , since, in general, the termà, vhas not always meaning whenàis a measure and v∈W01,r(Ω), withr≤N. On the other hand the solution cannot be expected to
belong to the energy spaceW01,p(Ω), as simple examples show. Thus, it is necessary to change the functional setting in order to prove existence result.
In the linear case, i.e., if p = 2 and a(x,∇u) = A(x)∇u, where A is a N ×N matrix such that
|A(x)ξ| ≤C0|ξ| and A(x)ξξ≥C1|ξ|2, ∀ξ∈IRN, for a.e.x∈Ω, (2.7) with C0 and C1 two positive real constants, problem (2.6) was studied by G.
Stampacchia in [33].
Definition 2.2. Assume that Ω satisfies the following regularity condition: there exists a constantγ >0 such that
meas(Br(x)\Ω)≥γmeas(Br(x)),
for everyx∈∂Ω and for everyr >0, whereBr(x) denotes the open ball with centre xand radiusr. A functionuà ∈L1(Ω) is a solution in the sense of Stampacchia (also called solution by duality) of the equation (2.6) if
Ω
uàg dx=
Ω
u∗gdà, ∀g∈L∞(Ω), (2.8) whereu∗g is the solution of
A∗u∗g=g in Ω u∗g∈H01(Ω), andA∗ is the adjoint ofA.
Existence and uniqueness ofuà are proved in [33]. It is easy to prove that, for everyj >0,Tj(uà)∈H01(Ω) and
Ω
|∇Tj(uà)|2dx≤j|à|(Ω). (2.9) In the case p = 2, if A is strongly monotone and Lipschitz continuous, Stampacchia’s ideas continue to work. In this special case, indeed, F. Murat (in [28]) proved the existence and uniqueness of a solution (called reachable solution) of (2.6).
Assuming that Ω is a regular set (in the sense of Definition 2.2), we consider a : Ω×IRN → IRN a Carath´eodory function such that for every ξ, η in IRN (ξ=η), and for almost everyx∈Ω,
|a(x, ξ)−a(x, η)| ≤C0|ξ−η|, (2.10) (a(x, ξ)−a(x, η))(ξ−η)≥C1|ξ−η|2, (2.11)
a(x,0) = 0, (2.12)
whereC0 andC1 are two positive real constants.
Obstacle Problems for Monotone Operators with Measure Data 297 Definition 2.3. We say that a functionu∈T01,p(Ω) is a reachable solution of the problem (2.6) if there exist two sequencesàn andun such that
(i) àn ∈ Mb(Ω)∩W−1,p(Ω) and àn converges toà in the ∗-weak topology of Mb(Ω);
(ii) un ∈W01,p(Ω) andunsolves the Dirichlet problem (2.5) relative to the datum àn;
(iii) un converges to ua.e. in Ω.
Remark 2.4. Note that this definition can be given also for a general monotone operatorA(see [15], where the existence of a reachable solution has been proved in this generality).
Actually, in the general nonlinear case, when à ∈ Mb,0p (Ω), other types of solutions to (2.6) have been proposed. The notion of entropy solution, of SOLA, and of renormalized solution were introduced respectively in [2], [11], and [26].
These three frameworks, which are actually equivalent, are successful since they allow to prove existence and uniqueness results.
Definition 2.5. Letà∈Mb,0p (Ω); we say that a functionu∈T01,p(Ω) is an entropy solution of (2.6) if
Ω
a(x,∇u)∇Tk(u−ϕ)dx=
Ω
Tk(u−ϕ)dà, (2.13) for everyϕ∈W01,p(Ω)∩L∞(Ω) and for everyk >0.
Remark2.6.Let us point out that these three types of solutions coincide when each make sense. This means, for example, that ifAis linear the Stampacchia’s solution is also the reachable solution, as well as, ifp= 2 and Ais strongly monotone and Lipschitz continuous, andà is a measure inMb,0p (Ω), then the reachable solution is also the entropy solution.
Thanks to these notions of solutions we will arrive to a suitable definition of obstacle problems with measure data (see [13], [22], and [21]).
Roughly speaking, we choose the minimum element among those functionsvabove the obstacle, such thatA(v)−àis not only nonnegative in the sense of distributions but it is actually a nonnegative bounded Radon measure, and the equation is solved in the sense of Stampacchia or in the sense of the reachable solutions, ifAis linear orAis strongly monotone and Lipschitz continuous, respectively, and in the sense of entropy in the general nonlinear case when à is a bounded Radon measure vanishing on all sets ofp-capacity zero.
Actually, to define the solution for obstacle problems with measure data, we will distinguish two cases. The first one regards the case where à is a general bounded Radon measure. This case can be treated only consideringp= 2 and the operatorA Lipschitz continuous and strongly monotone. The second case deals with bounded Radon measuresàvanishing on all sets ofp-capacity zero, so that we are able to handle with a general nonlinear monotone operatorA. In this case,
we are not able to drop the assumption thatàis absolutely continuous with respect to thep-capacity, since in our approach we need a suitable notion of solution to equations with measure data which ensures existence and uniqueness results. As a matter of fact, for a general monotone operatorA the question of existence of a solution to (2.6) has been faced in [16] where the authors extend the notion of renormalized solution (see [26]) to the case of a general measureà∈Mb(Ω). In that paper they proved the existence of such a solution and introduced other equivalent definitions, which show that all the renormalized solutions are constructed by approximating, in an appropriate way, the measureà(with respect to the∗-weak convergence of measures), so that they are reachable solutions. On the other hand the question of uniqueness of a reachable solution, in this case, is still an open problem.
Definition 2.7. Let p = 2 and let a satisfy (2.10)–(2.12); we say that u is the solution of the Obstacle Problem with datum à ∈ Mb(Ω) and the obstacle ψ (denoted byOP(A, à, ψ)) if
1. there exists a measureλ∈ Mb+(Ω) such that u is the reachable solution of (2.6) relative toà+λ, andu∈Kψ(Ω).
2. for anyν ∈ Mb+(Ω) such that the reachable solution v of (2.6) relative to à+ν is inKψ(Ω), we haveu≤v a.e. in Ω.
Remark 2.8. This definition includes, of course, also the case whereAis linear. In this case, as we already observed in Remark 2.6, the notion of reachable solution and of Stampacchia’s solution are equivalent.
Definition 2.9.Letasatisfy (2.1)–(2.4); we say thatuis the solution of the Obstacle Problem with datumà∈Mb,0p (Ω) and the obstacleψ(denoted byOP0(A, à, ψ)) if 1. there exists a measureλ ∈Mb,0p,+(Ω) such that u is the entropy solution of
(2.6) relative toà+λ, andu∈Kψ(Ω).
2. for any ν ∈ Mb,0p,+(Ω) such that the entropy solution v of (2.6) relative to à+ν is inKψ(Ω), we haveu≤v a.e. in Ω.
By definition, it is clear that, if such a solution (ofOP(A,à,ψ) orOP0(A,à,ψ)) exists, it is unique.
3. Main results
First of all we want to treat the question of existence (the uniqueness being implicit in the definition itself) of solutions toOP(A, à, ψ) andOP0(A, à, ψ).
We have the following theorems (see [22] and [21] for the proofs).
Theorem 3.1. Let p= 2and leta satisfy(2.10)–(2.12). If ψ: Ω→IRis such that
ψ≤uρ C2−q.e. in Ω, (3.1)
Obstacle Problems for Monotone Operators with Measure Data 299 whereρ∈Mb(Ω) anduρ is the reachable solution of
A(u) =ρ inΩ, u= 0 on∂Ω;
then, for every à ∈Mb(Ω), there exists a unique solution of OP(A, à, ψ) in the sense of Definition2.7. Moreover the corresponding obstacle reactionλsatisfies
λMb(Ω)≤ (à−ρ)−Mb(Ω).
Theorem 3.2. Let asatisfy (2.1)–(2.4)and letψ: Ω→IRbe such that
ψ≤uρ Cp−q.e. in Ω, (3.2)
whereρ∈W−1,p(Ω)∩Mb(Ω) anduρ is the variational solution of A(u) =ρ inΩ,
u= 0 on∂Ω;
then, for everyà∈Mb,0p (Ω), there exists a unique solution ofOP0(A, à, ψ)in the sense of Definition2.9. Moreover the corresponding obstacle reactionλsatisfies
λMb(Ω)≤ (à−ρ)−Mb(Ω).
The proofs of existence are based on an approximation technique. The obsta- cle reactions associated with the solutions for regular data are shown to satisfy an estimate on the masses, which allows to pass to the limit and obtain the solution in the general case.
In our setting we are able to prove (see [22] and [21]) the Lewy-Stampacchia inequality: first proved in [24] it has been extended by various authors to different cases. It has become a powerful tool for proving existence and regularity results.
Theorem 3.3. Let p= 2 and let a satisfy (2.10)–(2.12). Let à∈Mb(Ω) and ube the solution of OP(A, à, uρ) (uρ defined in (3.1)). If we denote byλ the obstacle reaction associated withu, it holds
λ≤(à−ρ)−.
Theorem 3.4. Let a satisfy (2.1)–(2.4), à ∈ Mb,0p (Ω) and u be the solution of OP0(A, à, uρ) (uρ defined in (3.2)). If we denote by λthe obstacle reaction asso- ciated withu, it holds
λ≤(à−ρ)−.
Furthermore, as in the classical framework, we study the interaction between obstacles and data, and in particular the complementarity conditions. More pre- cisely, the following theorem (see [22] for the proof) shows that the solutionuof OP0(A, à, ψ), whenà∈Mb,0p (Ω), is the only entropy solution of (2.6) relative to à+λsuch thatu=ψ λ-almost everywhere in Ω, andu≥ψ Cp-q.e. in Ω. We also find a more technical characterization of the solution of the Obstacle Problem, which turns out to be similar to (1.1).
Theorem 3.5. Let asatisfy (2.1)–(2.4),à∈Mb,0p (Ω) andψ satisfy(3.2); then the following statements are equivalent:
(1) uis the solution ofOP0(A, à, ψ)andλis the associated obstacle reaction;
(2) u≥ψ Cp-q.e. inΩ,λ∈Mb,0p,+(Ω), uis the entropy solution of(2.6) relative toà+λ, and
ΩTk(u−ϕ)dλ≤
ΩTk(v−ϕ)dλ,
∀ϕ∈W01,p(Ω)∩L∞(Ω), ∀v∈T01,p(Ω), v≥ψ Cp-q.e. inΩ; (3.3) (3) u≥ψ Cp-q.e. inΩ,λ∈Mb,0p,+(Ω), uis the entropy solution of(2.6) relative
toà+λ, and
u=ψ λ-a.e. in Ω.
Remark 3.6. Observe that ifψisCp-q.e. upper bounded, we can consider in (3.3) ϕ∈W01,p(Ω)∩L∞(Ω),ϕ≥ψ Cp-q.e. in Ω andv=ϕ, so that, taking into account thatuis the entropy solution of (2.6) relative toà+λ, for everyk >0,usatisfies
Ω
a(x,∇u)∇Tk(u−ϕ)dx≤
Ω
Tk(u−ϕ)dà, (3.4) which turns out to be quite similar to the usual variational formulation (1.1).
The previous formula was already obtained in [4] when the datumàis a function inL1(Ω).
This fact is non longer true when we pass to consider general data inMb(Ω) (whenp= 2 andAis strongly monotone and Lipschitz continuous). The following example, which is a variant of an example studied by L. Orsina and A. Prignet, shows that the solution of the obstacle problem with right-hand side measure does not touch the obstacle, though it is not the solution of the equation.
Example. Let N ≥ 2, Ω be the ball B1(0), A = −∆, à = −δ0 (the Dirac mass at the origin), andψ = −h a negative constant. Let ube the solution of OP(−∆,−δ0,−h), then−∆u=−δ0+λ(and the equation is satisfied in the sense of Stampacchia or, equivalently, in the setting of reachable solutions). We want to show thatλ=δ0.
Takingν =δ0 in condition 2 of Definition 2.7, we obtainu≤0 C2-q.e. in Ω.
Asu≥ −h, we haveu=Th(u) and henceu∈H01(Ω) by (2.9). This implies that the measure−δ0+λbelongs toMb(Ω)∩H−1(Ω), which is contained inMb,02 (Ω).
In other wordsλ=δ0+λ0, withλ0∈Mb,02 (Ω). Sinceλis nonnegative andδ0⊥λ0, the measureλ0 is nonnegative. Then u= uλ0 ≥0, and finally u= 0. Thus the solution can be far above the obstacle, but the obstacle reaction is nonzero, and is exactlyδ0.
The previous example can be explained by the following theorem that shows that, when the obstacle is controlled from above and from below in an appropriate way, it is possible to “isolate” the effect of the singular negative part of the data.
Namely, the reaction λ will be written as λ = λ0+à−s, where λ0 belongs to Mb,02,+(Ω). Moreover the “regular part” λ0 is concentrated on the coincidence set
Obstacle Problems for Monotone Operators with Measure Data 301 {x ∈ Ω : u(x) = ψ(x)} whenever ψ is C2-quasi upper semicontinuous, and a complementarity condition holds.
Theorem 3.7. Let p = 2, and let a satisfy (2.10)–(2.12). Let à ∈Mb(Ω) and let ψ: Ω→IR beC2-quasi upper semicontinuous satisfying
u−ρ−τ≤ψ≤uρ,
where ρ ∈ Mb,02 (Ω) and τ ∈ Mb(Ω), with τ⊥à−s (here u−ρ−τ and uρ are the reachable solutions of(2.6) relative to −ρ−τ andρ, respectively). Let uand u0 be the solutions of OP(A, à, ψ) and OP(A, àa+à+s, ψ), respectively, and λ and λ0 be the corresponding obstacle reactions. Then u=u0 a.e. inΩ,λ0∈Mb,02,+(Ω), λ=λ0+à−s, andu=ψ λ0-a.e. in Ω.
In [12] this theorem was proved in the linear case, investigating the behavior of the potential of two mutually singular measures near their singular points.
Actually in [21] we extend this result to our (nonlinear) context giving alternative proofs.
Finally we deal with the behavior of the Obstacle Problem in the sense of Definition 2.9 under perturbation of the operator, of the forcing term, and of the obstacle.
The study of the properties of the solutions to the Obstacle Problem when the operators vary is based on a notion of convergence for strictly monotone operators, calledG-convergence. Actually, to deal with this type of convergence we have to restrict our class of functionsasatisfying (2.1)–(2.4).
In particular, given two constantsc0,c1>0 and two constantsαandβ, with 0≤α≤1∧(p−1) andp∨2≤β <+∞, we consider the familyL(c0, c1, α, β) of Carath´eodory functionsa(x, ξ) : Ω×IRN →IRN such that:
|a(x, ξ)−a(x, η)| ≤c0(1 +|ξ|+|η|)p−1−α|ξ−η|α, (3.5) (a(x, ξ)−a(x, η))(ξ−η)≥c1(1 +|ξ|+|η|)p−β|ξ−η|β, (3.6)
a(x,0) = 0, (3.7)
for almost everyx∈Ω, for everyξ, η∈IRN.
Definition 3.8. We say that a sequence of functions ah(x, ξ) ∈ L(c0, c1, α, β) G- converges to a functiona(x, ξ) satisfying the same hypotheses (possibly with dif- ferent constants ˜c0,˜c1,α,˜ β) if for any˜ F ∈W−1,p(Ω), the solutionuh of
Ah(uh) =F in Ω
u= 0 on∂Ω
satisfies
uh uweakly inW01,p(Ω) and
ah(x,∇uh) a(x,∇u) weakly inLp(Ω)N, whereuis the unique solution of (2.5).
The following theorem justifies the definition ofG-convergence.
Theorem 3.9. Any sequenceah(x, ξ)of functions belonging toL(c0, c1, α, β)admits a subsequence which G-converges to a function a(x, ξ) ∈L(˜c0,˜c1,β−αα, β), where
˜
c0,c˜1 depend only onN, p, α, β, c0, c1
This compactness theorem was obtained by L. Tartar (see [34] and Theorem 1.1 of [17]) in the case of nonlinear monotone operators defined fromH01(Ω) into H−1(Ω), when p= 2 and the functionsah∈L(c0, c1,1,2), and then extended in the version of Theorem 3.9 in [10] (see Theorem 4.1).
Concerning the perturbation of the obstacles we consider a notion of conver- gence for sequences of convex sets introduced by U. Mosco in [27].
Definition 3.10. LetKhbe a sequence of subsets of a Banach spaceX. The strong lower limit
s−lim inf
h→+∞Kh
of the sequenceKhis the set of allv∈Xsuch that there exists a sequencevh∈Kh, forhlarge, converging tov strongly inX.
The weak upper limit
w−lim sup
h→+∞Kh
of the sequenceKh is the set of all v ∈ X such that there exists a sequence vk converging tovweakly inXand a sequence of integershk converging to +∞, such thatvk ∈Khk.
The sequence Kh converges to the set K in the sense of Mosco, shortly Kh
→MK, if
s−lim inf
h→+∞Kh=w−lim sup
h→+∞Kh=K.
Mosco proved that this type of convergence is the right one for the stability of variational inequalities with respect to obstacles. This is the main result of his theory.
Theorem 3.11. Let Kψh := Kψh(Ω)∩W01,p(Ω) and Kψ := Kψ(Ω)∩W01,p(Ω) be nonempty. Then
Kψh
→MKψ
if and only if, for anyF ∈W−1,p(Ω),
uh→ustrongly in W01,p(Ω),
whereuhanduare the solutions ofV I(A, F, ψh) (the variational inequality relative toA,F, andψh)andV I(A, F, ψ) (the variational inequality relative toA,F, and ψ), respectively.
Several stability results can be proved as corollaries of this theorem by Mosco.
In particular, the strong convergence
ψh→ψstrongly inW1,p(Ω)
Obstacle Problems for Monotone Operators with Measure Data 303 easily implies the convergence ofKψh toKψ in the sense of Mosco, but the weak convergence
ψh ψweakly inW1,r(Ω), r > p, also implies the same result (see [7], [1]).
Now, we consider a sequenceah of functions in L(c0, c1, α, β), a sequence of measuresρh∈Mb(Ω)∩W−1,p(Ω), and the variational solutionuρh of
Ah(uρh) =ρh in Ω uρh ∈W01,p(Ω).
We assume that
sup
h
ρhMb(Ω)<+∞ (3.8)
and that the functionψh satisfies:
ψh≤uρh Cp-q.e. in Ω. (3.9)
Moreover we suppose that
ψ≤0Cp-q.e. in Ω. (3.10)
Theorem 3.12. Let ah be a sequence in L(c0, c1, α, β), which G-converges to a functiona, and letAhandAbe the operators associated toah anda, respectively.
Let us assume(3.8),(3.9), and (3.10), withKψh converging toKψ in the sense of Mosco. Finally, we consider àh,à∈Mb,0p (Ω)such that
àh(B)→à(B), for every Borel setB⊆Ω.
Then the solutions uh and u of the obstacle problems OP0(Ah, àh, ψh) and OP0(A, à, ψ), respectively, satisfy
Tj(uh) Tj(u) weakly inW01,p(Ω), for everyj >0, ah(x,∇uh) a(x,∇u)weakly inLq(Ω)N, for everyq < NN−1,
Ω
ah(x,∇uh)∇Tj(uh)dx→
Ω
a(x,∇u)∇Tj(u)dx, for everyj >0.
Remark 3.13. By formal modifications we can prove Theorem 3.12 (see [23]) re- placing (3.10) with (3.2) and
ψ≤M Cp−q.e.in Ω, whereM is a positive constant.
Theorem 3.12 is the analogous of Theorem 3.1 of [14] proved in the classical setting of variational inequalities.
References
[1] H. Attouch, C. Picard, Probl`emes variationnels et th´eorie du potential non lin´eaire, Ann. Facult´e Sci. Toulouse1(1979), 89–136.
[2] L. Benilan, L. Boccardo, T. Gallo¨uet, R. Gariepy, M. Pierre, J.L. Vazquez, AnL1- theory of existence and uniqueness of nonlinear elliptic equations,Ann. Scuola Norm.
Sup. Pisa22(1995), 240–273.
[3] L. Boccardo, G.R. Cirmi, Nonsmooth unilateral problems.Nonsmooth optimization:
methods and applications (Erice, 1991), F. Giannessi ed., 1–10, Gordon and Breach, Amsterdam 1992.
[4] L. Boccardo, G.R. Cirmi, Existence and uniqueness of solution of unilateral problems withL1-data,J. Convex Anal.6(1999), 195–206
[5] L. Boccardo, T. Gallo¨uet, Probl`emes unilat´eraux avec donn´ees dansL1,C. R. Acad.
Sci. Paris S´er. I Math.311(1990), 617–619.
[6] L. Boccardo, T. Gallo¨uet, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data,Ann. Inst. H. Poincar´e13(1996), 539–551.
[7] L. Boccardo, F. Murat, Nouveaux r´esultats de convergence dans des probl`emes uni- lat´eraux,Nonlinear partial differential equations and their applications (Coll`ege de France s´eminaire), Pitman Res. Notes Math.60(1962), 64–85.
[8] H. Brezis, S. Serfaty, A variational formulation for the two-sided obstacle problem with measure data,Commun. Contemp. Math.4(2002), 357–374.
[9] H. Brezis, A. Ponce, Reduced measures for obstacle problems,Adv. Differential Equa- tions,10 (2005), 1201–1234.
[10] V. Chiad`o Piat, A. Defranceschi, Asymptotic behaviour of quasi linear problems with Neumann boundary conditions on perforated domains,Appl. Anal.36(1990), 65–87.
[11] A. Dall’Aglio, Approximated solutions of equations withL1 data,Ann. Mat. Pura Appl.170(1996), 207–240.
[12] P. Dall’Aglio, G. Dal Maso, Some properties of the solutions of obstacle problems with measure data,Ricerche Mat.48(1999), 99–116.
[13] P. Dall’Aglio, C. Leone, Obstacle problems with measure data and linear operators, Potential Anal.17(2002), 45–64.
[14] G. Dal Maso, A. Defranceschi, Convergence of unilateral problems for monotone operators,J. Analyse Math.53(1989), 269–289.
[15] G. Dal Maso, A. Malusa, Some properties of reachable solutions of nonlinear elliptic equations with measure data,Ann. Scuola Norm. Sup. Pisa Cl. Sci.25(1997), 375–
396.
[16] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), 741–808.
[17] T. Del Vecchio, On the homogenization of a class of pseudomonotone operators in divergence form,Boll. Un. Mat. Ital.7(1991), 369–388.
[18] M. Fukushima, K. Sato, S. Taniguchi, On the closable part of pre-Dirichlet forms and the fine support of the underlying measures,Osaka J. Math.28(1991), 517–535.