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Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems

  • We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.

    Citation: Fernando Farroni, Gioconda Moscariello, Gabriella Zecca. Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems[J]. Mathematics in Engineering, 2023, 5(4): 1-23. doi: 10.3934/mine.2023071

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  • We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.



    To Giuseppe Mingione, on the occasion of his 50th birthday, with regard and admiration.

    The aim of this paper is to study a nonlinear and noncoercive parabolic variational inequality with constraint and homogeneous Dirichlet boundary condition. The Lewy-Stampacchia inequality associated with it is addressed. After the first results of H. Lewy and G. Stampacchia [19] concerning inequalities in the context of superharmonic problems, there is by now a large literature concerning the theory of elliptic obstacle problems as well as of elliptic variational inequalities. We refer to [3,16,25] for a classical overview. For a more recent treatment related to nonlinear elliptic operators see also [23]. The obstacle problem for nonlocal and nonlinear operators has been cosidered in [17,26]. An abstract and general version of the Lewy-Stampacchia inequality is given in [13]. Concerning the parabolic case, first existence results related to problems with time independent obstacles have been treated in [20] in the linear case and in [5] for the more general parabolic problems. The case of obstacles functions regular in time has been considered in [2,5]. Existence and regularity theory for solutions of parabolic inequalities involving degenerate operators in divergence form have been established in [4,18]. More recently in [15], the Authors prove Lewy-Stampacchia inequality for parabolic problems related to pseudomonotone type operators. In this paper we study a variational parabolic inequality for noncoercive operators that present singularities in the coeffcients of the lower order terms in the same spirit of [9,12,14].

    Let us state the functional setting and the assumptions on the data.

    Let ΩRN, N2, be a bounded open Lipschitz domain and let ΩT:=Ω×(0,T) be the parabolic cylinder over Ω of height T>0. We shall denote by v and tv (or vt) the spatial gradient and the time derivative of a function v respectively. We consider the class

    Wp(0,T):={vLp(0,T,W1,p0(Ω)):vtLp(0,T,W1,p(Ω))}, (1.1)

    where

    2NN+2<p<N. (1.2)

    and p is the conjugate exponent of p, i.e., 1p+1p=1. In (1.1), Lp(0,T,W1,p0(Ω)) and Lp(0,T,W1,p(Ω)) denote parabolic Banach spaces defined according to (2.7).

    Given a measurable function ψ:ΩTΩ×{0}R, we are interested in finding functions u:ΩTR in the convex subset Kψ(ΩT) of Wp(0,T) defined as

    Kψ(ΩT):={vWp(0,T):vψa.e. in ΩT}

    and satisfying the following variational inequality

    T0ut,vudt+ΩTA(x,t,u,u)(vu)dxdtT0f,vudtvKψ(ΩT), (1.3)

    where

    fLp(0,T,W1,p(Ω)) (1.4)

    and , denotes the duality between W1,p(Ω) and W1,p0(Ω). The vector field

    A=A(x,t,u,ξ):ΩT×R×RNRN

    is a Carathéodory function, i.e., A measurable w.r.t. (x,t)ΩT for all (u,ξ)R×RN and continuous w.r.t. (u,ξ)R×RN for a.e. (x,t)ΩT, and such that for a.e. (x,t)ΩT and for any uR and ξ,ηRN,

    A(x,t,u,ξ)ξα|ξ|p(b(x,t)|u|)pH(x,t) (1.5)
    [A(x,t,u,ξ)A(x,t,u,η)](ξη)>0if ξη (1.6)
    |A(x,t,u,ξ)|β|ξ|p1+(˜b(x,t)|u|)p1+K(x,t) (1.7)

    hold true. Here α,β are positive constants, while H, K, b and ˜b are nonnegative measurable functions defined on ΩT such that HL1(ΩT), KLp(ΩT) and

    b,˜bL(0,T,LN,(Ω)), (1.8)

    where LN,(Ω) is the Marcinkiewicz space. For definitions of LN,(Ω) and L(0,T,LN,(Ω)) see Sections 2.2 and 2.3, respectively.

    We assume that the obstacle function fulfills

    ψC0([0,T],L2(Ω))Lp(0,T,W1,p(Ω)) (1.9)
    ψ0a.e. in Ω×(0,T) (1.10)
    ψtLp(ΩT) (1.11)
    ψ(,0)W1,p0(Ω). (1.12)

    For

    u0L2(Ω) (1.13)

    we impose the following compatibility condition

    u0ψ(,0)a.e. in Ω. (1.14)

    In the following, we will refer to a function uKψ(ΩT) satisfying (1.3) and such that u(,0)=u0 as a solution to the variational inequality in the strong form with initial value u0.

    Under previous assumptions the existence of a solution in the weak form can be proved, see [12]. However the existence of a solution in the sense stated above is not guaranteed even in simpler cases. Then we assume that the source term and the obstacle function are such that

    g:=fψt+div A(x,t,ψ,ψ)=g+gwithg+,gLp(0,T,W1,p(Ω))+. (1.15)

    Here Lp(0,T,W1,p(Ω))+ denotes the non-negative elements of Lp(0,T,W1,p(Ω)). Following the terminology of [7] or [15], (1.15) is equivalent to say that g is an element of the order dual Lp(0,T,W1,p0(Ω)) defined as

    Lp(0,T,W1,p0(Ω)):={g=g+g,g±Lp(0,T,W1,p(Ω))+}.

    Then, our main result reads as follows

    Theorem 1.1. Let (1.2) and (1.4)–(1.15) be in charge. Assume further that

    Db:=distL(0,T,LN,(Ω))(b,L(ΩT))<α1/pSN,p, (1.16)

    where SN,p=ω1/NNpNp and ωN denotes the measure of the unit ball of RN. Then, there exists at least a solution uKψ(ΩT) of the strong form of the variational inequality (1.3) satisfying u(,0)=u0. Moreover, the following Lewy-Stampacchia inequality holds

    0tudiv A(x,t,u,u)fg=(ftψ+div A(x,t,ψ,ψ)). (1.17)

    In (1.16), Db denotes the distance of b from L(ΩT) in the space L(0,T,LN,(Ω)) defined in (2.8) below.

    Assumptions (1.8) on the coefficients of the lower order terms allow us to consider diffusion models in which the boundedness of the convective field with respect to the spatial variable is too restrictive (see [8]). The corresponding bounded case has been treated in [15].

    We discuss condition (1.16) through an example. It's easy to verify that the operator

    A(x,t,u,ξ)=|ξ|p2ξ+et|u|p2u(γ|x|+1γarctan|x|)p1x|x|

    satisfies (1.5)–(1.8). According to (2.2) and (2.3) below, we get that

    Db=(11p)1/pω1/NNγ

    and so (1.16) holds true whenever γ is small enough. On the other hand, we notice that (1.16) does not imply smallness of the norm of the coefficient b. Indeed

    bL(0,T,LN,(Ω))Cγ

    for a constant C independent of γ.

    Theorem 1.1 also applies in the case b and ˜b lie in a functional subspace of weak–LN in which bounded functions are dense. For more details see also [10]. For other examples of operators satisfying conditions above we refer to [12].

    We remark that for f,ψt,div A(x,t,ψ,ψ)Lp(ΩT) condition (1.15) is satisfied. Then, Theorem 1.1 is comparable with the existence result of Lemma 3.1 in [4]. In order to prove our result, we consider a sequence of suitable penalization problems for which an existence result holds true (see [12]). Then we are able to construct a solution u to (1.3) as limit of solutions of such problems despite the presence of unbounded coefficients in the lower order terms.

    In this section we provide the notation and several preliminary results that will be fundamental in the sequel.

    The symbol C (or C1,C2,) will denote positive constant, possibly varying from line to line. For the dependence of C upon parameters, we will simply write C=C(,,). The positive and the negative part of a real number z will be denoted by z+ and z, respectively, and are defined by z+:=max{z,0} and z:=min{z,0}. Given z1,z2R, we often use the notation z1z2 and z1z2 in place of min{z1,z2} and max{z1,z2} respectively.

    Let Ω be a bounded domain in RN. For any 1<p< and 1q<, the Lorentz space Lp,q(Ω) is the set of real measurable functions f on Ω such that

    fqLp,q:=p0[λf(k)]qpkq1dk<.

    Here λf(k):=|{xΩ:|f(x)|>k}| is the distribution function of f. When p=q, the Lorentz space Lp,p(Ω) coincides with the Lebesgue space Lp(Ω). When q=, the space Lp,(Ω) is the set of measurable functions f on Ω such that

    fpLp,:=supk>0kpλf(k)<.

    This set coincides with the Marcinkiewicz space weak-Lp(Ω). The expressions above do not define a norm in Lp,q or Lp, respectively, in fact triangle inequality generally fails. Nevertheless, they are equivalent to a norm, which make Lp,q(Ω) and Lp,(Ω) Banach spaces when endowed with them. An important role in the potential theory is played by these spaces as pointed out in [22].

    For 1q<p<r, the following inclusions hold

    Lr(Ω)Lp,q(Ω)Lp,r(Ω)Lp,(Ω)Lq(Ω).

    For 1<p<, 1q and 1p+1p=1, 1q+1q=1, if fLp,q(Ω), gLp,q(Ω) we have the Hölder–type inequality

    Ω|f(x)g(x)|dxfLp,qgLp,q. (2.1)

    Since L(Ω) is not dense in Lp,(Ω), for fLp,(Ω) in [6] the Authors stated the following

    distLp,(Ω)(f,L(Ω)):=infgL(Ω)fgLp,(Ω). (2.2)

    As already observed in [10,11], we have

    distLp,(Ω)(f,L(Ω))=limm+fχ{|f|>m}Lp, (2.3)

    and

    distLp,(Ω)(f,L(Ω))=limm+fTmfLp,,

    where, for all m>0, Tm is the truncation operator at levels ±m, i.e.,

    Tmy:=min{m,max{m,y}}for yR. (2.4)

    Another useful estimate is provided by the following sort of triangle inequality

    f+εgLp,(1+ε)fLp,+ε(1+ε)gLp, (2.5)

    which holds true for f,gLp,(Ω) and ε>0.

    For 1q<, any function in Lp,q(Ω) has zero distance to L(Ω). Indeed, L(Ω) is dense in Lp,q(Ω), the latter being continuously embedded into Lp,(Ω).

    Assuming that 0Ω, b(x)=γ/|x| belongs to LN,(Ω), γ>0. For this function, we have

    distLN,(Ω)(b,L(Ω))=γω1/NN.

    The Sobolev embedding theorem in Lorentz spaces [1,24] reads as

    Theorem 2.1. Let us assume that 1<p<N, 1qp, then every function uW1,10(Ω) verifying |u|Lp,q(Ω) actually belongs to Lp,q(Ω), where p:=NpNp is the Sobolev conjugate exponent of p and

    uLp,qSN,puLp,q, (2.6)

    where SN,p is the Sobolev constant given by SN,p=ω1/NNpNp.

    Let T>0 and X be a Banach space endowed with a norm X. Then, the space Lp(0,T,X) is defined as the class of all measurable functions u:[0,T]X such that

    uLp(0,T,X):=(T0u(t)pXdt)1/p< (2.7)

    whenever 1p<, and

    uL(0,T,X):=esssup0<t<Tu(t)X<

    for p=. The space C0([0,T],X) represents the class of all continuous functions u:[0,T]X with the norm

    uC0([0,T],X):=max0tTu(t)X.

    We essentially consider the case where X is either a Lorentz space or Sobolev space W1,p0(Ω). This space will be equipped with the norm gW1,p0(Ω):=gLp(Ω) for gW1,p0(Ω).

    For fL(0,T,Lp,(Ω)) we define

    distL(0,T,Lp,(Ω))(f,L(ΩT))=infgL(ΩT)fgL(0,T,Lp,(Ω)) (2.8)

    and as in (2.3) we find

    distL(0,T,Lp,(Ω))(f,L(ΩT))=limm+fχ{|f|>m}L(0,T,Lp,(Ω)). (2.9)

    In the class Wp(0,T) defined in (1.1) and equipped with the norm

    uWp(0,T):=uLp(0,T,W1,p(Ω))+utLp(0,T,W1,p(Ω)),

    the following inclusion holds (see [27,Chapter III, page 106]).

    Lemma 2.2. Let p>2N/(N+2). Then Wp(0,T) is contained into the space C0([0,T],L2(Ω)) and any function uWp(0,T) satisfies

    uC0([0,T],L2(Ω))CuWp(0,T)

    for some constant C>0.

    Moreover, the function t[0,T]u(,t)2L2(Ω) is absolutely continuous and

    12ddtu(,t)2L2(Ω)=ut(,t),u(,t)for a.e. t[0,T].

    The compactness result due to Aubin–Lions reads as follows.

    Lemma 2.3. Let X0,X,X1 be Banach spaces with X0 and X1 reflexive. Assume that X0 is compactly embedded into X and X is continuously embedded into X1. For 1<p,q< let

    W:={uLp(0,T,X0):tuLq(0,T,X1)}.

    Then W is compactly embedded into Lp(0,T,X).

    As an example, we choose q=p, X0=W1,p0(Ω), X1=W1,p(Ω) and X=Lp(Ω) if p2 or X=L2(Ω) for 2NN+2<p<2. Therefore, we deduce

    Lemma 2.4. If p>2N/(N+2) then Wp(0,T) is compactly embedded into Lp(ΩT) and into L2(ΩT).

    Let δ>0. We introduce the following initial–boundary value problem

    {tuδdiv [A(x,t,max{uδ,ψ},uδ)]=1δ[(ψuδ)+]q1+fin ΩT,uδ=0on Ω×(0,T),uδ(,0)=u0in Ω, (3.1)

    where

    q:=min{2,p}.

    Moreover, in this section we assume that

    ψ0a.e. in ΩT. (3.2)

    We introduce the notation

    ˜A(x,t,w,ξ):=A(x,t,max{w,ψ},ξ).

    By the elementary inequality

    |aa||a|aRa(,0] (3.3)

    and recalling (1.5), (1.6) and (1.7), we easily deduce

    ˜A(x,t,u,ξ)ξα|ξ|p(b(x,t)|u|)pH(x,t)[˜A(x,t,u,ξ)˜A(x,t,u,η)](ξη)>0if ξη|˜A(x,t,u,ξ)|β|ξ|p1+(˜b(x,t)|u|)p1+K(x,t)

    for a.e. (x,t)ΩT and for any uR and ξ,ηRN.

    For u0L2(Ω) and fLp(0,T,W1,p(Ω)), a solution to problem (3.1) is a function

    uδC0([0,T],L2(Ω))Lp(0,T,W1,p0(Ω))

    such that

    ΩTuδφtdxds+ΩT˜A(x,s,uδ,uδ)φdxds=1δΩT[(ψuδ)+]q1φdxds+Ωu0φ(x,0)dx+T0f,φds

    for every φC(ˉΩT) such that suppφ[0,T)×Ω.

    By using the elementary inequality

    (a+a)θaθ+aθa,a[0,+)θ(0,1)

    and Young inequality we see that

    p<2[(ψu)+]p1|ψ|p1+|u|p1(p1)(|u|+|ψ|)+2(2p).

    Hence, by Theorem 4.2 and Remark 4.5 in [12] we get the following existence result.

    Proposition 3.1. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. For every fixed δ>0, problem (3.1) admits a solution uδC0([0,T],L2(Ω))Lp(0,T,W1,p0(Ω)).

    The arguments of [12] lead to some estimates for the sequence {uδ}δ>0. We propose here a proof that carefully keeps trace of the constants in the estimates.

    Lemma 3.2. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. Any solution uδC0([0,T],L2(Ω))Lp(0,T,W1,p0(Ω)) to problem (3.1) satisfies the following estimate

    uδ2L(0,T,L2(Ω))+uδpLp(ΩT)C(b,N,p,α)[u02L2(Ω)+fpLp(0,T,W1,p(Ω))+HL1(ΩT)+(u02L2(Ω)+fpLp(0,T,W1,p(Ω))+bpLp(ΩT))pbpLp(ΩT)]. (3.4)

    Proof. We fix t(0,T) and we set Ωt:=Ω×(0,t). We choose φ:=T1(uδ)χ(0,t) as a test function. If we let Φ(z):=z0T1(ζ)dζ for zR, we have

    ΩΦ(uδ(x,t))dx+Ωt˜A(x,s,uδ,uδ)T1(uδ)dxds=1δΩt[(ψuδ)+]q1T1(uδ)dxds+ΩΦ(u0)dx+t0f,T1(uδ)ds.

    Assumption (3.2) implies that [(ψuδ)+]q1T1(uδ)0 a.e. in ΩT, so we have

    ΩΦ(uδ(x,t))dx+Ωt{|uδ|1}˜A(x,s,uδ,uδ)uδdxdsΩΦ(u0(x,0))dx+t0f,T1(uδ)ds.

    By (1.5) and (1.7) we deduce

    ΩΦ(uδ(x,t))dx+αΩt{|uδ|1}|uδ|pdxdsΩΦ(u0)dx+t0f,T1(uδ)ds+Ωt{|uδ|1}(b|uδψ|)pdxds+Ωt{|uδ|1}Hdxds. (3.5)

    Now, as 0Φ(z)z22 for all zR, we have

    ΩΦ(u0)dx12u02L2(Ω). (3.6)

    By Hölder and Young inequality we get

    t0f,T1(uδ)dsfLp(0,T,W1,p(Ω))T1(uδ)Lp(Ωt)=fLp(0,T,W1,p(Ω))(Ωt{|uδ|1}|(uδ)|pdxds)1/pα2Ωt{|uδ|1}|uδ|pdxds+C(α,p)fpLp(0,T,W1,p(Ω)). (3.7)

    Finally, by (3.3)

    Ωt{|uδ|1}(b|uδψ|)pdxdsΩt{|uδ|1}(b|uδ|)pdxdsbpLp(ΩT). (3.8)

    Gathering (3.6), (3.7), and (3.8) and using Hölder inequality, by (3.5) we have

    ΩΦ(uδ(x,t))dxM0,

    where

    M0:=C(N,p,α)[u02L2(Ω)+fpLp(0,T,W1,p(Ω))+bpLp(ΩT).] (3.9)

    It is easily seen that

    |u|2Φ(u) for |u|1

    and so

    sup0<t<T|{xΩ:|uδ(x,t)|>k}|C(N,p,α,β)M0kk1. (3.10)

    We fix t(0,T) and choose φ:=uδχ(0,t) as a test function in (3.1). Again, assumption (3.2) implies that [(ψuδ)+]q1uδ0 a.e. in ΩT, then

    12uδ(,t)2L2(Ω)+Ωt˜A(x,s,uδ,uδ)uδdxds12u02L2(Ω)+t0f,uδds.

    By Young inequality for ε>0

    t0f,uδdsεΩt|uδ|pdxds+p1ppε1pfpLp(0,T,W1,p(Ω)).

    Then, by (1.5) we further have

    uδ(,t)2L2(Ω)+αΩt|uδ|pdxdsu02L2(Ω)+εΩt|uδ|pdxds+C(ε,p)fpLp(0,T,W1,p(Ω))+Ωt(b|uδψ|)pdxds+ΩtHdxds. (3.11)

    For m>0 to be chosen later, we have from (3.3)

    Ωt(b|uδψ|)pdxdsΩt(b|uδ|)pdxds=Ωt(bχ{bm}|uδ|)pdxds+Ωt(bχ{b>m}|uδ|)pdxds. (3.12)

    We estimate separately the two terms in the right–hand side of (3.12). For k>1 fixed, we obtain

    Ωt(bχ{bm}|uδ|)pdxdsmpt0ds{|uδ(,s)|>k}|uδ|pdx+kpt0dsΩb(x,s)pdx. (3.13)

    Now we apply Hölder inequality (2.1), estimates (2.6) and (3.10) to get

    t0ds{|uδ(,s)|>k}|uδ|pdx=t0dsΩ|uδχ{|uδ(,s)|>k}|pdxt0χ{|uδ(,s)|>k}pLN,(Ω)uδpLp,p(Ω)dsSpN,pMp/N0kp/NΩt|uδ|pdxds, (3.14)

    where M0 is the constant in (3.9). On the other hand, using again Hölder inequality (2.1) and estimate (2.6)

    we have

    Ωt(bχ{b>m}|uδ|)pdxdsSpN,pbχ{b>m}pL(0,T,LN,(Ω))Ωt|uδ|pdxds. (3.15)

    Inserting (3.13), (3.14) and (3.15) into (3.12) we obtain

    Ωt(b|uδψ|)pdxds[mpSpN,pMp/N0kp/N+SpN,pbχ{b>m}pL(0,T,LN,(Ω))]uδpLp(Ωt)+kpt0dsΩb(x,s)pdx. (3.16)

    Observe that (3.11) and (3.16) imply

    12uδ(,t)2L2(Ω)+αuδpLp(Ωt)12u02L2(Ω)+kpbpLp(ΩT)+p1ppε1pfpLp(0,T,W1,p(Ω))+HL1(ΩT)+[ε+mpSpN,pMp/N0kp/N+SpN,pbχ{b>m}pL(0,T,LN,(Ω))]uδpLp(Ωt).

    Now we choose m>0 so large to guarantee

    SpN,pbχ{b>m}pL(0,T,LN,(Ω))<α.

    The existence of such a value of m is a direct consequence of (1.16) and the characterization of distance in (2.9). It is also clear that m is a positive constant depending only on b, N, p and α. So we get

    12uδ(,t)2L2(Ω)+α1uδpLp(Ωt)12u02L2(Ω)+kpbpLp(ΩT)+p1ppε1pfpLp(0,T,W1,p(Ω))+HL1(ΩT)+[ε+mpSpN,pMp/N0kp/N]uδpLp(Ωt)

    for some α1=α1(b,N,p,α). We may also choose ε=α12. Then the latter relation becomes

    12uδ(,t)2L2(Ω)+α12uδpLp(Ωt)12u02L2(Ω)+kpbpLp(ΩT)+C1(b,N,p,α)fpLp(0,T,W1,p(Ω))+HL1(ΩT)+C2(b,N,p,α)(M0k)p/NuδpLp(Ωt).

    We choose k=M0(α14C2)N/p so that C2(M0k)p/N=α14 and therefore

    12uδ(,t)2L2(Ω)+α14uδpLp(Ωt)12u02L2(Ω)+C3(b,N,p,α)Mp0bpLp(ΩT)+C1(b,N,p,α)fpLp(0,T,W1,p(Ω))+HL1(ΩT).

    Taking into account the definition of M0, the latter leads to the estimate (3.4).

    Lemma 3.3. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. Assume further that g defined in (1.15) is such that

    gLq(ΩT). (3.17)

    Then, for every δ>0, every solution uδ of problem (3.1) satisfies

    (uδψ)q1Lq(ΩT)δgLq(ΩT). (3.18)

    Moreover, there exists a positive constant C depending only on the data and independent on δ such that

    tuδLp(0,T;W1,p(Ω))C. (3.19)

    Proof. We use the function ϕ=(ψuδ)+ as a test function in the equation of Problem (3.1). Then, we get

    T0tuδ,(ψuδ)+dt+ΩTA(x,t,max{uδ,ψ},u)(ψuδ)+dxdt=1δΩT[(ψuδ)+]qdxdt+T0f,(ψuδ)+dt.

    Recalling (1.15), this implies

    1δΩT[(ψuδ)+]qdxdt=ΩTg(ψuδ)+dxdtT0g+,(ψuδ)+dtT0t(ψuδ),(ψuδ)+dtΩT{ψ>uδ}[A(x,t,ψ,ψ)A(x,t,ψ,uδ)](ψuδ)dxdt.

    By (1.14) we observe that

    T0t(ψuδ),(ψuδ)+dt=12(uδψ)(T)2L2(Ω)

    hence, by (1.6) we get

    1δΩT[(ψuδ)+]qΩTg(ψuδ)+dxdt.

    Then, using Hölder inequality and dividing both sides of the inequality by (ψuδ)+)Lq((ΩT) we obtain (3.18). To obtain (3.19) we fix φLp(0,T;W1,p0(Ω)) and then we observe that

    |T0tuδ,φdt|(A(,,max{uδ,ψ},uδ)Lp(ΩT)+fLp(ΩT))φLp(0,T;W1,p0(Ω))+1δ(ψuδ)+q1Lq(ΩT)φLq(ΩT).

    At this point we observe that the definition of q and Holder inequality imply

    φLq(ΩT)C(p,|Ω|,T)φLp(ΩT).

    Finally, using (3.18) and Poncaré inequality slicewise, we conclude that

    |T0tuδ,φdt|C(p,|Ω|,T)φLp(0,T;W1,p0(Ω)),

    where C is a positive constant independent of δ. This immediately leads to (3.19).

    We proceed step by step. We first prove the result under regularity assumptions on g and sign conditon (3.2) on the obstacle function ψ. Then we address the general case.

    Proposition 4.1. Let (1.2), (1.4)–(1.16), (3.2) and (3.17) be in charge. There exists at least solution uKψ(ΩT) to the variational inequality (1.3) such that u(,0)=u0 in Ω and satisfying the following estimate

    u2L(0,T,L2(Ω))+upLp(ΩT)C(b,N,p,α)[u02L2(Ω)+fpLp(0,T,W1,p(Ω))+HL1(ΩT)+(u02L2(Ω)+fpLp(0,T,W1,p(Ω))+bpLp(ΩT))pbpLp(ΩT)]. (4.1)

    Proof. By Proposition 3.1, for every δ>0 there exists a solution uδC0([0,T],L2(Ω))Lp(0,T,W1,p0(Ω)) to problem (3.1) satisfying (3.4). Hence we have that, by Lemma 3.3 and Lemma 2.2, there exists uC0([0,T],L2(Ω))Lp(0,T,W1,p0(Ω)) such that

    uδustrongly in Lp(ΩT) (4.2)
    uδuweakly in Lp(ΩT,RN) (4.3)
    uδuweakly  in L(0,T;L2(Ω))tuδtuweakly in Lp(0,T,W1,p(Ω))

    as δ0+. By semicontinuity, (3.4) implies (4.1)

    We claim that the limit function u solves the variational inequality (1.3) in the strong form.

    It is immediate to check that

    u(,0)=u0a.e. in Ω, (4.4)
    uψ a.e. inΩT. (4.5)

    Indeed, (4.4) holds since uδ(,0)=u0 a.e. in Ω for every δ>0. On the other hand, if we pass to the limit as δ0+ in (3.18) and take into account (4.2) we have (uψ)L2p(ΩT)=0 which clearly implies (4.5).

    Our next goal is to prove that

    uδua.e. in ΩT (4.6)

    as δ0+. We test the penalized equation by T1(uδu) and since condition (4.5) implies

    ΩT[(ψuδ)+]q1T1(uδu)dxdt0

    we get the following inequality

    T0tuδ,T1(uδu)dt+ΩTA(x,t,uδψ,uδ)T1(uδu)dzT0f,T1(uδu)dt. (4.7)

    If we set Φ(z):=z0T1(ζ)dζ, by (4.4) we obtain

    T0tuδ,T1(uδu)dt=ΩΦ(uδu)(x,T)dx+T0tu,T1(uδu)dt.

    Because of (4.3), the latter term in the last inequality vanishes in the limit as δ0. So, as Φ is nonnegative, we get

    lim supδ0T0tuδ,T1(uδu)dt0.

    Again by (4.3), the right hand side of (4.7) vanishes in the limit as δ0, and so (4.7) implies

    lim supδ0ΩT{|uδu|1}A(x,t,uδψ,uδ)(uδu)dxdt0. (4.8)

    By (1.7), (3.2) and (3.3) we have

    |A(x,t,uδψ,u)|χ{|uδu|1}β|u|p1+(˜b|uδ|)p1χ{|uδu|1}+Kβ|u|p1+C(p)˜bp1+C(p)(˜b|u|)p1+K

    therefore, by the dominated convergence theorem and by (4.2), we get

    limδ0ΩT{|uδu|1}A(x,t,uδψ,u)(uδu)dxdt=0. (4.9)

    Combining (4.8) and (4.9) and by (1.6) we get

    limδ0ΩT[A(x,t,uδψ,uδ)A(x,t,uδψ,u)]T1(uδu)dxdt=0. (4.10)

    Using again (1.6), relation (4.10) gives

    [A(x,t,uδψ,uδ)A(x,t,uδψ,u)](uδu)χ{|uδu|1}0a.e. in ΩT

    and so by (4.2) we get

    [A(x,t,uδψ,uδ)A(x,t,uδψ,u)](uδu)0a.e. in ΩT

    as δ0. By Lemma 3.1 in [21] we deduce that (4.6) holds.

    We let vKψ(ΩT). It is clear that [(ψuδ)+]q1Tλ(uδv)0 a.e. in ΩT and for every λ>0. For this reason, if we use Tλ(uδv) as a test function in (3.1) we deduce

    T0tuδ,Tλ(uδv)dt+ΩT[A(x,t,uδψ,uδ)A(x,t,uδψ,v)]Tλ(uδv)dxdtT0f,Tλ(uδv)dtΩTA(x,t,uδψ,v)Tλ(uδv)dxdt. (4.11)

    We set Φλ(z):=z0Tλ(ζ)dζ and we have

    T0tuδ,Tλ(uδv)dt=T0tv,Tλ(uδv)dt+T0tuδtv,Tλ(uδv)dt=T0tv,Tλ(uδv)dt+ΩΦλ(uδv)(x,T)dxΩΦλ(u0v(x,0))dx. (4.12)

    We observe that Lemma 2.2 applies because of (3.4) and (3.19), so

    uδ(,t)u(,t)weakly in L2(Ω) for all t[0,T].

    This convergence and the Lipschitz continuity of Φλ gives Φλ(uδv)(,T)Φλ(uv)(,T) weakly in L2(Ω), then

    limδ0ΩΦλ(uδv)(x,T)dx=ΩΦλ(uv)(x,T)dx. (4.13)

    On the other hand, by Fatou lemma, we are able to pass to the limit as δ0 in the third term on the left–hand side of (4.11). Indeed, for this term we know by the monotonicity condition (1.6) that the integrand is nonnegative and we have already observed that uδ and uδ converge a.e. according to (4.2) and (4.6) respectively. We only need to handle the term

    ΩTA(x,t,uδψ,v)Tλ(uδv)dxdt.

    This can be done arguing similarly as for the case λ=1. By (1.7) we have

    |A(x,t,uδψ,v)|χ{|uδv|λ}β|v|p1+K+C(p)λp1(˜bp1+(˜b|v|)p1).

    By (4.2) and (4.5) we obtain A(x,t,uδψ,v)A(x,t,u,v) a.e. in ΩT, Therefore, by the dominated convergence theorem, A(x,t,uδψ,v)A(x,t,u,v) strongly in Lp(ΩT,RN), and this yields

    limδ0ΩTA(x,t,uδ,v)Tλ(uδv)dxdt=ΩTA(x,t,u,v)Tλ(uv)dxdt.

    Taking into account the latter relation and also (4.12) and (4.13), we can now pass to the limit as δ0 in (4.11) and obtain

    T0tv,Tλ(uv)dt+ΩΦλ(uv)(x,T)dxΩΦλ(u0v(x,0))dx+ΩTA(x,t,u,u)Tλ(uv)dxdtT0f,Tλ(uv)dt.

    Since

    Tλ(uv)uvstrongly in Lp(0,T,W1,p0(Ω)) as λ,Φλ(uv)(,T)12|u0v(,0)|2strongly in L1(Ω) as λΦλ(u0v(,0))12|u(,0)v(,0)|2strongly in L1(Ω) as λ

    and also observing that

    T0tv,uvdt=T0tu,uvdt+12Ω|u0v(,0)|2dx12Ω|u(,T)v(,T)|2dx

    we conclude that (1.3) holds.

    Next result shows that a Lewy–Stampacchia inequality can be derived under some suitable assuption, that we are going to remove later.

    Proposition 4.2. Let (1.2), (1.4)–(1.16), (3.2) and (3.17) be in charge. If we also assume that

    gLp(ΩT)Lp(0,T,W1,p0(Ω))g0a.e. in ΩTtgLq(ΩT)

    the solution u of the obstacle problem constructed in Proposition 4.1 satisfies the Lewy–Stampacchia inequality (1.17).

    Proof. We define

    zδ:=g1δ[(ψuδ)+]q1.

    For k1 we also define

    ηk(y):=(q1)y+0min{k,sq2}dsΨk(x,t,λ):=(g1δηk(λ))Λk(x,t,λ):=λ0Ψk(x,t,σ)dσ.

    Thanks to Lemma 4.3 in [15] we are able to test (3.1) by \Psi_k(x, s, u_\delta-\psi) \chi_{(0, t)} for t\in (0, T) , obtaining

    \begin{equation} \begin{split} -\int_{ \Omega_t} & \partial _ t \Lambda_k (x, s, u_\delta-\psi) \, \mathrm d x \, \mathrm d s + \int_{ \Omega} \Lambda_k (x, t, (u_\delta-\psi)(x, t)) \, \mathrm d x - \int_{ \Omega} \Lambda_k (x, 0, (u_\delta-\psi)(x, 0)) \, \mathrm d x \\& \quad - \int_{ \Omega_t} \left[A(x, s, u_\delta\vee \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right]\cdot \nabla \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) \right)^- \, \mathrm d x \, \mathrm d s \\& \quad - \int_{ \Omega_t} z_\delta \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) \right)^- \, \mathrm d x \, \mathrm d s \\ & = -\int_0^t \langle g^+, \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) \right)^- \rangle \, \mathrm d s \leqslant 0. \end{split} \end{equation} (4.14)

    By (1.14) we have

    \int_{ \Omega} \Lambda_k (x, 0, (u_\delta-\psi)(x, 0)) \, \mathrm d x = 0.

    We also have

    \begin{equation*} \begin{split} -\int_{ \Omega_t} \partial _ t \Lambda_k (x, s, u_\delta-\psi) \, \mathrm d x \, \mathrm d s & = -\int_{ \Omega_t} \partial_t g^- \int_0^{u_\delta-\psi} \chi_{ \left\{g^- - \frac 1\delta \eta_k(\tau^-) < 0 \right\} } \, \mathrm d \tau \, \mathrm d x \, \mathrm d s \\ & = -\int_{ \Omega_t} \partial_t g^- \int_0^{-(u_\delta-\psi)^-} \chi_{ \left\{g^- - \frac 1\delta \eta_k(\tau^-) < 0 \right\} } \, \mathrm d \tau \, \mathrm d x \, \mathrm d s \\ & \geqslant -\int_{ \Omega_t} | \partial_t g^- | |(u_\delta-\psi)^-| \, \mathrm d x \, \mathrm d s. \end{split} \end{equation*}

    So, taking into account (4.14), we have

    - \int_{ \Omega_t} | \partial_t g^- | |(u_\delta-\psi)^-| \, \mathrm d x \, \mathrm d s + \\ \int_{ \Omega} \Lambda_k (x, t, (u_\delta-\psi)(x, t)) \, \mathrm d x - \int_{ \Omega_t} z_\delta \left(g^- - \frac 1 \delta \eta_k\left((u_\delta-\psi)^-\right) \right)^- \, \mathrm d x \, \mathrm d s \\ -\int_{ \Omega_t} \left[A(x, s, u_\delta\vee \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right]\cdot \nabla \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) \right)^- \, \mathrm d x \, \mathrm d s \leqslant 0 . (4.15)

    We remark that

    \begin{equation*} \begin{split} - \int_{ \Omega_t} & z_\delta \left(g^- - \frac 1 \delta \eta_k\left((u_\delta-\psi)^-\right) \right)^- \, \mathrm d x \, \mathrm d s \\ & = - \int_{ \Omega_t} \left( g^- - \frac 1 \delta \left[ (\psi-u_\delta)^+ \right]^{q-1}\right)\left(g^- - \frac 1 \delta \eta_k\left((u_\delta-\psi)^-\right) \right)^- \, \mathrm d x \, \mathrm d s. \end{split} \end{equation*}

    Since we have \left\{ g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) < 0\right\} \subset \{ u_\delta < \psi \} then

    - \int_{ \Omega_t} \left[A(x, s, u_\delta\vee \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right]\cdot \nabla \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) \right)^- \, \mathrm d x \, \mathrm d s \\ \quad = \int_{ \Omega_t} \chi_{ \left\{ g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^- < 0 \right\} } \left[A(x, s, \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right]\cdot \nabla \\ \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-)) \right) \, \mathrm d x \, \mathrm d s.

    By (1.6) it follows that

    \begin{equation*} \begin{split}& \left[A(x, s, \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right]\cdot \nabla \left(g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-)) \right) \\ & \quad \geqslant \frac 1 \delta \eta_k^\prime ( (u_\delta-\psi)^-) \left[A(x, s, \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right]\cdot \nabla (u_\delta-\psi) \\ & \qquad \qquad - | \left[A(x, s, \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right] | | \nabla g^- | \\ &\quad \geqslant - \left| A(x, s, \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right| | \nabla g^- |. \end{split} \end{equation*}

    Hence, we deduce from (4.15)

    \begin{equation*} \begin{split} -&\int_{ \Omega_t} | \partial_t g^- | |(u_\delta-\psi)^-| \, \mathrm d x \, \mathrm d s + \int_{ \Omega} \Lambda_k (x, t, (u_\delta-\psi)(x, t)) \, \mathrm d x \\ & - \int_{ \Omega_t} \left( g^- - \frac 1 \delta \left[ (\psi-u_\delta)^+ \right]^{q-1}\right)\left(g^- - \frac 1 \delta \eta_k\left((u_\delta-\psi)^-\right) \right)^- \, \mathrm d x \, \mathrm d s\\ & -\int_{ \Omega_t} \left| A(x, s, \psi, \nabla u_\delta)-A(x, s, \psi, \nabla \psi) \right||\nabla g^-| \, \mathrm d x \, \mathrm d s \leqslant 0 . \end{split} \end{equation*}

    Now, we pass to the limit as k\rightarrow \infty . In particular, by using the monotone convergence theorem, we have

    \lim\limits_{k\rightarrow \infty} \int_{ \Omega} \Lambda_k (x, t, (u_\delta-\psi)(x, t)) \, \mathrm d x = -\int_ \Omega \, \mathrm d x \int_0^{(u_\delta-\psi)(x, t)} \left(g^- - \frac 1 \delta \left[\sigma^-\right]^{q-1}\right)^- \, \mathrm d\sigma \geqslant0

    and also

    -\lim\limits_{k\rightarrow \infty} \int_{ \Omega_t} \left( g^- - \frac 1 \delta \left[ (\psi-u_\delta)^+ \right]^{q-1}\right)\left(g^- - \frac 1 \delta \eta_k\left((u_\delta-\psi)^-\right) \right)^- \, \mathrm d x \, \mathrm d s = \|z_\delta^-\|^2_{L^2( \Omega_t)}

    We gather the previous relations, and (since t\in (0, T) is arbitrary) we get

    -\int_{ \Omega_T} | \partial_t g^- | |(u_\delta-\psi)^-| \, \mathrm d x \, \mathrm d s + \|z_\delta^-\|^2_{L^2( \Omega_T)} \\ \leqslant \int_{ \Omega_T} \chi_{ \{ \psi > u_\delta \} } \left| A(x, t, \psi, \nabla u_\delta)-A(x, t, \psi, \nabla \psi) \right||\nabla g^-| \, \mathrm d x \, \mathrm d s.

    Since it is clear that

    \lim\limits_{\delta\to 0} \int_{ \Omega_T} | \partial_t g^- | |(u_\delta-\psi)^-| \, \mathrm d x \, \mathrm d s = 0

    we obtain

    \begin{equation} \begin{split} \limsup\limits_{\delta\to 0} \|z_\delta^-\|^2_{L^2( \Omega_T)} \leqslant \limsup\limits_{\delta\to 0} \int_{ \Omega_t} \chi_{ \{ \psi > u_\delta \} } \left| A(x, t, \psi, \nabla u_\delta)-A(x, t, \psi, \nabla \psi) \right||\nabla g^-| \, \mathrm d x \, \mathrm d s. \end{split} \end{equation} (4.16)

    Observing that (4.2), (4.5) and (4.6) hold, then

    F_\delta: = \chi_{ \{ \psi > u_\delta \} } \left| A(x, t, \psi, \nabla u_\delta)-A(x, t, \psi, \nabla \psi) \right| \rightarrow 0 \qquad \text{a.e. in } \Omega_T

    as \delta \rightarrow 0 . By (1.7), (3.2) and (3.4), F_\delta is also bounded in L^{p^\prime}(\Omega_T) , hence F_\delta\rightharpoonup 0 in L^{p^\prime}(\Omega_T) . We deduce

    \lim\limits_{\delta\to 0} \int_{ \Omega_T} \chi_{ \{ \psi > u_\delta \} } \left| A(x, t, \psi, \nabla u_\delta)-A(x, t, \psi, \nabla \psi) \right||\nabla g^-| \, \mathrm d x \, \mathrm d s = 0.

    By (4.16) we obtain

    \lim\limits_{\delta\to 0} \|z_\delta^-\|^2_{L^2( \Omega_T)} = 0.

    Hence we have

    0 \leqslant \frac 1 \delta \left[ (u_\delta-\psi)^- \right]^{q-1} = \partial_t u_\delta - { \rm div } A(\cdot, \cdot, u_\delta\vee \psi, \nabla u_\delta)-f

    and so

    0 \leqslant \partial_t u - { \rm div } A(\cdot, \cdot, u , \nabla u )-f.

    Similarly, rewriting (3.1) as follows

    z_\delta^+ + \partial_t u_\delta - { \rm div } A(\cdot, \cdot, u_\delta\vee \psi, \nabla u_\delta)-f = g^-+z_\delta^-

    then

    \partial_t u - { \rm div } A(\cdot, \cdot, u , \nabla u )-f \leqslant g^-

    and the proof is completed.

    Next result provides the one of Theorem 1.1 under the assumption (3.2) but removing condition (3.17).

    Proposition 4.3. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. There exists at least solution u \in \mathcal K_{\psi}(\Omega_T) to the variational inequality (1.3) satisfying u(\cdot, 0) = u_0 in \Omega , the estimate (4.1) and the Lewy–Stampacchia inequality (1.17).

    Proof. We know that

    \begin{equation*} g : = f - \psi_t + \text{div } A(x, t, \psi, \nabla \psi) = g^+-g^-, \end{equation*}

    where g^\pm are nonnegative elements of L^{p'}(0, T, W^{-1, p'}(\Omega)). By using a regularization procedure, due to [7] Lemma p. 593, and Lemma 4.1 in [15], we find a sequence \{g_n^-\}_{n\in \mathbb N} of nonnegative functions such that

    \begin{align*} &g_n^- \in L^{p^\prime}( \Omega_T) \cap L^p(0, T, W^{1, p}_0(\Omega)) \\ &g_n \geqslant 0 \quad \text{a.e. in } \Omega_T \\ &\partial_t g_n^- \in L^{q^\prime}( \Omega_T) \end{align*}

    and

    g_n^- \rightarrow g^- \quad \text{in } L^{p'}(0, T, W^{-1, p'}(\Omega)) \text{ as }n\to \infty .

    We define

    \begin{equation*} f_n = \psi_t - \text{div } A(x, t, \psi, \nabla \psi) + g^+-g_n^- . \end{equation*}

    It is clear that

    \begin{equation*} f_n \rightarrow f \quad \text{in } L^{p'}(0, T, W^{-1, p'}(\Omega)) \end{equation*}

    as n\to \infty . Due to the regularity assumptions on g_n^- , we get the existence of u_n \in \mathcal K_{\psi}(\Omega_T) with u_n(\cdot, 0) = u_0 in \Omega such that for every v \in \mathcal K_{\psi}(\Omega_T) we have

    \begin{equation} \begin{split} \int_0^T \langle \partial_t u_n, v-u_n \rangle\, dt & + \int_{\Omega_T} A(x, t, u_n, \nabla u_n) \cdot \nabla (v-u_n) \, dxdt \geqslant \int_0^T \langle f_n, v-u_n \rangle\, dt. \qquad \end{split} \end{equation} (4.17)

    Moreover, the subsequent estimate holds

    \begin{equation*} \begin{split} & \| u_n (\cdot, t) \| ^2_{L^2( \Omega)} + \|\nabla u_n \|_{L^p(\Omega_t)}^p \leqslant C(b, N, p, \alpha) \biggr[ \| u_0\| ^{2}_{L^2( \Omega)}+ \|f_n\|^{p^\prime}_{ L^{p'}(0, T, W^{-1, p'}(\Omega))} +\|H\|_{L^1(\Omega_T)} \\ &\quad + \left(\|u_0 \|^2_{L^2( \Omega)} + \|f_n\|^{p^\prime}_{ L^{p'}(0, T, W^{-1, p'}(\Omega))} + \|b\|^p_{L^p( \Omega_T)} \right)^p \|b\| _{L^p( \Omega_T)}^p \biggr] \end{split} \end{equation*}

    and the following Lewy-Stampacchia inequality holds

    \begin{equation} 0\leq \partial_t u_n - \text{div } A(x, t, u_n , \nabla u_n ) -f_n \leq g_n^-. \end{equation} (4.18)

    Since the sequence \{f_n\}_{n \in \mathbb N} is strongly converging (and hence bounded) in L^{p'}(0, T, W^{-1, p'}(\Omega)) , we obtain

    \begin{equation*} \sup\limits_{0 < t < T} \int_\Omega |u_n (\cdot, t) |^2\, dx + \int_{\Omega_T} | \nabla u_n |^p\, dxdt \leqslant C \end{equation*}

    for some positive constant C independent of n . Moreover, the Lewy–Stampacchia inequality (4.18) implies a uniform bound of this kind

    \begin{equation*} \|\partial _t u_n \|_{L^{p'}(0, T;W^{-1, p'} ( \Omega ))}\leq C \end{equation*}

    again for some positive constant C independent of n . Therefore, there exists u\in C^0\left([0, T], L^2(\Omega)\right) \cap L^p(0, T, W^{1, p}_0(\Omega)) with u(\cdot, 0) = u_0 in \Omega such that

    \begin{align} u_n & \rightarrow u \quad\text{strongly in } L^p(\Omega_T) \\ \nabla u_n & \rightharpoonup \nabla u \quad\text{weakly in } L^p\left(\Omega_T, \mathbb R^N\right) \\ u_n & \stackrel{\ast}\rightharpoonup u \quad\text{weakly}^{\ast}\text{ in }L^\infty(0, T;L^2(\Omega)) \\ \partial_t u_n & \rightharpoonup \partial_t u \quad\text{weakly in } L^{p'}(0, T, W^{-1, p'}(\Omega)) \end{align} (4.19)

    as n\to \infty . Obviously (4.19) implies u \geqslant \psi a.e. in \Omega_T . If we summarize, we have u \in \mathcal K_{\psi}(\Omega_T) and then v_n: = u_n-\mathcal T_1(u_n-u) \in \mathcal K_{\psi}(\Omega_T). Hence, we use v_n as a test function in (4.17) and, arguing as in the proof of Proposition 4.1, we obtain

    \nabla u_n \rightarrow \nabla u \qquad \text{a.e. in } \Omega_T

    as n\to \infty . For fixed \lambda > 0 and v \in \mathcal K_{\psi}(\Omega_T) we also have v_{n, \lambda}: = u_n-\mathcal T_\lambda(u_n-v) \in \mathcal K_{\psi}(\Omega_T). Arguing again as in the proof of Proposition 4.1, we get (1.3) passing to the limit (first as n \rightarrow \infty and then as \lambda\rightarrow \infty ) in the inequality obtained by testing (4.17) by v_{n, \lambda} .

    Finally, we remove condition (3.2), i.e., we are able to prove Theorem 1.1.

    Proof of Theorem 1.1. The convex set \mathcal K_{\psi}(\Omega_T) is nonempty and one can find w \in \mathcal K_{\psi}(\Omega_T) such that w(\cdot, 0) = \psi(\cdot, 0) in \Omega (see for details Remark 2.1 in [15]). Let us define

    \begin{align*} \hat A(x, t, u, \eta)&: = A(x, t, u+w , \eta+\nabla w ) \\ \hat f&: = f-\partial _ t w \\ \hat \psi&: = \psi-w \\ \hat u_0&: = u_0-w(\cdot, 0). \end{align*}

    Hence \hat f \in L^{p'}(0, T, W^{-1, p'}(\Omega)) and \hat \psi and \psi share the same trace on \partial \Omega \times (0, T) . Therefore, one can conclude

    \begin{align*} & \hat \psi \leqslant 0 \quad \text{a.e. in } \Omega_T \\ & \hat \psi (\cdot, 0) = 0 \quad \text{a.e. in } \Omega . \end{align*}

    Moreover, the vector field \hat A enjoys similar properties as A . This is trivial for conditions (1.6) and (1.7). As in [12], properties of A and Young inequality, we have for \varepsilon > 0

    \hat A(x, t, u, \xi)\cdot \xi \geqslant (\alpha-\beta\, \varepsilon^p)\, |\xi+\nabla w|^p - \left( b^p+ \varepsilon^p\, \tilde b^p\right) |u+w| ^p - H_1

    with a suitable H_1\in L^1(\Omega_T) . Moreover, as an elementary consequence of the convexity of |\; |^p , for 0 < \vartheta < 1 we find a constant C = C(\vartheta, p) > 0 such that

    |\xi+\nabla w|^p \geqslant \vartheta^p\, |\xi|^p-C\, |\nabla w|^p\, , \qquad |u+w|^p \leqslant \vartheta^{-p}\, |u|^p+C\, |w|^p.

    Hence, we get coercivity condition for \hat A :

    \hat A(x, u, \xi)\cdot \xi \geqslant \hat\alpha\, |\xi|^p-(\hat b\, |u|)^p-\hat H,

    where we set

    \hat\alpha = (\alpha-\beta\, \varepsilon^p)\, \vartheta^p\, , \qquad \hat b = \frac{b+ \varepsilon\, \tilde b}{\vartheta}

    and denoted by \hat H a suitable nonnegative function in L^1(\Omega_T) . Obviously, we can make \hat\alpha arbitrarily close to \alpha , by choosing \varepsilon close to 0 and \vartheta close to 1 . Using inequality (2.5) for b and \tilde b in place of f and g , respectively, we can easily show that also \mathscr D_{\hat b} is arbitrarily close to \mathscr D_{ b} , again by choosing \varepsilon close to 0 and \vartheta close to 1 . Indeed, we have

    \begin{equation*} \begin{split} & \mathrm{dist}_{L^\infty(0, T, L^{N, \infty} ( \Omega))} (\hat b, L^\infty( \Omega_T)) \\ & \qquad \leqslant \frac{1+\sqrt \varepsilon}\vartheta\, \mathrm{dist}_{L^\infty(0, T, L^{N, \infty} ( \Omega))}(b, L^\infty( \Omega_T))+\frac{\sqrt \varepsilon(1+\sqrt \varepsilon)}\vartheta\, \|\tilde b\|_{L^\infty(0, T, L^{N, \infty} ( \Omega))}. \end{split} \end{equation*}

    By (1.16) we can also have

    \begin{equation*} \mathscr D_{\hat b} < \frac {\hat\alpha^{1/p}} {S_{N, p}}. \end{equation*}

    We observe that

    \begin{equation*} \hat f - \hat \psi_t + \text{div } A(x, t, \hat \psi, \nabla \hat \psi) = f - \psi_t + \text{div } \hat A(x, t, \psi, \nabla \psi) . \end{equation*}

    We can apply Proposition 4.3 for the operator \hat A . Therefore, we obtain the existence of a function \hat u \in \mathcal K_{\hat \psi} (\Omega_T) such that

    \begin{equation} \hat u(\cdot, 0) = \hat u_0 \qquad \text{in }\Omega \end{equation} (4.20)

    and the following parabolic variational inequality

    \begin{equation*} \begin{split} \int_0^T & \langle \hat u _t, \hat v -\hat u \rangle \, \mathrm d t + \int_{\Omega_T} \hat A(x, t, \hat u, \nabla \hat u) \cdot \nabla (\hat v -\hat u) \, \mathrm d x \, \mathrm d t \geqslant \int_0^T \langle \hat f, \hat v -\hat u \rangle \, \mathrm d t \qquad \end{split} \end{equation*}

    holds true for every admissible function \hat v \in \mathcal K_{\hat \psi} (\Omega_T) . Since any v \in \mathcal K_{ \psi} (\Omega_T) can be rewritten as v = \hat v +w for some \hat v \in \mathcal K_{\hat \psi} (\Omega_T) , by (4.20), by the definitions of \hat A , \hat f and \hat \psi , we see that the variational inequality (1.3) holds true with u: = \hat u +w and for any admissible function v \in \mathcal K_{\psi}(\Omega_T) . The fact that u \in \mathcal K_{\psi}(\Omega_T) and u(\cdot, 0) = u_0 in \Omega is obvious, and this concludes the proof.

    The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). F. Farroni also acknowledges support by project Starplus 2020 Unina Linea 1 "New challenges in the variational modeling of continuum mechanics'' from the University of Naples Federico II and Compagnia di San Paolo. G. Zecca also acknowledges support by Progetto FRA 2022 "Groundwork and OptimizAtion Problems in Transport'' from the University of Naples Federico II.

    The authors declare no conflict of interest.



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