UNILATERAL PROBLEMS FOR THE p-LAPLACE OPERATOR IN PERFORATED MEDIA INVOLVING LARGE PARAMETERS

We address homogenization problems for variational inequalities issue from unilateral constraints for the p-Laplacian posed in perforated domains of R, with n ≥ 3 and p ∈ [2, n]. ε is a small parameter which measures the periodicity of the structure while aε ε measures the size of the perforations. We impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter βε which may be very large, namely, βε →∞ as ε→ 0. We first consider the case where p < n and the domains periodically perforated by tiny balls and we obtain homogenized problems depending on the relations between the different parameters of the problem: p, n, ε, aε and βε. Critical relations for parameters are obtained which mark important changes in the behavior of the solutions. Correctors which provide improved convergence are also computed. Then, we extend the results for p = n and the case of non periodically distributed isoperimetric perforations. We make it clear that in the averaged constants of the problem, the perimeter of the perforations appears for any shape. 1991 Mathematics Subject Classification. 35B27, 35J60, 35J87, 35B25. ...


Introduction
Homogenization problems in perforated media for the p-Laplace operator have been considered in the literature over the last decades.We mention [8,28,32] for Dirichlet boundary conditions, [14] for Neumann conditions, [36] for Signorini conditions, [38][39][40] for some generalized Robin type boundary conditions, [15] for perforations along a manifold, [26,41] for obstacles in perforated domains, and [3,4,12] for different abstract frameworks involving perforated media: see also references therein.Different assumptions on the geometry and the distribution of the perforations are made in the above-mentioned papers; also different assumptions on p are considered.See [13,29,43] and references therein in connection with models related to p-Laplacian, for different values of p, arising e.g. in glaciology, torsional creep and flows through porous media.
The problems under consideration in this paper are different from those in previous papers.We consider the p-Laplace operator in a perforated domain by "tiny cavities", with constraints for the solution in a general framework (cf., e.g., [42]), and further specifying, constraints for the solution and its normal derivative (associated with the p-Laplacian) on the boundary of the perforations, involving a nonlinear function of the solution, σ, and a parameter.This parameter may depend on the period and it can be either a very large parameter or a very small parameter; also σ can be a quite general monotonic increasing function (cf.(2.1)-(2.3)and Section 8).
We focus on obtaining critical sizes of perforations and critical relations between parameters which give rise to a strange term in the homogenized problem, at the same time as we describe all the possible homogenized media depending on the parameters of the problem.In addition, we construct correcting terms which provide strong convergence in the corresponding Sobolev spaces and obtain precise bounds for convergence rates.Strange terms issue simultaneously from the constraint for the solution and the constraint for the normal derivative on the boundary of the cavities, and correctors for solutions, are obtained for the first time in homogenization theory of the p-Laplace operator.These strange terms can appear in a boundary value problem or in an obstacle problem; their nonlinear character being very different.As one might expect, considering all the possible relations between parameters and the improved convergence provided by the correctors may restrict the geometrical configuration of the problem as well as the properties of the nonlinear function σ which however seems to be optimal to obtain all the results (cf.Section 8).
More precisely, we consider the domain Ω ε which is obtained by removing small domains G ε , the cavities/perforations, of diameter 2a ε ε, from a fixed domain Ω of R n (see Figure 1).The cavities are distributed over the whole domain at a distance O(ε) between them, ε being a small parameter that we shall make to go to 0. We denote by S ε the boundary of the cavities, namely, S ε ≡ ∂G ε .For n ≥ 3 and p ∈ [2, n], we study the asymptotic behavior of the solution u ε , as ε → 0, of the following problem: where f ∈ L q (Ω) with q = p/(p − 1), ∆ p u ≡ div(|∇u| p−2 ∇u), ∂ νp u ≡ |∇u| p−2 (∇u, ν), ν denotes the unit outward normal to Ω ε on S ε , β ε > 0 is a ε-dependent constant, σ = σ(x, u) is a continuously differentiable function defined in Ω × R, strongly monotone with respect to u (cf.(2.1)-(2.3)).Note that β ε is referred to as the adsorption parameter, and the variational formulation of (1.1) is (2.5).
We distinguish two ranges of p: p ∈ [2, n) and p = n.For 2 ≤ p < n we assume that the adsorption parameter takes the value ε −γ , with γ ∈ R, and that the cavities G ε are balls of radius a ε = C 0 ε α (with α > 1 and C 0 > 0), which are periodically distributed over Ω.It should be emphasized that this geometrical configuration is essential over all for the relations α = n/(n − p) and γ = n(p − 1)/(n − p) = α(p − 1) (see the intersection point in Figures 2 and 3) since the solution of the local problem obtained from the microstructure of the model is somewhat related with the fundamental solution of the p-Laplace operator.The solution of the local problem can be computed via a nonlinear equation that recalls the functional equation (1.3) (cf.Section 8).We relax the above geometrical configuration for the case where p = n, a case where a certain non periodically distribution of the cavities is allowed while they can have arbitrary shapes with a fixed perimeter (see the different cells in Figure 1 and the functional equation (1.10)- (1.11)): a comparison result makes it difficult to extend the result to p ∈ [2, n).
Below, cf.Section 1.1, we relate all the homogenized problems and the main results that we obtain as well as the structure of the paper.

The homogenized problems
For p ∈ [2, n), we obtain the homogenized problem, as ε → 0, for different relations between α and γ (see a sketch of all the possible situations in Figure 2).Among these relations, two of them provide asymptotic relations between adsorption, size and periodicity parameters which are related to as critical size and critical relation for the adsorption.Let us explain this in further detail.By comparison with the p-Laplace operator in perforated media, the classical critical size for the perforations with the p-Laplacian and Dirichlet boundary conditions on S ε (see [28]) is given by α = n/(n − p).For α > n/(n − p) the cavities are very small and they, as well as the constraints with any adsorption parameter, do not influence the average process (cf. the region in green color in Figure 2, problem (1.8), Theorem 6.4, and Theorem 6.5 for improved convergence).A strange term appears for α = n/(n − p).
It should be mentioned that the terminology of strange term here used appears in [9] for linear problems with Dirichlet boundary conditions on the perforations; see the reference to the original work introducing this terminology and further references in [9]; see also [31] in connection with the above-mentioned term in these linear problems; see [28] for Dirichlet conditions with the p-Laplacian, and [21,24] for the Laplacian with nonlinear Robin boundary conditions on the perforations.In our problem, it happens that this strange term also depends on the adsorption parameter and it ranges from a classical reaction term associated with the p-Laplacian, namely of the type |u| p−2 u or |u − | p−2 u − , to the reaction term σ(x, u) by multiplicative constants of the problem or a reaction term given by a function implicitly defined in a functional equation of the type (2.19).Also the character of the homogenized problem can change including boundary value problems (cf.(1.2), (1.4), (1.5) and (1.8)) and obstacle problems (cf.(1.6) and (1.7)).In fact, for each value of α, 1 < α ≤ n/(n − p), the relation γ = α(n − 1) − n provides the so-called critical relation for the adsorption parameter which implies that the total area of the perforations multiplied by the adsorption parameter is of order O (1).
In order to make more comprehensible the entire results for p ∈ [2, n), which we summarize in Figures 2 and  3, we introduce here a table with all the possible limit situations: I.When α = n/(n − p) and γ = n(p − 1)/(n − p), the homogenized problem is: IV.When α ∈ (1, n/(n − p)) and γ = α(n − 1) − n, the homogenized problem is where VI.When α ∈ (1, n/(n − p)) and γ > α(n − 1) − n, u ≡ 0, that is, as ε → 0, the solution u ε vanishes asymptotically in the whole Ω.VII.When α > n/(n − p) and γ ∈ R, the homogenized problem is Above, and throughout the paper ω n denotes the area of the unit sphere in R n , and u + and u − denote u + = sup(u(x), 0) and u − = u − u + .The existence and uniqueness of the solution for all the homogenized problems holds as does that for the ε-dependent problem (1.1) (cf.Theorem 2.1).Point I is referred to as the most critical case, where we have the critical size of perforations and the critical relation for the adsorption parameter.Points II and III deal with the the classical critical size of perforations.Case IV fits into the case of the critical relation for the adsorption.This case is of great interest, since for each size of the holes (namely, for each α) we have a critical relation for the adsorption (namely, of γ) giving rise to the strange term.Of course the role of α and γ inverts (see the red discontinuous line in Figure 2).In points V-VII some extreme relations for parameters hold.
Hence, the most critical relation between parameters is provided by the intersection, in the plane αγ, cf.3)) issued from the microstructure of the problem.The function defining this new term satisfies the same properties of monotonicity as σ and it is in good agreement with the existing results in the literature for p = 2 (cf.[23]).As a matter of fact, the strange term is the sum of two terms related to the contribution both of the constraints u ε ≥ 0 and ∂ νp u ε ≥ −ε −γ σ(x, u ε ) on the boundary of the perforations.The first term is given by a somewhat classical reaction term |u − | p−2 u − multiplied by averaged constants; the other one involves a nonlinear function of u, H(x, u) implicitly defined from (1.3): see [28], [40] and [41] to compare with strange terms when we have a Dirichlet condition or a generalized Robin boundary condition with a more restrictive datum σ.See (8.1) for some explicit computation of H.
We also note that the above mentioned fact (on the double contribution for the strange term) has already been detected in [17,23] for variational inequalities for the Laplacian (p = 2) in perforated media depending on whether the perforations are placed over the whole domain or along a manifold.We mention [17] for an extensive bibliography on variational inequalities in homogenization problems.Also, [21] should be mentioned as the first work in the literature where a nonlinear strange term appears defined implicitly via a functional equation, and [24,25] as works which consider for the first time homogenization problems for the Laplace operator and semilinear boundary conditions leaving as an open question the most critical case (namely, the one homologous to the big point in Figure 2 when p = 2), problem which remained unsolved for a long time even for the Laplace operator (cf.Section 8 in this connection).In the case of the critical relation for the adsorption parameter γ = α(n − 1) − n and α smaller than for the critical size, the nonlinear strange term is σ(x, u) multiplied by some averaged constants.It accompanies the p-Laplacian in Ω and the homogenized problem is now an obstacle problem; namely, it is an obstacle problem associated to the corresponding homogenized medium (cf.(1.6) and the discontinuous red line in Figure 2).The convergence and bounds for convergence rates are in Theorems 5.1 and 5.2.
Finally, for the extreme relations, that is, the very large size of perforations and very large adsorption parameters, the solution of the ε-dependent problem is approached by 0 (cf.Theorem 6.3, and the blue region in Figure 2) as if the adsorption parameter becomes a small parameter accompanying the normal derivative, and therefore, as if Dirichlet conditions are imposed on very big perforations (see [28]).This is quite in contrast with the case of large size of perforations and small adsorption where we obtain an obstacle problem for the p-Laplace operator in Ω which ignores both the nonlinear term σ and the adsorption parameter: see problem (1.7), Theorems 6.1 and 6.2 and the region below the red discontinuous line in Figures 2 and 3.In the case where p = n (cf.Section 7), we consider the most critical situation which is somewhat homologous to that of the big point in Figure 2.More specifically, for the geometry of the cavities described by (7.1) and the relations between sizes of cavities a ε and adsorption parameter β ε described by (7.2), the homogenized problem reads: where 1) , and for every (x, τ ) ∈ Ω × R, H(x, τ ) is the solution of the functional equation with l, α 2 and C 2 0 are positive constants (see their precise definitions in (7.1) and (7.2) and (1.12)), l the perimeter of the cavities (cf. the different cells in Figure 1).
For brevity, in order to outline the extra difficulties when dealing with cavities which are not balls, in Section 7 we consider only the above homogenized problem (1.9) when p = n, leaving the whole map of possible limit situations and proofs to be published in a forthcoming publication by the authors.However, it should be noted that for p = n, due to the fact that logarithmic scale appears (cf.(7.2)), a graphic of the type of Figure 2 summarizing all the possible homogenized problem becomes more complicated (even unthinkable), and, as occurs in [18] for the Laplacian and perforation by tubes, the graphics should be performed for well defined dependence of a ε and β ε in terms of ε.In this respect, as a sample, we outline that (1.9)-(1.11)provide the homogenized problem of (1.1) when we have the following relations: with k = j ≥ 0, C 2 0 > 0, α 2 > 0, and ι = n/(n − 1).However, other choices of order functions for a ε and β ε could lead to the same homogenized problem.
As happens for p ∈ [2, n), in the most critical case, the homogenized problem (1.9) is a boundary value problem in Ω containing the strange term in the partial differential equation which is the sum of two terms as a consequence of a double contribution (cf.[28], [41] and [39] to compare with other boundary conditions).The contribution due to the constraint for the flux leads to the nonlinear function |H| n−2 H in the strange term, H being implicitly defined by (1.10), where the perimeter l of the cavities arises now in the averaged constant (1.11), due exclusively to the influence of the adsorption parameter independently of the shape.
Note that in the case where the nonlinear function σ is the classical one arising in the Robin boundary condition, namely σ = b(x)|u| p−2 u with p ∈ [2, n], H can be defined explicitly in terms of b(x) and u and we observe that H depends on b(x) in a quite unusual way (see (8.1)).
As regards the technique, we mainly use the energy method to show the convergence of the solutions.Nevertheless, since we are dealing with homogenization of variational inequalities, and constraints involving nonlinear functions on the boundary of the perforations, proofs rely on monotonic operator theory, on extension operators, on suitable transformations of certain surface integrals on S ε into volume integrals, on convergence of measures, and on the appropriate choice of test functions which allows us to pass to the limit in the weak formulations.These choices imply introducing auxiliary problems in the periodicity cell (cf.(2.13), (5.8), (7.6), (7.8) and (7.26)).As a matter of fact, somehow five auxiliary functions are used in the process depending on the range of p and on the relations between the parameters β ε and a ε .Functions w j ε and w j ε deal with the classical test functions used in the literature when the perforations are balls; both functions can be explicitly constructed.q j ε deal with the test functions for more general geometries; also, the sets of functions {M j ε } and {m j ε } (j ∈ Z n ), which are solutions of the non-homogeneous Neumann problems for the p-Laplacian, (5.8) and (7.26) respectively, become crucial in the identification of certain homogenized problems.
For the most critical case, we construct the test functions (cf.(3.2) and (7.15)) using w j ε and q j ε and the function H arising in the strange term (see (1.3) and (1.10) depending on p).To show the improved convergence for solutions, we construct correctors using the auxiliary functions, the implicitly defined function H, and some intermediate singularly perturbed problem (cf.(5.16)): under the assumption of W 1,∞ -smoothness of the solution of the homogenized problems, allows us to obtain precise bounds for convergence rates in the W 1,pnorm (see a map of the different situations in Figure 3).
As regards the structure of the paper: Sections 2-6 are devoted to the case where p ∈ [2, n) and Section 7 contains the case p = n.Figure 2 summarizes the cluster of possible homogenized problems for different relations between the parameters α and γ, once we set p and n for p ∈ [2, n). Figure 3 provides a sketch of the corrector terms and improved convergence for p ∈ [2, n).The proofs are distributed in the paper as follows.Section 3 contains results for the most critical case (cf. the big point in Figures 2 and 3, and case I of the table).Section 4 contains results for the critical size of the perforations (cf. the vertical half-lines α = n/(n − p) in Figures 2  and 3, and cases II and III in the table).Section 5 addresses the critical relation for the adsorption parameter (cf.discontinuous red line in Figures 2 and 3, and case IV of the table).Section 6 addresses the rest of extreme relations (see colors blue, purple and green in Figures 2 and 3; cases V-VII of the table).Section 2 deals with the setting of the problem and some preliminary results useful for proofs throughout Sections 3-6; some technical proofs of these results are in the Appendix.Section 8 contains some final remarks on our results and on possible extensions to this paper.
Finally, in short, we emphasize that this paper provides a very general framework for variational inequalities with the p-Laplacian and constraints on the boundary of the perforations.The entire results imply improving and extending results in former papers in the literature (cf.Remark 8.1): only the results in Theorems 3.1 and 7.4 have been stated in [16,19] under stronger restrictions on σ, we provide here their complete proofs.Also, we extend the results in [23] for the Laplace operator; namely, in [23] only items I and IV of the table for p = 2 have been addressed.We consider a more general σ at the same time that cover the rest of the cases for p = 2 and the rest of p.In all the cases we provide correctors or improved convergence with precise bounds for convergence rates.We also note that depending on the situation in the general map of Figures 2 and 3, the result obtained can be extended to more general geometries of the cavities and other nonlinear data arising in the constraints (see Section 8 in this connection).

The homogenization problem and preliminaries
In this section we introduce the variational inequality for the p-Laplace operator associated with (1.1), and the precise geometry and notations used throughout Sections 3-6 for p ∈ [2, n).Each section or subsection contains different relations between parameters.We extend notations and the geometry of the problem in Section 7 for p = n.
We denote by G 0 the ball of radius 1 centered at the origin of coordinates.Let ω n be the area of the unit sphere in R n , that is, ω n = |∂G 0 |.For a domain B and for δ > 0, we define the sets δB where a ε ε, and Υ ε = {j ∈ Z n : (a ε G 0 + εj) ∩ Ω ε = ∅}; Z n is the set of vectors z with integer coordinates (see Figure 1).Note that Also we consider the space W 1,p (Ω ε , ∂Ω) (W 1,p (Ω, ∂Ω), respect.) to be the completion with respect to W 1,p (Ω ε )norm (W 1,p (Ω)-norm, respect.) of the set of infinitely differentiable functions in Ω ε (Ω, respect.),vanishing in a neighborhood of ∂Ω.For a function u in W 1,p (Ω), u + and u − denote u + = sup(u(x), 0) and u − = u − u + respectively.Let us consider σ(x, u) a continuously differentiable function of variables (x, u) ∈ Ω × R satisfying: for all x ∈ Ω, u, v ∈ R, and certain constants The variational formulation of problem (1.1) is: find where the set K ε is defined by For p ∈ [2, n) we set the values We have the following result: , and a ε and β ε given by (2.7).For fixed ε, problem (2.5)-(2.6)has a unique solution u ε ∈ K ε which also satisfies the inequality In addition, for u ε the solution of (2.5)-(2.6),there exists P ε u ε an extension of u ε to Ω, P ε u ε ∈ W 1,p (Ω, ∂Ω) with the following properties and

.10)
In all the estimates above, K > 0 denotes a constant independent of ε.
The existence of a function P ε u ε ∈ W 1,p (Ω, ∂Ω) which extends u ε to Ω and satisfies properties (2.9) is a consequence of Lemma 2.7 (see below).
Let us show estimate (2.10).Setting ψ ≡ 0 in (2.5) and v ≡ 0 in (2.2), we have Then, from the Poincaré inequality for the elements W 1,p (Ω, ∂Ω) and (2.9), we obtain the estimates . Therefore, (2.10) follows and the estimates above conclude the proof of the theorem.
Considering (2.10), there is a subsequence ( still denoted by ε ) such that, as ε → 0, for a certain function u which, once identified, provides the convergences (2.12) for the whole sequence of ε.
Note that such an extension provides a bound of the Poincaré constant independent of ε.
Throughout Sections 3-6, we show that this homogenized function u is the unique solution of a homogenized problem which depends on the relation between the parameters α, γ, p and n.That is, depending on the dimension of the space, the value of p, and the different relations between the ε-dependent parameters (the radius of the cavities O(ε α ) and the adsorption parameter O(ε −γ )), we have very different limit behaviors for the solution of problem (1.1).For fixed n ≥ 3 and p ∈ [2, n), Figure 2 shows a graph of γ versus α in such a way that for each α ∈ (1, n/(n − p)], the values of γ above, below or equal to α(n − 1) − n provide different homogenized problems.In the case where α > n/(n − p), the size of the cavities is very small and the solution u ε ignores asymptotically their influence.In addition, depending on the relations between the parameters α, γ, p and n, we also construct different correctors which provide estimates for convergence rates of solutions (cf. Figure 3).

Preliminary results
In this section, we introduce results which we shall use throughout Sections 3-6.We provide either precise references for their proof or a detailed proof in the Appendix.First, we introduce a function, related to the solution of the microscopic problem, which allows us to construct the test functions to pass to the limit in (2.8), as ε → 0. Also, we obtain certain estimates that we need for proofs in Sections 3-6.Here and in what follows, K denotes a constant independent of ε.
Let us denote by P j ε the center of the ball G j ε , j ∈ Υ ε .We denote by T j ε/4 the ball of radius ε/4 with center P j ε .Let w j ε be the solution of the following problem (2.13) It can be easily verified that for p ∈ [2, n) we have . (2.14) We define the function W ε ∈ W 1,p (Ω, ∂Ω) by setting extended by 1 inside G j ε , j ∈ Υ ε , and by 0 in R n \ j∈Υε T j ε/4 .Thus, we compute and, consequently, as ε → 0, we conclude that (2.17) Next, it will prove useful to introduce a well-known result on the monotonicity of the function |λ| p−2 λ with respect to λ ∈ R n for p ≥ 2: there exists a constant (cf., e.g., [6]).Note that here and throughout the paper we write λ 1 λ 2 as the scalar product in R n .
Using this result, we introduce a proposition which provides existence and uniqueness of solution of the functional equation arising in the homogenized problem (1.2): see the Appendix for its proof.Also, see for example [20] and references therein for different functional equations when p = 2. Proposition 2.2.Let p be p ≥ 2. Let be a strictly positive constant and let σ be the function σ(x, u) defined from Ω × R into R which is assumed to be a continuously differentiable function in Ω × R satisfying (2.1)-(2.2).Then, the equation has a unique solution H(x, τ ) which is a continuously differentiable function in Ω × (R \ {0}) and continuous in Ω × R, and satisfies H(x, 0) = 0 and ) The following result simplifies the computations throughout the paper: see the Appendix for its proof.
Finally, we introduce Lemmas 2.4-2.8 which we need for the proofs throughout Sections 3-6.Applying the technique in Lemmas 1 and 2 in [33], and Lemma 3 in [34] for p = 2 we obtain Lemmas 2.4, 2.5 and 2.6 respectively (see also [40] in this connection).See Theorem 1 of [39] and references therein for the proof of Lemma 2.7 (cf. also in this connection [1] and [38], and [10] and [33] when p = 2).We refer to Lemma 1 in [44] for the proof of Lemma 2.8.In these lemmas, the constant K does not depend on ε nor on the functions ϕ appearing in their statements.
Lemma 2.6.Let Y ε be the domain defined in Lemma 2.4 and let Y ε denote the domain 2ε ].

The most critical case for
In this case, the homogenized problem is the boundary value problem (1.2).We show that the nonlinear function arising in the strange term is defined through a functional equation (cf. the reaction term in (1.2) and (1.3)).The properties of this function allow us to obtain a corrector: see (3.2) for u = v, and the point intersection of all the lines in Figures 2 and 3.The convergence and the corrector results are in Theorem 3.1 and 3.2 respectively.
and let u ε be the weak solution of (1.1).Then, the limit function u of the extension of u ε , defined by (2.12), is the weak solution of problem (1.2).
Let us consider the function where v ∈ C ∞ 0 (Ω), W ε is the function defined by (2.15) and H(x, τ ) is the solution of the functional equation (1.3).Let us prove that ψ ≥ 0 on S ε , and thus it belongs to K ε .Suppose that for some point x 0 ∈ S ε we have ψ(x 0 ) < 0.Then, we get . Thus, we obtain a contradiction.
We now take ψ defined by (3.2) as a test function in (2.8); since and we pass to the limit when ε → 0. We denote by L ε the left hand side of (3.3).Let us show that In order to do that, we take into account that which is obtained from formula (2.14).Then, we apply Proposition 2.3 with where P ε u ε is the extension defined in Theorem 2.1.This is possible since on account of (2.10), (3.5), (2.12) and (2.17), we can check that ∇ϕ ε L p (Ω) is bounded independent of ε and Thus, we obtain lim where On account of (3.6) and the fact that We study the limit of L 2 ε + L 3 ε when ε → 0. From (3.5), (3.6) and (2.10), it follows Moreover, by the properties of H(x, z), we have H(x, v + )v − = 0 and, hence, Thus, using the definition of W ε and the Green formula, we get In order to compute (3.10), we use the explicit form of the normal derivatives of the auxiliary functions w j ε given by (3.12) where α ε = a By the definition of ϕ ε and W ε and the fact that v − (v + − H(x, v + )) = 0, u ε ≥ 0 on ∂G j ε and (3.11), we obtain In addition, from (3.11)-(3.12), it follows Now, taking into account that H is the solution of the equation (1.3) and using the Hölder inequality, (2.10) and the size of S ε we get Moreover, on account of (3.6), we apply Lemma 2.8 and have lim and, consequently, Finally, we use (3.4) and (3.6) to pass to the limit in (3.3), as ε → 0, and obtain that the limit function u satisfies the following inequality for all v ∈ W 1,p (Ω, ∂Ω).As usual, taking v = u ± λφ in (3.15)where φ ∈ W 1,p (Ω, ∂Ω) and passing to the limit as λ → +0, we obtain that u satisfies the integral identity (3.1), which concludes the proof.Theorem 3.2.Let α = n/(n − p), γ = n(p − 1)/(n − p) and p ∈ [2, n).Let u ε be the weak solution of (1.1), u ∈ W 1,p (Ω, ∂Ω) the weak solution of the boundary value problem (1.2) with the additional regularity u ∈ W 1,∞ (Ω), and W ε defined by (2.15).Then, as ε → 0, we have Proof.Let us consider problems (2.5) and (3.1) and take as test functions 12).Subtracting both expressions and taking into account the definition of W ε on G ε , we obtain Let us denote by A 1 ε the first integral on the left hand side of (3.17) and by H the function H(x, u + ) + u − .Then, we can rewrite it in the following way Using the monotonicity of the functions σ(x, u) and |λ| p−2 λ (see (2.2) and (2.18)), from (3.18) and (3.17), we deduce where We study the limit of I 1 ε + I 2 ε + I 3 ε when ε → 0. From (2.10) and (3.5), we apply Proposition 2.3 with η ε ≡ −W ε H and ϕ = ϕ ε ≡ u − W ε H − P ε u ε , and have that lim ε→0 Moreover, rewriting the computations (3.9)-(3.13)with minor modifications, we have Thus, using the definition of H, (2.10), Lemma 2.8, (2.12) and (2.17 To get (3.16) from (3.22), we consider the Poincaré inequality for the and consequently, we have and (3.16) also holds.Thus, Theorem 3.2 is proved.

4.
Critical size for perforations when p ∈ [2, n) and γ = n(p−1) n−p When α = n/(n − p) and γ = n(p − 1)/(n − p), we show that the homogenized problem does not depend on σ although its properties are somewhat present in the homogenization process.For a very small (large, respectively) adsorption the asymptotic behavior of the solution of (1.1) is the same as if Signorini (Dirichlet, respectively) conditions had been imposed on the boundary of the cavities (cf.[11] when p = 2, and [28], respectively).Correctors are given by W ε u − and W ε u depending on whether we have small or large adsorption (see line α = n/(n − p) in Figures 2 and 3).The results for small adsorption are in Section 4.1 whereas those for large adsorption are in Section 4.2.Then, the limit function u of the extension of u ε , defined by (2.12), is the weak solution of problem (1.4).
Let us take in (2.8) the test function 2) and we pass to the limit when ε → 0.
Proof.The variational formulation of (1.5) reads: find u ∈ W 1,p (Ω, ∂Ω) such that From the monotonicity of the function |λ| p−2 λ, the existence and uniqueness of solution of (4.19) holds (cf., e.g., Section II.8.2 in [30]).Let us take in (2.8) the test function and we pass to the limit when ε → 0. On account of (2.12) and (2.17), we deduce Let us show that Using (3.5), we apply Proposition 2.3 with which is obtained rewriting the proof for (3.6).Thus, we obtain where By (4.23) and the fact that |G ε | → 0, we have Moreover, using (3.5), (4.23), (2.10), the definition of W ε and the Green formula, we get Besides, from (3.12), Lemma 2.8 and (4.23), we have lim  4.22) to pass to the limit in (4.20), as ε → 0, and obtain that the limit function u satisfies the following inequality As usual, taking v = u ± λφ in (4.29)where φ ∈ W 1,p (Ω, ∂Ω) and passing to the limit as λ → +0, we obtain that u satisfies the integral identity (4.19), which concludes the proof.

Critical relation for the adsorption
In this section, we deal with sizes of cavities larger than the critical size.Because of the adsorption parameter, the constraints on the boundary of the cavities in (1.1) transform asymptotically into an obstacle problem with a nonlinear strange term D n σ(x, u) that also contains information on the geometrical configuration of the original problem, namely, the area of the unit sphere and the scaling factor C n−1 0 (cf.(1.6)).We show the convergence of the extension of the solution of (1.1), as ε → 0, towards that of (1.6) in the W 1,p -norm and compute bounds for discrepancies in the way stated by Theorem 5.2.
and let u ε be the weak solution of (1.1).Then, the limit function u of the extension of u ε , defined by (2.12), is the weak solution of problem (1.6).
Proof.First, we observe that the variational formulation of problem (1.6) is: find u ∈ K 0 such that where K 0 is defined by K 0 = {v ∈ W 1,p (Ω, ∂Ω) : v ≥ 0 a.e. in Ω}. (5. 2) The existence and uniqueness of solution u of (5.1)-(5.2) follows from (2.2) and (2.18) (see the technique in Theorem 2.1).Besides, by Minty Lemma, problem (5.1) is equivalent to finding u ∈ K 0 such that Let us prove that the negative part of the limit function u, u − , is equal to zero a.e. in Ω and, consequently, u ∈ K 0 .Applying Lemma 2.6 and using that u − ε = 0 on S ε and (2.10), we conclude Thus, from (2.12) and the fact that |G ε | → 0, we have In order to prove that the limit function u satisfies (5.3), we pass to the limit in (2.8) with ψ = v ∈ K 0 .On account of (2.12) and the volume of G ε , it follows that lim ε→0 Ωε (5.6) Let us show that, under the assumptions α ∈ (1, n/(n − p)) and γ = α(n − 1) − n, the following equality holds: (5.7) To do this, we introduce the function M ε defined by and We assume that Taking as a test function M j ε in the integral identity for M j ε and applying the Hölder inequality, we obtain Besides, using Lemma 2.5 and Lemma 2.4, we get .

Extreme cases for p ∈ [2, n)
We consider the rest of possible relations between the parameters α and γ which have not been considered in previous sections.Section 6.1 contains the results for the case of big cavities and small adsorption; the constraints on the boundary of the cavities in (1.1) transform asymptotically into an obstacle problem for the p-Laplacian in Ω, which ignores the adsorption parameter (which in fact can converge towards ∞); that is, as if Signorini conditions had been imposed (cf.[11] when p = 2).Section 6.3 contains the results for the case of small cavities; also the solution ignores asymptotically the adsorption parameter.In both cases, the convergence of the extension of the solution in the W 1,p -norm is proved along with bounds for discrepancies as stated in Theorems 6.2 and 6.5 respectively.Section 6.2 contains the case of large sizes of cavities and adsorption parameters; the solution of (1.1) vanishes asymptotically and we obtain estimates of the W 1,p -norm (cf.(6.8)).Then, the limit function u of the extension of u ε , defined by (2.12), is the weak solution of problem (1.7).
Proof.We rewrite the proof of Theorem 5.1 with minor modifications; we briefly outline the main differences here.

The case
Then, the extension P ε u ε of the weak solution of (1.1), defined by Theorem 2.1, verifies and, consequently, P ε u ε converges to zero in W 1,p (Ω) when ε → 0.

The case
and let u ε be the weak solution of (1.1).Then, the limit function u of the extension of u ε , defined by (2.12), is the weak solution of the Dirichlet problem (1.8).
Proof.Let us take in (2.8) the test function and we pass to the limit when ε → 0. On account of (2.12) and (2.17), we deduce Besides, using (3.5), (2.12) and (2.17), we apply Proposition 2.3 with Thus, we get that u satisfies the following inequality As usual, taking v = u ± λφ in (6.9)where φ ∈ W 1,p (Ω, ∂Ω) and passing to the limit as λ → +0, we obtain that u satisfies the integral identity for problem (1.8), which concludes the proof.

The most critical relation when p = n
In this section, we consider the case where p = n, n ≥ 3, and a more general geometry than that in Sections 3-6 (see Figure 1).For the sake of brevity, we only provide the homogenized problem of (1.1) and the corresponding corrector in the most critical situation, namely, what can be the analogous case to the big point in Figures 2-3.Further specifying, among all the possible relations between the parameters β ε , ε and a ε we consider the critical size of the perforations provided by the relation ε n/(n−1) ln(a −1 ε ) = O(1), and the critical relation for the adsorption parameter which is obtained when β ε multiplied by the total area of the perforations is of order 1.Conditions (7.2) give the mentioned relations while (1.12) give particular choices or a ε and β ε satisfying (7.2).
Considering problem (1.1) in perforated domains Ω ε , with isoperimetric perforations of arbitrary shape (cf.(7.1)), in Theorem 7.4, we prove the convergence of the solution towards that of the homogenized problem in (1.9), which is a boundary value problem in Ω with the strange term in the partial differential equation containing a double contribution on the boundary of the perforations, namely, the contribution due to the constraint u ε ≥ 0 and ∂ νn u ε ≥ −β ε σ(x, u ε ).Due to the last constraint, the function H(x, u) arising in the strange term is implicitly defined from a functional equation (cf.(1.10)) in which also the perimeter of the perforations l appears for any shape.We refer to Proposition 2.2 for the existence and uniqueness of the solution H = H(x, u) of (1.10) and its properties, as well as Section 8 for examples of explicit solutions for certain data σ.The result on the corrector and improved convergence is in Theorem 7.5.We follow the scheme of proofs in Section 3.
Let us first introduce the geometrical configuration of the problem, the new test functions that we need to prove convergence and some preliminary results.
Let M be a finite subset of Z which we can identify with {1, We define where G j coincides with one of the domains D m , m ∈ M , and 1).Obviously, we have |Υ ε | ∼ = dε −n , with some d > 0, and where T j aε and T j ε/4 are balls with radius a ε and ε/4, respectively, and center P j ε , which coincides with the center of Y j ε .Now we can define Let us consider (1.1) when p = n, and the ε-depending parameters a ε and β ε satisfy where C 0 and α are some constants different from zero.Recall that σ arising in (1.1) satisfies (2.1)-( 2.3) with δ ∈ [n − 1, ∞).Also, its variational formulation reads (2.5)-(2.6).
Theorem 7.1.Let ε > 0, f ∈ L n/(n−1) (Ω), and a ε and β ε given by (7.2).Then, problem (2.5)-(2.6)has a unique solution u ε ∈ K ε which also satisfies the inequality In addition, for u ε the solution of (2.5)-(2.6),there exists an extension P ε u ε of u ε to Ω, P ε u ε ∈ W 1,n (Ω, ∂Ω) with the following properties and L n/(n−1) (Ω) . (7.4) The proof of Theorem 7.1 holds by rewriting the proof of Theorem 2.1 with minor modifications.Considering (7.4), for each sequence of ε we can extract a subsequence (still denoted by ε) such that as ε → 0 for a certain function u which, once identified, provides the convergences (7.5) for the whole sequence of ε.The aim of the section is to obtain the homogenized problem satisfied by the function u in (7.5) (see Theorem 7.4).
To do this, we introduce the functions Q ε and W ε ∈ W 1,n (Ω, ∂Ω) as follows: For j ∈ Υ ε , let q j ε (x) be the solution of the problem and we introduce the function extended by 1 inside G j ε , j ∈ Υ ε , and by 0 in R n \ j∈Υε T j ε/4 .Similarly, for j ∈ Υ ε , let w j ε (x) be the solution of the problem It can be easily verified that We define the function W ε ∈ W 1,n (Ω, ∂Ω) by setting extended by 1 inside T j aε , j ∈ Υ ε , and by 0 in R n \ j∈Υε T j ε/4 .Thus, we compute and, since ε n/(n−1) ln(4a ε /ε) → − α 2 as ε → 0, we have and 12) It should be noted that because of the geometry of the G j , in general, the function q j ε , defined by (7.6), cannot be explicitly constructed.Lemma 7.2 provides us some properties for Q ε by means of comparison with W ε in Ω (see Lemma 2 in [39] for the proof).Lemma 7.2.Let us assume that ε n/(n−1) ln(4a ε /ε) → − α 2 as ε → 0. Let Q ε and W ε be defined by (7.7) and (7.10) respectively.Then, we have Also for the sake of completeness, we introduce the following result.
The proof of Lemma 7.3 holds applying the technique in Lemma 2 in [33] for p = 2 (cf.Lemma 2.5).
Theorem 7.4.Let a ε and β ε satisfy (7.2) and let u ε be the weak solution of problem (1.1) with p = n.Then, the limit function u of the extension of u ε , defined by (7.5), is the weak solution of the problem (1.9)-(1.10).
Proof.First, let us note that on account that Proposition 2.2, equation (1.10) has a unique solution H ≡ H(x, u), which is a continuously differentiable function in Ω×(R\{0}) and continuous in Ω×R, and satisfies H(x, 0) = 0, (2.20) and (2.21) with p = n.Also, we observe that the weak solution of problem (1.9) is the solution in W 1,n (Ω, ∂Ω) of the integral equation (7.14) From the monotonicity of the function |λ| n−2 λ and (2.20) with p = n, the existence and uniqueness of solution of (7.14) holds (cf.[30]).
Let us consider the function where v ∈ C ∞ 0 (Ω), Q ε is the function defined by (7.7) and H(x, τ ) is the solution of the functional equation (1.10).Because of (7.7) and (2.21), we can check that φ ≥ 0 on S ε and, hence, it belongs to K ε .We now take φ as a test function in (7.3); by definition of Q ε , we get 16) and we pass to the limit in (7.16) when ε → 0.
We denote by T ε the first integral on the left hand side of (7.16) and by H the function H ≡ H(x, v + ) + v − .Thus, we have where Using Hölder inequality, (7.13), (7.11) and (7.4), it follows , which converge towards zero as ε → 0.Moreover, taking into account the inequalities (9.3) and (9.4) with p = n (see, for the technique, the estimate |R a ε | for p > 3 in the proof of Proposition 2.3), we have where From Proposition 2.3, (7.11) and (7.4), we obtain |Z a ε | → 0 and |Z c ε | → 0 as ε → 0. Besides, on account of (7.12), (7.5) and the size of G ε , we deduce Finally, by (7.11), we get Thus, gathering the above convergences, we obtain Now, let us consider the second term on the the right hand side of (7.16) and let us prove that By the definition of W ε and the Green formula, we have Moreover, using the properties of H(x, u), we get H(x, v + )v − = 0 and, hence, where On account of (7.5) and (7.2), we apply Lemma 2.8 and have Therefore, the proof of (7.18) is completed by showing To prove (7.23), we have to introduce a set of functions {m j ε } j∈Υε : for each j ∈ Υ ε , we consider the problem which has a unique solution defined up to an additive constant.We note that because for j ∈ Υ ε , G j ∈ M, we are dealing with a finite number of different functions (7.24).For j ∈ Υ ε , we set It is easy to see that m j ε (x) is a solution of the following problem We take as a test function in the integral identity for problem (7.26) and we get Now, by (7.25), it follows and, hence, Thus, from (7.27), (7.28) and (7.4), we derive Let us prove (7.23).To do it, we write where Besides, applying Lemma 7.3, we obtain

Final comments
Here, we gather some comments and remarks about the extensions of the results throughout the paper.
As regards the most critical situation (point I in the table of Section 1), let us note that in the case where σ(x, u) = b(x)|u| p−2 u, with b(x) a strictly positive continuously differentiable function in Ω, we can solve explicitly the functional equation (2.19) for p ∈ [2, n]; namely, we can define the solution H of (2.19) explicitly in terms of b(x) and u.As a matter of fact, we obtain where contains information on the averaged constant of the problem (cf.also (2.21) and (2.22)).Since all the results of the paper apply to this case, σ(x, u) = b(x)|u| p−2 u, we observe that the dependence of the nonlinear strange term on b(x) ranges from linear to nonlinear or no dependence (cf. the table in Section 1); the nonlinear dependence appearing for the most critical case (cf.Sections 3 and 7).
Also, an important point to underline is that in the case where b(x) ≡ b is a positive constant, even for the most critical situation, arbitrary shapes of the cavities (periodically placed) can be considered and some kind of capacity constant will likely appear in the homogenized problem.The latter can be easily shown for p = 2 suitably modifying proofs in Section 3, although, to our knowledge, the result for variational inequalities is not found in the literature.
As regards the geometrical configuration of the problem, we observe that, for p ∈ [2, n), the limit behavior of the solution of (1.1) remains to be obtained in the cases where the cavities G ε are not balls or there is not periodicity of the structure.For the case where p = 2, different shapes of the domains have been considered in [24,25] for boundary value problems outside the most critical situation (namely, outside the big point in Figures 2 and 3).As a matter of fact, the local problem obtained from the microstructure of the original problem, strongly depends on the center of the cavities and makes it difficult to guess the homogenized problem.This fact has been observed in very different homogenization problems in perforated media, with linear partial differential equations and with different boundary conditions or constraints on the boundary condition, and always related with critical sizes of the cavities.Sometimes the difficulty can be overcome by considering periodicity of the coefficients arising in the partial differential equations or more restrictive parameters and functions arising on the Robin boundary conditions (cf., e.g., [11,35] and references therein).Also let us note that other techniques avoiding local problems could allow less restrictive geometrical configuration to be considered: see [7] and [35] for two second order elliptic operators with oscillating coefficients and two types of cavities, in each periodicity cell, with Signorini condition or nonlinear Robin condition on the boundary of one of these cavities.However, we highlight that, in the existing literature, the most critical case (cf.Point I on the table of Section 1.1) for arbitrary shapes of perforations has not been considered even when p = 2: cf.[24] for arbitrary shapes when σ = bu and p = 2; see the above paragraph in this connection.
For nonlinear Robin boundary conditions, even for the Laplace operator, in the most critical situation, the problem has been unsolved for a long time.[24] considers perforations that are not necessarily balls but the problem for the most critical relation remained as an open problem for any geometry of the perforations until [21].[21] appears as the first paper in the literature where an implicitly defined homogenized problem is outlined, the perforations being for balls.In fact [21,24,25,44] consider the Laplace operator in perforated media over the whole domain and their results complement each other.However [21,44] consider only spherical cavities while the cavities can be of different shapes in [24,25] but for relations between parameters outside the big point.See [18,20] for a long list of references on related problems.
In the most critical situation, for nonlinear Robin boundary conditions, the Laplacian and n = 2, namely, p = n = 2, we refer to [37] for general geometries of the cavities and to [18] when p = 2, n = 3 and the domain is perforated by tubes.An extension to p = n ≥ 3 can be found in [39].Here, in Section 7, we consider a different problem (cf.(1.1)), with unilateral constraints.Also, a more restrictive σ is considered in [18,37,39].However, it should be emphasized that, in any case, the perimeter of the perforations arises in the strange term instead of the shape of the perforations as one might expect for n ≥ 3.In these cases, the difference to broach the problem for the different values of p, namely 2 ≤ p < n and p = n, recalls the difference when n ≥ 3 and n = 2 for the Laplacian in [9], or the Stokes equations in [2], both with Dirichlet conditions on the boundary of the perforations.When considering the adsorption, the perimeter of the perforations also arises in the homogenized problem (cf.[37] for further details on the differences when p = n = 2.) Related with the nonlinear data of our problem, we note that the hypotheses (2.1)-(2.3)on the function σ(x, u) in this paper seem to be optimal and allow us to provide a general framework for results and proofs.However, many of the results hold true under weaker hypotheses for σ.Actually, the strong monotonicity outlined in (2.2) can be changed by the weaker hypothesis of strict monotonicity or only monotonicity depending on the relations for parameters or the appropriate improved convergence.This can be seen in a simple way when verifying proofs.
However, as noticed in [22], the adsorption isotherms used mostly in the literature are of the form σ(x, u) = g(u) with g a positive strictly increasing function in [0, ∞).In this connection, we also note that certain proofs can be adapted for functions σ both with less smoothness or increasing requirements.We refer to [5] for explicit definitions of σ arising in models from ecology, hydrogeology or chemical reactions, for comments on possible extensions when u ≤ 0, and for further references.
Remark 8.1.Note that, in this paper, we give results for the p-Laplacian on perforated domains, by tiny cavities, with constraints for solutions and their normal derivatives on the boundary of the cavities, which are completly new in the literature.Dealing with unilateral constraints for the p-Laplacian and the homogenization of perforated media, we mention very different problems and results in [36] for Signorini conditions (when α = 1) and [26,41] for obstacle problems.For different constraints and sizes of perforations, we provide a map of all possible homogenized problems and construct the corresponding correctors (see Figures 2 and 3).In particular, we obtain seven different limits when p ∈ [2, n), most than ever found for the p-Laplace operator in perforated media (see [28,40] to compare).In this connection, [40] considers a ε-dependent boundary value problem with generalized Robin condition (no constraints for solutions), without any corrector result and with a more restrictive σ.Except one, the strange terms in [40] are different since they cannot get the double influence coming from the constraints on the solutions and on their normal derivatives (cf.Section 1); among all the homogenized problems here obtained, only (1.5) and (1.8) coincide with some homogenized problems in [40] for some σ.In addition, to improve the weak convergence obtained in [40], it suffices to re-write the proofs for correctors in Sections 2-6 with the suitable modifications.Similar comments apply to the results in [37] and [39] when p = n = 2 and p = n ≥ 3, respectively, and to our new strange term in (1.9) and corrector in Theorem 7.5.Dealing with the results in [16,19,23], see the end of Section 1.

Appendix
To avoid introducing technical details in Section 2.1, we provide here the proof of Propositions 2.2 and 2.3 that we have not found in the literature.

Figure 1 .
Figure 1.The geometrical configuration of Ω ε and the periodicity cell.

Figure 2 .
Figure 2. Sketch of homogenized problems depending on the relations between α and γ (n and p fixed).

2 ,
• • • , m M } for m M ∈ Z. Assume that we have the set M of domains D m satisfying the following properties: for any m ∈ M , D m ⊂ T 1/4 ⊂ Y , where Y = (−1/2, 1/2) n , T 1/4 = {y ∈ R n : |y| < 1/4}, D m is diffeomorphic to a ball m ∈ M , and the area of D m is equal to a given number l > 0, i.e. |∂D m | = l, ∀m ∈ M. (7.1)