Abstract
We prove the Lewy–Stampacchia’s inequality for elliptic variational inequalities with obstacle involving Leray–Lions type operator whose simpler model case is given by the following
where \(\Omega \) is a smooth bounded domain of \(\mathbb {R}^N\) with \(N\geqslant 2\), \(\Delta _N u\) denotes the classical N–Laplacian operator and the coefficient \(B:\Omega \rightarrow \mathbb {R}^N\) belongs to a suitable Lorentz–Zygmund space. For this kind of obstacle problems, we also provide regularity results and amongst them we give sufficient conditions to get boundedness of solutions.
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1 Introduction
We let \(\Omega \) be a Lipschitz bounded domain of \({{\,\mathrm{\mathbb {R}}\,}}^N\) with \(N\geqslant 2\). In the present paper, we consider some obstacle problem involving a Leray–Lions operator of the type
where \(A:\Omega \times {{\,\mathrm{\mathbb {R}}\,}}\times {{\,\mathrm{\mathbb {R}}\,}}^N\rightarrow {{\,\mathrm{\mathbb {R}}\,}}^N\) is a Carathéodory vector field satisfying the following assumptions
The function space \(L^{N,\infty }\log ^{}L (\Omega )\) is the Lorentz–Zygmund space (see Sect. 2.2 for more precise definitions) which consists of real measurable functions u in \(\Omega \) such that
where \(u^*\) is the decreasing rearrangement of u. The structure assumptions (2), (3), (4) and the presence of coefficients in the lower order term fulfilling (5) are modeled on an operator of the form
where \(\gamma >0\), \(R:=\sup _{x\in \Omega } |x|\) and \(0\in \Omega \). In the right–hand side of (6) we adopt the usual notation \(\Delta _N u:=\text {div}\left( |\nabla u|^{N-2}\nabla u\right) \) for the N–Laplacian operator.
Let us now give a more precise statement of the obstacle problem we consider in the present paper. For a classical overview on topic we refer for instance to [25, 33]. We let \(\psi :\Omega \rightarrow [-\infty ,+\infty ]\) be a measurable function and consider the set
Assume
A function \(u \in {{\,\mathrm{\mathcal {K}}\,}}_\psi (\Omega )\) satisfying a variational inequality of the type
is called a solution to the obstacle problem involving the operator \({{\,\mathrm{\mathcal {A}}\,}}\). Hereafter \(\left\langle \cdot ,\cdot \right\rangle \) denotes the duality product between \(W^{-1,N^\prime }(\Omega )\) and \(W^{1,N}_0(\Omega )\) so assumption (7) means that
for some
In turn, we have
We point out that obstacle problems for monotone or pseudomonotone operators have been previously considered in [5,6,7, 17, 27, 29]. In the peculiar case \(\psi \equiv -\infty \), the solution of the obstacle problem is actually a solution of the Dirichlet problem
in the sense that \(u \in W^{1,N}_0(\Omega )\) is such that
The summability of the coefficients in the lower order term comes into play for the pairing in (8) to be well defined. Thanks to the embedding Sobolev type theorem of Brezis-Wainger [8] and Hansson [24], the membership of the coefficients a and b to \(L^{N,\infty }\log ^{}L(\Omega )\) provides a necessary and sufficient condition to get
See Corollary 2.3 below for more details.
Linear and nonlinear operators similar to (1) with growth exponent \(p\in (1,N)\) have been already considered in literature. We point out that the treatment of problems involving variational inequalities as in (8) in the limit case considered here differs from the case where the principal part behaves like the p–Laplacian with \(1<p<N\) and the analogous of the coefficients a and b in inequalities (2) and (3) are in \(L^N(\Omega )\) [4, 34] or in the weak-Lebesgue space \(L^{N,\infty }(\Omega )\) with distance to bounded functions not large enough [14, 18, 22, 30, 37]. We refer to Section 7 for a discussion on the optimality of our assumptions. We also remark that, when \(\Omega \) is bounded, \(L^{N,\infty }\) \((\Omega )\) is strictly larger than \(L^N (\Omega )\), while \(L^{N,\infty }\log L (\Omega )\) is strictly smaller than \(L^N (\Omega )\), and this aspect will be crucial in our context.
This paper concerns the existence and regularity issues of a solution to problem (8). Before we enter into details, we recall that \(L^\infty (\Omega )\) is not dense in \(L^{N,\infty }\log L(\Omega )\), so one can define the distance to \(L^\infty (\Omega )\) in \(L^{N,\infty }\log L (\Omega )\) by setting
where
with \(R_\Omega := \sup _{x\in \Omega }|x|\) and \(\omega _N\) denotes the measure of the unit ball in \(\mathbb {R}^N\).
We assume that
such that
We point out that this inequality has to be understood in the sense of traces, i.e. \((\psi -w ) ^+\in W_0^{1,N}(\Omega ) \) for all \( w \in W_0^{1,N}(\Omega ) \). This in turn implies that \(\mathcal {K} _{\psi }(\Omega )\) is nonempty, as the positive part \(\psi ^+\) of \(\psi \) is an element of \(\mathcal {K} _{\psi }(\Omega )\).
We define
and we assume that g is an element of the order dual of \(W^{1,N}_0(\Omega )\), that is
An element \(h\in W^{-1,N^\prime }(\Omega )\) belongs to \(\left( W^{-1,N^\prime }(\Omega )\right) ^+\) if \(\langle h, w\rangle \geqslant 0 \) for all \( w \in W_0^{1,p}(\Omega ) \) such that \(w \geqslant 0\) a.e. in \(\Omega \).
We are in a position to state some existence result for a solution to the obstacle problem satisfying a Lewy–Stampacchia type inequality.
Theorem 1.1
Let assumptions (2)–(4), (7) and (13)–(16) be in charge. If we suppose that
then there exists a solution \(u \in \mathcal {K}_\psi (\Omega )\) to the obstacle problem involving \({{\,\mathrm{\mathcal {A}}\,}}\) satisfying the following Lewy–Stampacchia inequality
The Lewy-Stampacchia’s inequality was established for the first time in [28] and it plays a crucial role in existence and regularity theory for variational inequalities driven by various kind of operators, see for instance [5, 11, 23, 29] and the reference therein. The existence of a solution to the problem (9) under the only assumptions (13) and (14) is not addressed in details in this paper, since it is a byproduct of the proof of Theorem 1.1 and it can be shown in a similar fashion as [14, 15]. We explicitly remark that condition (16) is satisfied when \(g \in L^{N^\prime }(\Omega )\) and it occurs just to show existence of a solution to the obstacle problem.
As a consequence of the interplay between a proper bound on the distance to \(L^\infty (\Omega )\) and the regularity of the obstacle function on one hand and the source term on the other, we are able to prove some improvement of the summability of solutions (in the scale of Lorentz–Zygmund spaces).
Theorem 1.2
Assume that (2)–(4) and (14) are in charge and let \(u \in {{\,\mathrm{\mathcal {K}}\,}}_\psi (\Omega )\) be a solution to the obstacle problem involving \({{\,\mathrm{\mathcal {A}}\,}}\). For \(\lambda >1\), let \( \phi , f \in L^{N,\lambda N}\log ^{\frac{\lambda -1}{\lambda }}L(\Omega )\) and \(F \in L^{N,\lambda N}\log ^{\frac{\lambda -1}{\lambda }}L(\Omega ,\mathbb {R}^N)\). Let also \(\psi \in W^1\,L^{N,\lambda N}\log ^{ \frac{\lambda -1}{\lambda }}L(\Omega ) \). If
then
In particular,
Concerning the existence of bounded solutions, we are able to prove the following result.
Theorem 1.3
Assume that (2)–(4) and (14) are in charge and let \(u \in {{\,\mathrm{\mathcal {K}}\,}}_\psi (\Omega )\) be a solution to the obstacle problem involving \({{\,\mathrm{\mathcal {A}}\,}}\). For \(\gamma >1\), let \( \phi ,f\in L^{N,\infty }\log ^{\gamma }L(\Omega )\), \(F\in L^{N,\infty }\log ^{\gamma }L(\Omega ,\mathbb {R}^N)\). Let also \(\psi \in W^1 L^{N,\infty }\log ^{\gamma }L(\Omega )\). Then \(u\in L^{\infty }(\Omega )\).
The proof of Theorem 1.1 is based upon a penalization argument, which is a tool so effective to provide both existence and validity of the Lewy–Stampacchia inequality (18). In few words, a solution to the obstacle problem is obtained as the limit of a sequence of solutions to suitable Dirichlet problems which are a generalization of (9). It is worth to mention here that condition (17) turns to be essential to get existence results for the Dirichlet problem (9). We are going to discuss this issue in details in Sect. 4. The proofs of Theorems 1.2 and 1.3 apply verbatim to the Dirichlet problem (9) (see Theorem 7.1 below for a precise statement).
In case the coefficients in the lower order term vanishes, an elliptic equation with right-hand side in a Zygmund space has been considered in [13]. N-Laplacian type equations have been also considered in [12, 20].
2 Preliminaries
2.1 Notation
If A and B are two quantities, we use \(A\lesssim B\) to denote that there exists a constant \(C>0\), depending on the appropriate parameters, such that \(A\leqslant CB\). We also say that a Banach space X is embedded in another Banach space Y if there exists a bounded linear immersion map \(\iota :X\rightarrow Y\). Equivalently, X can be identified with a subset of Y in a linear way - usually in a natural way - and \(\Vert \cdot \Vert _Y\lesssim \Vert \cdot \Vert _X\).
The truncation at a level \(\sigma >0\) will be denoted by \(T_\sigma (\cdot )\) and is defined by
2.2 Lorentz–Zygmund spaces
Throughout this section we let \(\Omega \) be a bounded domain in \({{\,\mathrm{\mathbb {R}}\,}}^N \). For a real measurable function u defined in \(\Omega \), we let \(\mu _u:[0,\infty )\rightarrow [0,|\Omega |]\) be the distribution function of u, namely
The decreasing rearrangement of u is denoted by \(u^*\) and defined through the formula
As an immediate consequence of the latter definition, one gets that u and \(u^*\) have the same distribution function. Also, it is well known that the following Hardy–Littlewood inequality holds
for all real measurable functions u and v defined in \(\Omega \).
For \(p,q\in (0,\infty ]\) and \(\alpha \in \mathbb {R}\) the Lorentz–Zygmund space \(L^{p,q}\log ^{\alpha }L(\Omega )\) consists of all real measurable functions u in \(\Omega \) such that the quantity
is finite, where we use the convention \(1/\infty =0\). In particular, if \(q<\infty \) we have
while
The quantity \(\Vert \cdot \Vert _{L^{p,q}\log ^{\alpha }L(\Omega ) }\) defines a quasinorm which is equivalent to a norm with respect to which \(L^{p,q}\log ^{\alpha }L(\Omega )\) is complete. Moreover, the space \({L^{p,q}\log ^{\alpha }L(\Omega )}\) includes the Lorentz spaces \(L^{p,q}(\Omega )\) (which corresponds to the case \(\alpha =0\)) and the Zygmund space \(L^p\log ^\beta (\Omega )\) (which corresponds to the case \(p=q\) and \(\alpha =p\beta \)).
It is worth to mention here that, for \(\gamma >0\) the class \(L^{\infty ,\infty }\log ^{1 / \gamma }L (\Omega )\) coincides with the space \(\textrm{EXP}^{\gamma }(\Omega )\) of exponentially integrable functions, which is defined as the class of real measurable functions u in \(\Omega \) for which there exists \(\lambda =\lambda (u)>0\) such that
Inclusion relations among these spaces can be described taking into account several different cases. We start by recalling that Lorentz–Zygmund spaces decrease with the primary index regardless of what the other exponents does, in the sense that
On the other hand, if the primary exponent does not change and is finite, then
while
It is also interesting to see the dual of \(L^{p,q}\log ^{\alpha }L(\Omega )\). It can be shown [3] that if \(1<p<\infty \) and \(1\leqslant q<\infty \) the duality relation
holds, where as usual, for an exponent \(p\in [1,\infty ]\) we denote by \(p'\in [1,\infty ]\) its conjugate exponent defined by the relation
with the convention \(1/\infty =0\).
We will later need to use the following generalized Hölder-type inequality for Lorentz–Zygmund spaces.
Lemma 2.1
Let \(\Omega \subset {{\,\mathrm{\mathbb {R}}\,}}^N\) be a bounded open set. Let \(1\leqslant p,p_1,p_2,q_1,q_2\leqslant \infty \) and \(\alpha \in {{\,\mathrm{\mathbb {R}}\,}}\) be such that
Then
for all \(f\in L^{p_1,q_1}\log ^{\alpha }L(\Omega )\) and all \(g\in L^{p_2,q_2}\log ^{-\alpha }L(\Omega )\).
Proof
From the Hardy–Littlewood inequality we know that \(\Vert fg\Vert _{L^p(\Omega )}\leqslant \Vert f^*(t)g^*(t)\Vert _{L^p(0,|\Omega |)}\). We therefore have
The proof is complete. \(\square \)
2.3 Lorentz–Zygmund–Sobolev spaces
Let \(0<p,q\leqslant \infty \) and \(\alpha \in {{\,\mathrm{\mathbb {R}}\,}}\). We define the Lorentz–Zygmund–Sobolev space \(W^1L^{p,q}\log ^{\alpha }L(\Omega )\) as the space of all \(f\in L^{p,q}\log ^{\alpha }L(\Omega )\cap W^{1,1}(\Omega )\) such that
endowed with the norm \(\Vert f\Vert _{W^1L^{p,q}\log ^{\alpha }L(\Omega )}:=\Vert f\Vert _{L^{p,q}\log ^{\alpha }L(\Omega )}+[f]_{W^1L^{p,q}\log ^{\alpha }L(\Omega )}\), and its subspace \(W^1_0L^{p,q}\log ^{\alpha }L(\Omega )\) as the closure of the space \(C^\infty _c(\Omega )\) in \(W^1L^{p,q}\log ^{\alpha }L(\Omega )\), endowed with the norm \(\left\| \left| \nabla \left( \cdot \right) \right| \right\| _{L^{p,q}\log ^{\alpha }L(\Omega )}\).
Now, we discuss some embedding results in the scale of the Lorentz–Zygmung spaces. First, we want to recall that Yudovich [36], Pohozaev [32] and Trudinger [35] independently proved that \(W^{1,N}_0(\Omega ) \hookrightarrow \textrm{EXP}^{N/(N-1)}(\Omega ) \). This embedding result have been later generalized in the independent papers Brezis–Wainger [8] and Hansson [24], where it is shown that \(W^{1,N}_0(\Omega ) \hookrightarrow L^{\infty ,N}\log ^{-1}L (\Omega )\). Such embedding is optimal in the context of rearrangement–invariant space and there is a relevant Sobolev type inequality naturally connected with this embedding. To provide a sharp form of such inequality, we follow [10]. Let \(\omega _N\) be the measure of the unit ball in \(\mathbb {R}^N\). If
for \(p,q\in (0,\infty ]\) and \(\alpha \in \mathbb {R}\) we define
For the cases we are interested in, (23) and (24) define quasinorms equivalent to \(\Vert \cdot \Vert _{L^{p,q}\log ^{\alpha }L(\Omega )}\).
With this notation at hand, we have the following Sobolev type inequality.
Theorem 2.2
Let \(\Omega \subset {{\,\mathrm{\mathbb {R}}\,}}^N\) be a bounded open set with Lipschitz boundary. If \(u \in W^{1,N}_0(\Omega )\) then \(u\in L^{\infty ,N}\log ^{-1}L (\Omega )\) and
Inequality (25) appears in [10] (see formula (1.13) there) and it is known to be equivalent to the Hardy inequality
which holds true for every \(v \in W^{1,N}_0(\Omega )\), provided \(\Omega \) is a smooth bounded open subset of \(\mathbb {R}^N\) with \(0\in \Omega \) and \(D\geqslant R_\Omega \).
Since Lemma 2.1 holds also for the \(\llbracket \cdot \rrbracket _{L^{p,q}\log ^{\alpha }L}\) norms, a consequence of Theorem 2.2 is the following corollary.
Corollary 2.3
Let \(\Omega \subset {{\,\mathrm{\mathbb {R}}\,}}^N\) be a bounded open set with Lipschitz boundary. Then
for all \(f\in L^{N,\infty }\log ^{}L(\Omega )\) and \(g\in W^{1,N}_0(\Omega )\), where \(S_N=(N-1)\omega _N^{1/N}\).
We remark that \(L^\infty (\Omega )\) is not dense in \(L^{p,\infty }\log ^{\alpha }L(\Omega )\) for \(1<p<\infty \). For such spaces, distance to \(L^\infty \) has been widely studied along with its applications - see for instance [1, 2, 9].
2.4 Fixed point theorems and approximation results
To prove existence results, we shall use the Leray–Schauder fixed point theorem in a version proposed in [19, Theorem 11.3 pg. 280].
Theorem 2.4
Let \(\mathcal {F}\) be a compact mapping of a Banach space X into itself, and suppose there exists a constant M such that \(\Vert x\Vert _{X}<M\) for all \(x\in X\) and \(\lambda \in [0,1]\) satisfying \(x=\lambda \mathcal {F}(x).\) Then, \(\mathcal {F}\) has a fixed point.
We recall that a continuous mapping between two Banach spaces is called compact if the images of bounded sets are precompact.
In the sequel, we will need the following approximation result (see [29]).
Theorem 2.5
Assume that \(q>1\) and \(\Omega \) is a Lipschitz bounded domain of \(\mathbb {R}^N\). Let \(h \in \left( W^{-1,q^\prime }(\Omega )\right) ^+\). Then, there exists a sequence \((h_n)_{n\in \mathbb {N}}\) of nonnegative functions in \(W_0^{1,q}(\Omega ) \) such that \(h_n \rightarrow h\) strongly in \( W^{-1,q^\prime }(\Omega ) \).
We mention here a weak compactness result proved in [14, Lemma 2].
Lemma 2.6
Let \(\mathcal {B}\) be a nonempty subset of \(W^{1,p}_0(\Omega )\) with \(p>1\). Assume that there exists a constant \(C>0\) such that
for any \(\sigma >0\) and \(u \in \mathcal {B}\), where \(\Omega _\sigma :=\{x\in \Omega :\,|u(x)| \geqslant \sigma \}\). Then, there exists a constant \(M>0\) such that
for any \(u \in \mathcal {B}\).
3 A technical tool
Throughout this section we let all the assumptions (2)–(4), (13)–(16) be in charge and for subsequent purposes, we want to describe the properties of Carathéodory vector field
The vector field \(\hat{A}\) satisfies some conditions similar to (2), (3) and (4). We want to discuss the properties of \(\hat{A}\) providing some details, as for instance (19) is made in terms of the constant \(\alpha \) and the coefficient a. As in [16], we prove the following.
Lemma 3.1
Let the assumptions (2)–(4), and (13)–(16) be in charge. For all \(\varepsilon >0\) and \(\vartheta \in (0,1)\), we obtain
where
and \(\hat{\phi }\) is a suitable nonnegative function in \(L^1(\Omega )\). Moreover, the following estimate holds
Proof
By Young’s inequality one gets
with a suitable \(\phi _1\in L^1(\Omega )\). As \( \mathbb {R}\ni t\mapsto |t|^N\) is convex, there exists \(C=C(\vartheta ,N)>0\) such that
Hence, (30) is proved. Estimate (31) follows by the definition of \(\hat{a}\). \(\square \)
4 Auxiliary Dirichlet problems
We consider the following
where \(B:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) is a Carathéodory function satisfying
with \(\gamma >0\) and \(M\in L^N(\Omega )\), and a sign condition of the type
Proposition 4.1
Let \(\Phi \in W^{-1,N'}(\Omega )\) and \(A:\Omega \times \mathbb {R}^N\rightarrow \mathbb {R}^N\) a Carathéodory vector field satisfying (2), (3) and (4). If
then there exists \(u\in W^{1,N}_0(\Omega )\) solving problem (32).
Proof
We divide this proof into several steps.
Step 1: the case of bounded coefficients.
Here we consider the case in which A satisfies (2) and (3) with \(a,b\in L^\infty (\Omega )\). As in [14] we consider, for a fixed \(v\in W^{1,N}_0(\Omega )\),
Thanks to a classical result by Leray and Lions [26] we obtain existence of a solution to
which is unique because of the monotonicity assumption (4). It is not difficult to prove that, for a fixed \(\Phi \in W^{-1,N'}(\Omega )\), the map
sending v to the solution u of (36) is compact.
We want to use Leray-Schauder fixed point theorem to the map \({{\,\mathrm{\mathcal {F}}\,}}\) to construct a solution \(u\in W^{1,N}_0(\Omega )\) to problem (9). Let \(0<t\leqslant 1\) and let u be a solution to \(u=t{{\,\mathrm{\mathcal {F}}\,}}(u)\), i.e.
By using condition (34) we have
We use \(T_\sigma u\) as test for some \(\sigma >0\) and use conditions (2) and (38) to obtain
Applying Young inequality and using the fact that \(t^{1-N}\geqslant 1\) we get
and therefore
and we apply Lemma 2.6 to obtain that \(\Vert u\Vert _{W^{1,N}(\Omega )}\leqslant K\) for some constant K independent of u.
Step 2: we prove the result in the general case by using an approximation argument.
For \(n\in {{\,\mathrm{\mathbb {N}}\,}}\) we define
and we see that the field \(A_n(x,u,\xi ):=A(x,\theta _nu,\xi )\) satisfies
for all \(\xi \ne \eta \) in \({{\,\mathrm{\mathbb {R}}\,}}^N\).
For a fixed \(\Phi \in W^{-1,N'}(\Omega )\), what we proved in Step 1 provides the existence of a solution \(u_n\in W^{1,N}_0(\Omega )\) to the problem
We want to apply again Lemma 2.6. We test problem (42) by \(T_\sigma u\) and we use (39) and (38) to obtain
The latter relation clearly implies
We now take \(M>0\) and write a as \(\left( a-T_Ma\right) +T_Ma\), so if we apply the triangle inequality we obtain
We can now use Corollary 2.3 to manage the first term in the right hand side
Using assumption (17) and the fact that
that can be proved as in [14], we can choose M large enough to have
so that we have
Applying Young inequality we show that the assumptions of Lemma 2.6 and obtain
for some \(K>0\) independent of n.
Due to (43) we can assume that there exists \(u\in W^{1,N}_0(\Omega )\) such that
We test equation (9) with \(T_1(u_n-u)\) and obtain
Since \(T_1(u_n-u)\rightharpoonup 0\) in \(W^{1,N}_0(\Omega )\), we have
Let us show that
Since \(\nabla u_n\overset{W^{1,N}_0(\Omega )}{\rightharpoonup }\nabla u\), it suffices to prove that \(A_n(x,u_n,\nabla u_n)\chi _{\left\{ |u_n-u|\leqslant 1\right\} }\) is compact. Using growth condition (3) we have
and since \(b\in L^{N,\infty }\log ^{}L(\Omega )\hookrightarrow L^N(\Omega )\) we can pass to the limit. We therefore have
and, up to a subsequence,
Reasoning as in [26, Lemma 3.3] we obtain \(\nabla u_n\rightarrow \nabla u\) a.e. in \(\Omega \) and
which concludes the proof. \(\square \)
To conclude this section, we provide an example which shows that a condition on the distance as in (35) is crucial to get existence of a solution to our problem.
Example 4.1
Let \(\gamma >0\) and consider the problem
for \(u\in W^{1,2}_0(B)\), where B is the unit ball in \({{\,\mathrm{\mathbb {R}}\,}}^2\). The underlying operator for problem (44) is given by
By using Young’s inequality we get
where
If we consider the adjoint problem
we have that a solution is given by
and in particular it satisfies
Let us now assume that there exists a solution u to problem (44). If \(\gamma >\frac{1}{2}\) we have that \(v\in W^{1,2}_0(B)\), so we can test it in (44) and we obtain
which is impossible and therefore problem (44) does not admit a solution if \(\gamma >\frac{1}{2}\). This result is also sharp in terms of the parameter \(\gamma \). For the function a it results that
and a direct computation (see for instance [31]) gives us
Taking into account (45), if we impose condition (35) we are forced to require \(\gamma <\frac{1}{2}\). \(\square \)
5 Regularity
In this section we prove Theorems 1.2 and 1.3.
5.1 Improving the summability in the scale of Lorentz–Zygmund spaces
Proof of Theorem 1.2
We divide the proof in several steps.
Step 1: we prove that
under the assumption
We notice that \(v=T_ku\) is a valid test function since if \(u>0\) then \(v>0\geqslant \psi \), while if \(u\leqslant 0\) then \(v\geqslant u\geqslant \psi \). Using the structural assumption (2) we obtain
where \(E_k:=\left\{ x\in \Omega :|u(x)|>k\right\} \). Applying a Young inequality we obtain
where \(\varepsilon >0\) is a parameter we will choose later and C is a constant depending on \(\varepsilon \). Multiplying by \(k^{(\lambda -1)N-1}\) and integrating on [0, K] gives, thanks to Fubini theorem,
which implies
Let \(M>0\). If we write \(a=T_Ma+(a-T_Ma)\), we can give an estimate of \(\left\| au\left| T_Ku\right| ^{\lambda -1}\right\| _{L^N {(\Omega )}}\):
To simplify this inequality, we use Corollary 2.3 on the first term on the right hand side to obtain
We now notice that \(\nabla \left( u\left| T_Ku\right| ^{\lambda -1}\right) \leqslant \lambda \left| \nabla u\right| \left| T_Ku\right| ^{\lambda -1}\), so that
and recalling the assumption (19) we have that by choosing M large enough and \(\varepsilon \) small enough we get
If we use Lemma 2.1 we can write
and similarly for F and u. We can then apply Corollary 2.3
to obtain
Taking the limit \(K\rightarrow +\infty \) we finally get
which concludes the proof, since \(u\in L^{N,\lambda N}\log ^{\frac{\lambda -1}{\lambda }}L(\Omega )\hookrightarrow W^{1,N}_0(\Omega )\).
Step 2: we get rid of assumption (48).
We consider the vector field \(\hat{A}\) as in (29) and we define \(\hat{\psi }:= \psi - \psi ^+\). As \(\hat{\psi } \leqslant 0\) a.e. in \(\Omega \), we obtain the existence of a function \( \hat{u} \in \mathcal {K}_{\hat{\psi }}(\Omega )\) such that the following variational inequality
holds true for every admissible function \(\hat{v} \in \mathcal {K}_{\hat{\psi }} (\Omega )\). Since any \( v \in \mathcal {K}_{ \psi } (\Omega )\) can be rewritten as \(v= \hat{v} + \psi ^+ \) for some \(\hat{v} \in \mathcal {K}_{\hat{\psi }} (\Omega )\). Hence, the variational inequality (8) holds true with \(u:=\hat{u} + \psi ^+ \) and any admissible function \(v \in \mathcal {K}^\prime _{\psi }(\Omega _T)\). It remains to show that u satisfies the required summability. By Step 1, we observe that \(|\hat{u}|^\lambda \in W^{1,N}_0(\Omega )\) and so \(\hat{u} \in L^{\infty ,\lambda N}\log ^{-\frac{1}{\lambda }}L(\Omega )\). On the other hand \(\psi ^+ \in L^{\infty ,\lambda N}\log ^{-\frac{1}{\lambda }}L(\Omega )\) by Theorem 4.2 in [21] and this proves the result. \(\square \)
5.2 Boundedness
Let us now prove the boundedness result.
Proof of Theorem 1.3
We divide the proof in several steps.
Step 1: we prove the result under the assumption (48).
Let \(k>0\) and let us define the function
We have \(\sigma _k(u) \in W^{1,N}_0(\Omega )\) and
where \(E_k:=\{|u|>k\}\). It is not difficult to prove that \(v_k:=u-\sigma _k(u) \in \mathcal {K} _\psi (\Omega )\). Indeed, \(\sigma _k(u)=0\) on the set where \(|u(x)|\leqslant k\). On the other hand, if \(u(x)<-k\) then \(\sigma _k(u)<0\). Finally, if \(u >k\) we notice that \(v_k=V(u)\) with
and \(V'(z)>0\) for \(z>k\), which gives \(v_k>0\geqslant \psi \). We can therefore test our problem with \(v_k\) and applying assumption (2) and Poincaré inequality we get
We observe that \(|\nabla \sigma _k(u)|^N \leqslant \frac{|\nabla u|^N}{(1+|u|)^N}\chi _{E_k}\). Hence, by using Young inequality and reabsorbing in the left–hand side, we obtain
For subsequent estimates, it is convenient to compute the norm \( \Vert \chi _{G}\Vert _{L^{\infty ,N}\log ^{-\gamma }L(\Omega )} \) of the characteristich function of a measurable set G contained in \(\Omega \). As \(\left( \chi _{G}\right) ^* = \chi _{(0,|G|)}\), the definition of \(\Vert \cdot \Vert _{L^{\infty ,N}\log ^{-\gamma }L(\Omega )}\)–norm leads to the following
In the last line we compute the exact value of the integral since \(\gamma > 1/N\). We proceed further and because of Lemma 2.1 we deduce
and similarly
We take into account previous estimates and we get
where
Let us define \(w(z):=\log (1+|z|) \textrm{sign} \,z \) and \(L:=\log (1+k)\). We set \(G_L(m):=m-T_L m\) for \(m\in \mathbb {R}\). Then \(G_L (w(u)) \in W^{1,N}_0(\Omega )\) and in particular \(G_L (w(u))=\textrm{sign}(u)\log \left( \frac{1+|u|}{1+k}\right) \chi _{E_k}\). Hence, Sobolev embedding theorem applied to \(G_L (w(u))\) provides the estimate
as long as \(h>k\). This implies
if \(h>k>0\). If we set \(h=\exp \eta -1\), \(k=\exp \kappa -1\) and define, for \(t>0\),
we obtain
with \(\beta =\frac{\gamma -\frac{1}{N}}{1-\frac{1}{N}}>1\), so \(\phi \) satisfies the assumptions [34, Lemme 4.1] and there is \(\theta >\tau \) such that \(\phi (\theta )=0\), or equivalently that \(|u(x)|\leqslant \log \theta \) almost everywhere.
Step 2: we get rid of assumption (48).
We adopt the same argument as in previous Step 2. By adopting the same notation, the only fact that we need to prove is that \(u=\hat{u}+\psi ^+\) is bounded in \(\Omega \). As \( \hat{u} \) is bounded in \(\Omega \) by previous step, we show that \(\psi ^+\) is bounded in \(\Omega \) as well. As \(\gamma >1\), we have \(\nabla \psi ^+ \in L^{N,1}(\Omega )\) and so \(\psi ^+\) is bounded in \(\Omega \) by Theorem 4.5 in [21]. \(\square \)
6 The Lewy–Stampacchia inequality
6.1 A penalized problem
In order to prove Theorem 1.1, we need to consider a penalized problem of the type
where \(\delta >0\) and \(s \vee t:=\max \left\{ s,t\right\} \) for all \(s,t\in \mathbb {R}\). We add the following assumption on the obstacle function
As a consequence of Theorem 4.1 we easily prove the following a priori estimates related to the penalized problem (51).
Corollary 6.1
Let assumptions (2)–(4), (19) (16) and (52) be in charge. For every \(\delta >0\), there exists a solution \(u_\delta \) to problem (51) satisfying the estimate
for some positive constant C independent of \(\delta \). In particular, there exists \(u\in \mathcal {K}_\psi (\Omega )\) which is a solution to the variational inequality (8) and
Proof
Estimate (53) can be proved by using the same argument which gives (43). It is clear that (53) implies the existence of \(u \in W^{1,N}_0(\Omega )\) such that (54) holds and \(\nabla u_\delta \rightharpoonup \nabla u\) weakly in \(L^N(\Omega ;{{\,\mathrm{\mathbb {R}}\,}}^N)\). Next stage of our proof consists in proving that u is greater than or equal to the obstacle function \(\psi \) a.e. in \(\Omega \). To this aim, we test equation (51) by \((\psi -u_\delta )^+\), and we get in turn
We have that
and using estimate (53) we obtain that
where in particular the right hand side does not depend on \(\delta \). Letting \(\delta \rightarrow 0\) we obtain that \((\psi -u)^+=0\) almost everywhere, i.e. \(u\in \mathcal {K}_\psi (\Omega )\).
We claim that \(\nabla u_\delta \rightarrow \nabla u\) a.e. in \(\Omega \). We test problem (51) with \(T_1(u_\delta -u)\) to obtain
Since \(u_\delta -u\leqslant u_\delta -\psi <0\) when \(\psi -u_\delta >0\) (up to a negligible set). Moreover, since \(u\geqslant \psi \), the remaining term on the right hand side goes to zero as well, so that
Using the fact that \(u_\delta \rightarrow u\) in \(L^N(\Omega )\) and assumption (2) we can apply the dominated convergence theorem to show that
and therefore
Assumption (4) implies that we must also have
and since \(u_\delta \rightarrow u\) in \(L^N(\Omega )\)
We conclude the proof of our claim by using [26, Lemma 3.1]. \(\square \)
Proposition 6.2
Let assumptions (2)–(4), (16), (19) and (52) be in charge. Assume further that
Then, the solution \(u \in \mathcal {K}_\psi (\Omega )\) provided by Corollary 6.1 satisfies the Lewy–Stampacchia inequality (18).
Proof
To simplify the proof, we set
and also we define
Therefore, the equation in problem (51) gives us
As \(g^-\in W^{1,N}_0(\Omega )\), also \(z_\delta \in W^{1,N}_0(\Omega )\), so we may test problem (51) by \(-z_\delta ^-\) and by using the obvious relation \(-z_\delta ^-z_\delta =|z_\delta ^-|^2\) we obtain
We observe that \(z_\delta ^-\) vanishes on the set where \(u_\delta \geqslant \psi \) and we recall that \(g^+\) is a nonnegative element of \(W^{-1,N^\prime }(\Omega )\). We deduce from (57) that
As already mentioned, to obtain the latter relation we used the fact that \(\psi >u_\delta \) a.e. in \(\{z_\delta ^-\ne 0\}\). Moreover, it is clear that \(g^- - \mu _\delta <0\) a.e. in \(\{z_\delta ^-\ne 0\}\), and so (58) gives us
By monotonicity, the first term in the right hand side of (59) is less than or equal to zero, so we have
By using (54) and (55) we deduce that \( \chi _{\{\psi >u_\delta \}} \left| A(x,\psi ,\nabla u_\delta ) - A(x,\psi ,\nabla \psi ) \right| \rightharpoonup 0 \) weakly in \(L^{N^\prime }(\Omega )\) and so \(z^-_\delta \rightarrow 0\) strongly in \(L^2(\Omega )\). Finally, we obtain the Lewy–Stampacchia inequality (18) passing to the limit as \(\delta \rightarrow 0\) in \( \mu _\delta - g^- \leqslant z_\delta ^- \). \(\square \)
We now consider the general case.
Proof of Theorem 1.1
We assume at first that (52) holds, namely \( \psi \leqslant 0\) a.e. in \(\Omega \). Let us consider a sequence \(\hat{g}_n \in W^{1,N}_0(\Omega )\) with \(\hat{g}_n \geqslant 0\) a.e. in \(\Omega \) such that \(\hat{g}_n \rightarrow g^-\) in \(W^{-1,N^\prime }_0(\Omega )\). Accordingly, we define
By Proposition 6.2 we easily see that there exists \(u_n\in W^{1,N}_0(\Omega )\) solving the obstacle problem
Moreover, the Lewy–Stampacchia inequality
holds. We remark that \(u_n-T_\sigma (u_n) \in \mathcal {K}_\psi (\Omega )\) and reasoning as in Proposition 4.1, the estimate
holds with a constant \(M>0\) independent from n, and satisfying a Lewy-Stampacchia inequality. As before, there exists \(u\in W^{1,N}_0(\Omega )\) such that \(u_n\rightarrow u\) in \(L^N(\Omega )\) and \(\nabla u_n\rightharpoonup \nabla u\) in \(L^N(\Omega ;{{\,\mathrm{\mathbb {R}}\,}}^N)\). Testing with \(w_n:=u_n-T_1(u_n-u)\) and reasoning as we did previously, we get \(\nabla u_n\rightarrow \nabla u\) a.e. in \(\Omega \).
We fix \(\lambda >0\) and observe that \(u_n-T_\lambda (u_n-v) \in \mathcal {K}_\psi (\Omega )\) for a fixed \(v \in \mathcal {K}_\psi (\Omega )\). Therefore we have
Using again the dominated convergence theorem, we are able to pass to the limit as \(n\rightarrow \infty \), to get
It follows that
by passing to the limit as \(\lambda \rightarrow \infty \). On the other hand, the Lewy–Stampacchia inequality claimed in the statement of Theorem 1.1 follows by passing to the limit in (61). To get rid of assumption (52) we construct an operator \(\tilde{A}\) as in Lemma 3.1 and use it to deduce a result for our case as done in the last parts of the proofs of Theorems 1.2 and 1.3. \(\square \)
7 Concluding remarks
A careful inspection of the proofs of both Theorems 1.2 and 1.3 allows us to conclude that those regularity results hold for all solutions to problem (9) under the same assumptions on \(f,F,\phi \). Precisely, we have the following result.
Proposition 7.1
Let \(u \in W^{1,N}_0(\Omega )\) solve the Dirichlet problem (11), under the assumptions (2)–(4). If \( \phi \in L^{N,\lambda N}\log ^{\frac{\lambda -1}{\lambda }}L(\Omega ), f \in L^{N,\lambda N}\log ^{\frac{\lambda -1}{\lambda }}L(\Omega ), F \in L^{N,\lambda N}\log ^{\frac{\lambda -1}{\lambda }}L(\Omega ,\mathbb {R}^N)\) for some \(\lambda >1\), then (20) holds true. Moreover, if \( \phi ,f\in L^{N,\infty }\log ^{\gamma }L(\Omega )\), \(F\in L^{N,\infty }\log ^{\gamma }L(\Omega ,\mathbb {R}^N)\) for some \(\gamma >1\), then \(u \in L^\infty (\Omega )\).
The first part of previous theorem would dare to say that boundedness of solutions to problem (9) can be obtained only by assuming that
Here \( \textrm{clos} \left( L^{N,\infty }\log L(\Omega ),L^\infty (\Omega )\right) \) denotes the closure of \(L^\infty (\Omega )\) in \(L^{N,\infty }\log ^{}L (\Omega )\), or equivalently the class of all \(g\in L^{N,\infty }\log ^{}L (\Omega )\) such that
so in particular, recalling that
then \( \textrm{clos} \left( L^{N,\infty }\log L(\Omega ),L^\infty (\Omega )\right) \) contains any function such that
with \(\rho \) being a positive function such that \(\lim \nolimits _{\lambda \rightarrow \infty }\rho (\lambda )=\infty \). The following counterexample shows that the set of assumptions we introduced before does not lead to boundedness of solutions, even if the stronger assumption
is in charge.
Example 7.1
Let \(\theta \in (0,1)\) and B the unit ball in \({{\,\mathrm{\mathbb {R}}\,}}^2\). Consider, for \(u\in W^{1,2}_0(B)\) the equation
with
for \(r\in (0,1)\). In particular, \(a\in L^{2,\infty }(\log L)^{}(\log \log L)^{}(B)\) and \(F\in L^{2,\infty }(\log L)(\log \log L)^{\theta }(B)\), which can be defined in an obvious way and are both contained in \(\textrm{clos} \left( L^{2,\infty }\log L(B),L^\infty (B)\right) \). It can be shown that the function
is the unique solution of problem (62). \(\square \)
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References
Angrisani, F., Ascione, G., Manzo, G.: Atomic decomposition of finite signed measures on compacts of \(\mathbb{R} ^n\). Ann. Fenn. Math. 46(2), 643–654 (2021)
Ascione, G., Manzo, G.: o-O structure of some rearrangement invariant Banach function spaces. J. Ellipt. Parabol. Equ. 6, 1–23 (2020)
Bennett, C., Rudnick, K.: On Lorentz–Zygmund spaces. Dissertationes Math. (Rozprawy Mat.) vol. 175, p. 67 (1980)
Boccardo, L.: Dirichlet problems with singular convection terms and applications. J. Differ. Equ. 258(7), 2290–2314 (2015)
Boccardo, L., Cirmi, G.R., Rodrigues, J.F.: On obstacle problems for non coercive linear operators. J. Ellip. Parabol. Equ. 8, 1–23 (2022)
Boccardo, L., Murat, F., Puel, J.P.: Existence of bounded solutions for non linear elliptic unilateral problems. Ann. Mat. 152(1), 183–196 (1988)
Brézis, H., Stampacchia, G.: Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. France 96, 153–180 (1968)
Brézis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Partial Differ. Equ. 5(7), 773–789 (1980)
Carozza, M., Sbordone, C.: The distance to \({L}^\infty \) in some function spaces and applications. Differ. Integr. Equ. 10(4), 599–607 (1997)
Cianchi, A., Ferone, A.: Hardy inequalities with non-standard remainder terms. Ann. l’IHP Anal. Non Linéaire 25(5), 889–906 (2008)
Donati, F.: A penalty method approach to strong solutions of some nonlinear parabolic unilateral problems. Nonlinear Anal. Theory Methods Appl. 6(6), 585–597 (1982)
Farroni, F., Greco, L., Migliaccio, L., Moscariello, G.: An integral form of the isoperimetric inequality. Ricerche mat. 63(1), 131–141 (2014)
Farroni, F., Greco, L., Moscariello, G.: Stability for p-Laplace type equation in a borderline case. Nonlinear Anal. Theory Methods Appl. 116, 100–111 (2015)
Farroni, F., Greco, L., Moscariello, G., Zecca, G.: Noncoercive quasilinear elliptic operators with singular lower order terms. Calc. Var. Partial. Differ. Equ. 60(3), 1–20 (2021)
Farroni, F., Greco, L., Moscariello, G., Zecca, G.: Nonlinear evolution problems with singular coefficients in the lower order terms. Nonlinear Differ. Equ. Appl. 28(4), 1–25 (2021)
Farroni, F., Greco, L., Moscariello, G., Zecca, G.: Noncoercive parabolic obstacle problems. Nonlinear Anal, Adv (2023). https://doi.org/10.1515/anona-2022-0322
Farroni, F., Moscariello, G., Zecca, G.: Lewy–Stampacchia inequality for noncoercive parabolic obstacle problems. Math. Eng. 5(4), 1–23 (2023)
Giannetti, F., Greco, L., Moscariello, G.: Linear elliptic equations with lower order terms. Differ. Integr. Equ. 26(5/6), 623–638 (2013)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, vol. 224, p. x+401. Springer, Berlin (1977)
Greco, L., Iwaniec, T., Moscariello, G.: Limits of the improved integrability of the volume forms. Indiana Univ. Math. J., 305–339, (1995)
Greco, L., Moscariello, G.: An embedding theorem in Lorentz–Zygmund spaces. Potent. Anal. 5(6), 581–590 (1996)
Greco, L., Moscariello, G., Zecca, G.: Regularity for solutions to nonlinear elliptic equations. Differ. Integr. Equ. 26(9/10), 1105–1113 (2013)
Guibé, O., Mokrane, A., Tahraoui, Y., Vallet, G.: Lewy–Stampacchia’s inequality for a pseudomonotone parabolic problem. Adv. Nonlinear Anal. 9(1), 591–612 (2019)
Hansson, K.: Imbedding theorems of Sobolev type in potential theory. Math. Scand. 45(1), 77–102 (1979)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications, Classics Appl. Math., 31 Society for Industrial and Applied Mathematics (SIAM), Philadelphia, pp. xx+313 (2000)
Leray, J., Lions, J.L.: Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93, 97–107 (1965)
Lewy, H., Stampacchia, G.: On the regularity of the solution of a variational inequality. Commun. Pure Appl. Math. 22, 153–188 (1969)
Lewy, H., Stampacchia, G.: On the Smoothness of Superharmonics which Solve a Minimum Problem. Istituto per le Applicazioni del Calcolo, Rome (1970)
Mokrane, A., Murat, F.: A proof of the Lewy–Stampacchia’s inequality by a penalization method. Potent. Anal. 9(2), 105–142 (1998)
Moscariello, G.: Existence and uniqueness for elliptic equations with lower-order terms. Adv. Calc. Var. 4(4), 421–444 (2011)
Pick, L., Kufner, A.: Function Spaces. Vol. 1. Second Revised and Extended Edition De Gruyter Ser. Nonlinear Anal. Appl., 14 Walter de Gruyter & Co., Berlin, pp. xvi+479 (2013)
Pohozaev, S.I.: The Sobolev embedding in the case pl= n. In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research, Vol. 1965, pp. 158–170 (1964)
Rodrigues, J.-F.: Obstacle Problems in Mathematical Physics. North-Holland Math. Stud., 134 Notas Mat., 114[Mathematical Notes]. North-Holland Publishing Co., Amsterdam, xvi+352 pp. (1987)
Stampacchia, G.: Le: problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. l’instit. Fourier 15(1), 189–257 (1965)
Trudinger, N.: On embeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)
Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. In: Doklady Akademii Nauk, vol. 138(4), pp. 805–808. Russian Academy of Sciences, Moscow (1961)
Zecca, G.: Existence and uniqueness for nonlinear elliptic equations with lower-order terms. Nonlinear Anal. Theory Methods Appl. 75(2), 899–912 (2012)
Acknowledgements
The Authors thank the anonymous Referee for carefully reading the manuscript and for providing valuable comments. 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. The work of F. Farroni is part of the project “Variational Analysis of Complex Systems in Materials Science, Physics and Biology” PRIN Project 2022HKBF5C, financed by European Union through NextGenerationEU.
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Farroni, F., Manzo, G. Regularity results for solutions to elliptic obstacle problems in limit cases. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 113 (2024). https://doi.org/10.1007/s13398-024-01608-w
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DOI: https://doi.org/10.1007/s13398-024-01608-w