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 Ω⊂RN, N⩾2, 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):={v∈Lp(0,T,W1,p0(Ω)):vt∈Lp′(0,T,W−1,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,W−1,p′(Ω)) denote parabolic Banach spaces defined according to (2.7).
Given a measurable function ψ:ΩT∪Ω×{0}→R, we are interested in finding functions u:ΩT→R in the convex subset Kψ(ΩT) of Wp(0,T) defined as
Kψ(ΩT):={v∈Wp(0,T):v⩾ψa.e. in ΩT} |
and satisfying the following variational inequality
∫T0⟨ut,v−u⟩dt+∫ΩTA(x,t,u,∇u)⋅∇(v−u)dxdt⩾∫T0⟨f,v−u⟩dt∀v∈Kψ(ΩT), | (1.3) |
where
f∈Lp′(0,T,W−1,p′(Ω)) | (1.4) |
and ⟨⋅,⋅⟩ denotes the duality between W−1,p′(Ω) and W1,p0(Ω). The vector field
A=A(x,t,u,ξ):ΩT×R×RN→RN |
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 u∈R and ξ,η∈RN,
A(x,t,u,ξ)⋅ξ⩾α|ξ|p−(b(x,t)|u|)p−H(x,t) | (1.5) |
[A(x,t,u,ξ)−A(x,t,u,η)]⋅(ξ−η)>0if ξ≠η | (1.6) |
|A(x,t,u,ξ)|⩽β|ξ|p−1+(˜b(x,t)|u|)p−1+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 H∈L1(ΩT), K∈Lp′(ΩT) and
b,˜b∈L∞(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
ψ∈C0([0,T],L2(Ω))∩Lp(0,T,W1,p(Ω)) | (1.9) |
ψ⩽0a.e. in ∂Ω×(0,T) | (1.10) |
ψt∈Lp′(ΩT) | (1.11) |
ψ(⋅,0)∈W1,p0(Ω). | (1.12) |
For
u0∈L2(Ω) | (1.13) |
we impose the following compatibility condition
u0⩾ψ(⋅,0)a.e. in Ω. | (1.14) |
In the following, we will refer to a function u∈Kψ(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:=f−ψt+div A(x,t,ψ,∇ψ)=g+−g−withg+,g−∈Lp′(0,T,W−1,p′(Ω))+. | (1.15) |
Here Lp′(0,T,W−1,p′(Ω))+ denotes the non-negative elements of Lp′(0,T,W−1,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,W−1,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=ω−1/NNpN−p and ωN denotes the measure of the unit ball of RN. Then, there exists at least a solution u∈Kψ(ΩT) of the strong form of the variational inequality (1.3) satisfying u(⋅,0)=u0. Moreover, the following Lewy-Stampacchia inequality holds
0≤∂tu−div A(x,t,u,∇u)−f≤g−=(f−∂tψ+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|p−2ξ+e−t|u|p−2u(γ|x|+1γarctan|x|)p−1x|x| |
satisfies (1.5)–(1.8). According to (2.2) and (2.3) below, we get that
Db=(1−1p)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
‖b‖L∞(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,z2∈R, we often use the notation z1∧z2 and z1∨z2 in place of min{z1,z2} and max{z1,z2} respectively.
Let Ω be a bounded domain in RN. For any 1<p<∞ and 1≤q<∞, the Lorentz space Lp,q(Ω) is the set of real measurable functions f on Ω such that
‖f‖qLp,q:=p∫∞0[λf(k)]qpkq−1dk<∞. |
Here λf(k):=|{x∈Ω:|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
‖f‖pLp,∞:=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 1≤q<p<r≤∞, the following inclusions hold
Lr(Ω)⊂Lp,q(Ω)⊂Lp,r(Ω)⊂Lp,∞(Ω)⊂Lq(Ω). |
For 1<p<∞, 1≤q≤∞ and 1p+1p′=1, 1q+1q′=1, if f∈Lp,q(Ω), g∈Lp′,q′(Ω) we have the Hölder–type inequality
∫Ω|f(x)g(x)|dx≤‖f‖Lp,q‖g‖Lp′,q′. | (2.1) |
Since L∞(Ω) is not dense in Lp,∞(Ω), for f∈Lp,∞(Ω) in [6] the Authors stated the following
distLp,∞(Ω)(f,L∞(Ω)):=infg∈L∞(Ω)‖f−g‖Lp,∞(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→+∞‖f−Tmf‖Lp,∞, |
where, for all m>0, Tm is the truncation operator at levels ±m, i.e.,
Tmy:=min{m,max{−m,y}}for y∈R. | (2.4) |
Another useful estimate is provided by the following sort of triangle inequality
‖f+εg‖Lp,∞⩽(1+√ε)‖f‖Lp,∞+√ε(1+√ε)‖g‖Lp,∞ | (2.5) |
which holds true for f,g∈Lp,∞(Ω) and ε>0.
For 1≤q<∞, 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 0∈Ω, 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, 1≤q≤p, then every function u∈W1,10(Ω) verifying |∇u|∈Lp,q(Ω) actually belongs to Lp∗,q(Ω), where p∗:=NpN−p is the Sobolev conjugate exponent of p and
‖u‖Lp∗,q≤SN,p‖∇u‖Lp,q, | (2.6) |
where SN,p is the Sobolev constant given by SN,p=ω−1/NNpN−p.
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
‖u‖Lp(0,T,X):=(∫T0‖u(t)‖pXdt)1/p<∞ | (2.7) |
whenever 1≤p<∞, and
‖u‖L∞(0,T,X):=esssup0<t<T‖u(t)‖X<∞ |
for p=∞. The space C0([0,T],X) represents the class of all continuous functions u:[0,T]→X with the norm
‖u‖C0([0,T],X):=max0≤t≤T‖u(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 ‖g‖W1,p0(Ω):=‖∇g‖Lp(Ω) for g∈W1,p0(Ω).
For f∈L∞(0,T,Lp,∞(Ω)) we define
distL∞(0,T,Lp,∞(Ω))(f,L∞(ΩT))=infg∈L∞(ΩT)‖f−g‖L∞(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
‖u‖Wp(0,T):=‖u‖Lp(0,T,W1,p(Ω))+‖ut‖Lp′(0,T,W−1,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 u∈Wp(0,T) satisfies
‖u‖C0([0,T],L2(Ω))≤C‖u‖Wp(0,T) |
for some constant C>0.
Moreover, the function t∈[0,T]↦‖u(⋅,t)‖2L2(Ω) is absolutely continuous and
12ddt‖u(⋅,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:={u∈Lp(0,T,X0):∂tu∈Lq(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=W−1,p′(Ω) and X=Lp(Ω) if p≥2 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
{∂tuδ−div [A(x,t,max{uδ,ψ},∇uδ)]=1δ[(ψ−uδ)+]q−1+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
ψ⩽0a.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
|a∨a′|⩽|a|∀a∈R∀a′∈(−∞,0] | (3.3) |
and recalling (1.5), (1.6) and (1.7), we easily deduce
˜A(x,t,u,ξ)⋅ξ⩾α|ξ|p−(b(x,t)|u|)p−H(x,t)[˜A(x,t,u,ξ)−˜A(x,t,u,η)]⋅(ξ−η)>0if ξ≠η|˜A(x,t,u,ξ)|⩽β|ξ|p−1+(˜b(x,t)|u|)p−1+K(x,t) |
for a.e. (x,t)∈ΩT and for any u∈R and ξ,η∈RN.
For u0∈L2(Ω) and f∈Lp′(0,T,W−1,p′(Ω)), a solution to problem (3.1) is a function
uδ∈C0([0,T],L2(Ω))∩Lp(0,T,W1,p0(Ω)) |
such that
−∫ΩTuδφtdxds+∫ΩT˜A(x,s,uδ,∇uδ)⋅∇φdxds=1δ∫ΩT[(ψ−uδ)+]q−1φdxds+∫Ωu0φ(x,0)dx+∫T0⟨f,φ⟩ds |
for every φ∈C∞(ˉΩT) such that suppφ⊂[0,T)×Ω.
By using the elementary inequality
(a+a′)θ⩽aθ+a′θ∀a,a′∈[0,+∞)∀θ∈(0,1) |
and Young inequality we see that
p<2⟹[(ψ−u)+]p−1⩽|ψ|p−1+|u|p−1⩽(p−1)(|u|+|ψ|)+2(2−p). |
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 uδ∈C0([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 uδ∈C0([0,T],L2(Ω))∩Lp(0,T,W1,p0(Ω)) to problem (3.1) satisfies the following estimate
‖uδ‖2L∞(0,T,L2(Ω))+‖∇uδ‖pLp(ΩT)⩽C(b,N,p,α)[‖u0‖2L2(Ω)+‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖H‖L1(ΩT)+(‖u0‖2L2(Ω)+‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖b‖pLp(ΩT))p‖b‖pLp(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 z∈R, we have
∫ΩΦ(uδ(x,t))dx+∫Ωt˜A(x,s,uδ,∇uδ)⋅∇T1(uδ)dxds=1δ∫Ωt[(ψ−uδ)+]q−1T1(uδ)dxds+∫ΩΦ(u0)dx+∫t0⟨f,T1(uδ)⟩ds. |
Assumption (3.2) implies that [(ψ−uδ)+]q−1T1(uδ)⩽0 a.e. in ΩT, so we have
∫ΩΦ(uδ(x,t))dx+∫Ωt∩{|uδ|⩽1}˜A(x,s,uδ,∇uδ)⋅∇uδdxds⩽∫ΩΦ(u0(x,0))dx+∫t0⟨f,T1(uδ)⟩ds. |
By (1.5) and (1.7) we deduce
∫ΩΦ(uδ(x,t))dx+α∫Ωt∩{|uδ|⩽1}|∇uδ|pdxds⩽∫ΩΦ(u0)dx+∫t0⟨f,T1(uδ)⟩ds+∫Ωt∩{|uδ|⩽1}(b|uδ∨ψ|)pdxds+∫Ωt∩{|uδ|⩽1}Hdxds. | (3.5) |
Now, as 0⩽Φ(z)⩽z22 for all z∈R, we have
∫ΩΦ(u0)dx⩽12‖u0‖2L2(Ω). | (3.6) |
By Hölder and Young inequality we get
∫t0⟨f,T1(uδ)⟩ds⩽‖f‖Lp′(0,T,W−1,p′(Ω))‖∇T1(uδ)‖Lp(Ωt)=‖f‖Lp′(0,T,W−1,p′(Ω))(∫Ωt∩{|uδ|⩽1}|∇(uδ)|pdxds)1/p⩽α2∫Ωt∩{|uδ|⩽1}|∇uδ|pdxds+C(α,p)‖f‖p′Lp′(0,T,W−1,p′(Ω)). | (3.7) |
Finally, by (3.3)
∫Ωt∩{|uδ|⩽1}(b|uδ∨ψ|)pdxds⩽∫Ωt∩{|uδ|⩽1}(b|uδ|)pdxds⩽‖b‖pLp(ΩT). | (3.8) |
Gathering (3.6), (3.7), and (3.8) and using Hölder inequality, by (3.5) we have
∫ΩΦ(uδ(x,t))dx⩽M0, |
where
M0:=C(N,p,α)[‖u0‖2L2(Ω)+‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖b‖pLp(ΩT).] | (3.9) |
It is easily seen that
|u|2⩽Φ(u) for |u|⩾1 |
and so
sup0<t<T|{x∈Ω:|uδ(x,t)|>k}|⩽C(N,p,α,β)M0k∀k⩾1. | (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 [(ψ−uδ)+]q−1uδ⩽0 a.e. in ΩT, then
12‖uδ(⋅,t)‖2L2(Ω)+∫Ωt˜A(x,s,uδ,∇uδ)⋅∇uδdxds⩽12‖u0‖2L2(Ω)+∫t0⟨f,uδ⟩ds. |
By Young inequality for ε>0
∫t0⟨f,uδ⟩ds⩽ε∫Ωt|∇uδ|pdxds+p−1ppε1−p‖f‖p′Lp′(0,T,W−1,p′(Ω)). |
Then, by (1.5) we further have
‖uδ(⋅,t)‖2L2(Ω)+α∫Ωt|∇uδ|pdxds⩽‖u0‖2L2(Ω)+ε∫Ωt|∇uδ|pdxds+C(ε,p)‖f‖p′Lp′(0,T,W−1,p′(Ω))+∫Ωt(b|uδ∨ψ|)pdxds+∫ΩtHdxds. | (3.11) |
For m>0 to be chosen later, we have from (3.3)
∫Ωt(b|uδ∨ψ|)pdxds⩽∫Ωt(b|uδ|)pdxds=∫Ωt(bχ{b⩽m}|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{b⩽m}|uδ|)pdxds⩽mp∫t0ds∫{|uδ(⋅,s)|>k}|uδ|pdx+kp∫t0ds∫Ωb(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=∫t0ds∫Ω|uδχ{|uδ(⋅,s)|>k}|pdx⩽∫t0‖χ{|uδ(⋅,s)|>k}‖pLN,∞(Ω)‖uδ‖pLp∗,p(Ω)ds⩽SpN,pMp/N0kp/N∫Ωt|∇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|)pdxds⩽SpN,p‖bχ{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|uδ∨ψ|)pdxds⩽[mpSpN,pMp/N0kp/N+SpN,p‖bχ{b>m}‖pL∞(0,T,LN,∞(Ω))]‖∇uδ‖pLp(Ωt)+kp∫t0ds∫Ωb(x,s)pdx. | (3.16) |
Observe that (3.11) and (3.16) imply
12‖uδ(⋅,t)‖2L2(Ω)+α‖∇uδ‖pLp(Ωt)⩽12‖u0‖2L2(Ω)+kp‖b‖pLp(ΩT)+p−1ppε1−p‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖H‖L1(ΩT)+[ε+mpSpN,pMp/N0kp/N+SpN,p‖bχ{b>m}‖pL∞(0,T,LN,∞(Ω))]‖∇uδ‖pLp(Ωt). |
Now we choose m>0 so large to guarantee
SpN,p‖bχ{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
12‖uδ(⋅,t)‖2L2(Ω)+α1‖∇uδ‖pLp(Ωt)⩽12‖u0‖2L2(Ω)+kp‖b‖pLp(ΩT)+p−1ppε1−p‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖H‖L1(ΩT)+[ε+mpSpN,pMp/N0kp/N]‖∇uδ‖pLp(Ωt) |
for some α1=α1(b,N,p,α). We may also choose ε=α12. Then the latter relation becomes
12‖uδ(⋅,t)‖2L2(Ω)+α12‖∇uδ‖pLp(Ωt)⩽12‖u0‖2L2(Ω)+kp‖b‖pLp(ΩT)+C1(b,N,p,α)‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖H‖L1(ΩT)+C2(b,N,p,α)(M0k)p/N‖∇uδ‖pLp(Ωt). |
We choose k=M0(α14C2)N/p so that C2(M0k)p/N=α14 and therefore
12‖uδ(⋅,t)‖2L2(Ω)+α14‖∇uδ‖pLp(Ωt)⩽12‖u0‖2L2(Ω)+C3(b,N,p,α)Mp0‖b‖pLp(ΩT)+C1(b,N,p,α)‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖H‖L1(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
g−∈Lq′(ΩT). | (3.17) |
Then, for every δ>0, every solution uδ of problem (3.1) satisfies
‖(uδ−ψ)−‖q−1Lq(ΩT)≤δ‖g−‖Lq′(ΩT). | (3.18) |
Moreover, there exists a positive constant C depending only on the data and independent on δ such that
‖∂tuδ‖Lp′(0,T;W−1,p′(Ω))≤C. | (3.19) |
Proof. We use the function ϕ=(ψ−uδ)+ as a test function in the equation of Problem (3.1). Then, we get
∫T0⟨∂tuδ,(ψ−uδ)+⟩dt+∫ΩTA(x,t,max{uδ,ψ},∇u)⋅∇(ψ−uδ)+dxdt=1δ∫ΩT[(ψ−uδ)+]qdxdt+∫T0⟨f,(ψ−uδ)+⟩dt. |
Recalling (1.15), this implies
1δ∫ΩT[(ψ−uδ)+]qdxdt=∫ΩTg−(ψ−uδ)+dxdt−∫T0⟨g+,(ψ−uδ)+⟩dt−∫T0⟨∂t(ψ−uδ),(ψ−uδ)+⟩dt−∫ΩT∩{ψ>uδ}[A(x,t,ψ,∇ψ)−A(x,t,ψ,∇uδ)]⋅∇(ψ−uδ)dxdt. |
By (1.14) we observe that
∫T0⟨∂t(ψ−uδ),(ψ−uδ)+⟩dt=12‖(uδ−ψ)−(T)‖2L2(Ω) |
hence, by (1.6) we get
1δ∫ΩT[(ψ−uδ)+]q≤∫ΩTg−(ψ−uδ)+dxdt. |
Then, using Hölder inequality and dividing both sides of the inequality by ‖(ψ−uδ)+)‖Lq((ΩT) we obtain (3.18). To obtain (3.19) we fix φ∈Lp(0,T;W1,p0(Ω)) and then we observe that
|∫T0⟨∂tuδ,φ⟩dt|≤(‖A(⋅,⋅,max{uδ,ψ},∇uδ)‖Lp′(ΩT)+‖f‖Lp′(ΩT))‖φ‖Lp(0,T;W1,p0(Ω))+1δ‖(ψ−uδ)+‖q−1Lq(ΩT)‖φ‖Lq(ΩT). |
At this point we observe that the definition of q and Holder inequality imply
‖φ‖Lq(ΩT)≤C(p,|Ω|,T)‖φ‖Lp(ΩT). |
Finally, using (3.18) and Poncaré inequality slicewise, we conclude that
|∫T0⟨∂tuδ,φ⟩dt|≤C(p,|Ω|,T)‖φ‖Lp(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 u∈Kψ(ΩT) to the variational inequality (1.3) such that u(⋅,0)=u0 in Ω and satisfying the following estimate
‖u‖2L∞(0,T,L2(Ω))+‖∇u‖pLp(ΩT)⩽C(b,N,p,α)[‖u0‖2L2(Ω)+‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖H‖L1(ΩT)+(‖u0‖2L2(Ω)+‖f‖p′Lp′(0,T,W−1,p′(Ω))+‖b‖pLp(ΩT))p‖b‖pLp(ΩT)]. | (4.1) |
Proof. By Proposition 3.1, for every δ>0 there exists a solution uδ∈C0([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 u∈C0([0,T],L2(Ω))∩Lp(0,T,W1,p0(Ω)) such that
uδ→ustrongly in Lp(ΩT) | (4.2) |
∇uδ⇀∇uweakly in Lp(ΩT,RN) | (4.3) |
uδ∗⇀uweakly ∗ in L∞(0,T;L2(Ω))∂tuδ⇀∂tuweakly in Lp′(0,T,W−1,p′(Ω)) |
as δ→0+. 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) |
u≥ψ 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 δ→0+ in (3.18) and take into account (4.2) we have ‖(u−ψ)−‖L2∧p(ΩT)=0 which clearly implies (4.5).
Our next goal is to prove that
∇uδ→∇ua.e. in ΩT | (4.6) |
as δ→0+. We test the penalized equation by T1(uδ−u) and since condition (4.5) implies
∫ΩT[(ψ−uδ)+]q−1T1(uδ−u)dxdt⩽0 |
we get the following inequality
∫T0⟨∂tuδ,T1(uδ−u)⟩dt+∫ΩTA(x,t,uδ∨ψ,∇uδ)⋅∇T1(uδ−u)dz⩽∫T0⟨f,T1(uδ−u)⟩dt. | (4.7) |
If we set Φ(z):=∫z0T1(ζ)dζ, by (4.4) we obtain
∫T0⟨∂tuδ,T1(uδ−u)⟩dt=∫ΩΦ(uδ−u)(x,T)dx+∫T0⟨∂tu,T1(uδ−u)⟩dt. |
Because of (4.3), the latter term in the last inequality vanishes in the limit as δ→0. So, as Φ is nonnegative, we get
lim supδ→0∫T0⟨∂tuδ,T1(uδ−u)⟩dt⩾0. |
Again by (4.3), the right hand side of (4.7) vanishes in the limit as δ→0, and so (4.7) implies
lim supδ→0∫ΩT∩{|uδ−u|⩽1}A(x,t,uδ∨ψ,∇uδ)⋅∇(uδ−u)dxdt⩽0. | (4.8) |
By (1.7), (3.2) and (3.3) we have
|A(x,t,uδ∨ψ,∇u)|χ{|uδ−u|⩽1}⩽β|∇u|p−1+(˜b|uδ|)p−1χ{|uδ−u|⩽1}+K⩽β|∇u|p−1+C(p)˜bp−1+C(p)(˜b|u|)p−1+K |
therefore, by the dominated convergence theorem and by (4.2), we get
limδ→0∫ΩT∩{|uδ−u|⩽1}A(x,t,uδ∨ψ,∇u)⋅∇(uδ−u)dxdt=0. | (4.9) |
Combining (4.8) and (4.9) and by (1.6) we get
limδ→0∫ΩT[A(x,t,uδ∨ψ,∇uδ)−A(x,t,uδ∨ψ,∇u)]⋅∇T1(uδ−u)dxdt=0. | (4.10) |
Using again (1.6), relation (4.10) gives
[A(x,t,uδ∨ψ,∇uδ)−A(x,t,uδ∨ψ,∇u)]⋅∇(uδ−u)χ{|uδ−u|⩽1}→0a.e. in ΩT |
and so by (4.2) we get
[A(x,t,uδ∨ψ,∇uδ)−A(x,t,uδ∨ψ,∇u)]⋅∇(uδ−u)→0a.e. in ΩT |
as δ→0. By Lemma 3.1 in [21] we deduce that (4.6) holds.
We let v∈Kψ(ΩT). It is clear that [(ψ−uδ)+]q−1Tλ(uδ−v)⩽0 a.e. in ΩT and for every λ>0. For this reason, if we use Tλ(uδ−v) as a test function in (3.1) we deduce
∫T0⟨∂tuδ,Tλ(uδ−v)⟩dt+∫ΩT[A(x,t,uδ∨ψ,∇uδ)−A(x,t,uδ∨ψ,∇v)]⋅∇Tλ(uδ−v)dxdt⩽∫T0⟨f,Tλ(uδ−v)⟩dt−∫ΩTA(x,t,uδ∨ψ,∇v)⋅∇Tλ(uδ−v)dxdt. | (4.11) |
We set Φλ(z):=∫z0Tλ(ζ)dζ and we have
∫T0⟨∂tuδ,Tλ(uδ−v)⟩dt=∫T0⟨∂tv,Tλ(uδ−v)⟩dt+∫T0⟨∂tuδ−∂tv,Tλ(uδ−v)⟩dt=∫T0⟨∂tv,Tλ(uδ−v)⟩dt+∫ΩΦλ(uδ−v)(x,T)dx−∫ΩΦλ(u0−v(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(uδ−v)(⋅,T)⇀Φλ(u−v)(⋅,T) weakly in L2(Ω), then
limδ→0∫ΩΦλ(uδ−v)(x,T)dx=∫ΩΦλ(u−v)(x,T)dx. | (4.13) |
On the other hand, by Fatou lemma, we are able to pass to the limit as δ→0 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,uδ∨ψ,∇v)⋅∇Tλ(uδ−v)dxdt. |
This can be done arguing similarly as for the case λ=1. By (1.7) we have
|A(x,t,uδ∨ψ,∇v)|χ{|uδ−v|⩽λ}⩽β|∇v|p−1+K+C(p)λp−1(˜bp−1+(˜b|v|)p−1). |
By (4.2) and (4.5) we obtain A(x,t,uδ∨ψ,∇v)→A(x,t,u,∇v) a.e. in ΩT, Therefore, by the dominated convergence theorem, A(x,t,uδ∨ψ,∇v)→A(x,t,u,∇v) strongly in Lp′(ΩT,RN), and this yields
limδ→0∫ΩTA(x,t,uδ,∇v)⋅∇Tλ(uδ−v)dxdt=∫ΩTA(x,t,u,∇v)⋅∇Tλ(u−v)dxdt. |
Taking into account the latter relation and also (4.12) and (4.13), we can now pass to the limit as δ→0 in (4.11) and obtain
∫T0⟨∂tv,Tλ(u−v)⟩dt+∫ΩΦλ(u−v)(x,T)dx−∫ΩΦλ(u0−v(x,0))dx+∫ΩTA(x,t,u,∇u)⋅∇Tλ(u−v)dxdt⩽∫T0⟨f,Tλ(u−v)⟩dt. |
Since
Tλ(u−v)→u−vstrongly in Lp(0,T,W1,p0(Ω)) as λ→∞,Φλ(u−v)(⋅,T)→12|u0−v(⋅,0)|2strongly in L1(Ω) as λ→∞Φλ(u0−v(⋅,0))→12|u(⋅,0)−v(⋅,0)|2strongly in L1(Ω) as λ→∞ |
and also observing that
∫T0⟨∂tv,u−v⟩dt=∫T0⟨∂tu,u−v⟩dt+12∫Ω|u0−v(⋅,0)|2dx−12∫Ω|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
g−∈Lp′(ΩT)∩Lp(0,T,W1,p0(Ω))g−⩾0a.e. in ΩT∂tg−∈Lq′(ΩT) |
the solution u of the obstacle problem constructed in Proposition 4.1 satisfies the Lewy–Stampacchia inequality (1.17).
Proof. We define
zδ:=g−−1δ[(ψ−uδ)+]q−1. |
For k⩾1 we also define
ηk(y):=(q−1)∫y+0min{k,sq−2}dsΨk(x,t,λ):=−(g−−1δηk(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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1. | Fernando Farroni, Gianluigi Manzo, Regularity results for solutions to elliptic obstacle problems in limit cases, 2024, 118, 1578-7303, 10.1007/s13398-024-01608-w |