Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization

We consider the intersection of a convex surface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ with a periodic perforation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document}Rd, which looks like a sieve, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\varepsilon = \bigcup _{k\in \mathbb {Z}^d}\{\varepsilon k+a_\varepsilon T\}$$\end{document}Tε=⋃k∈Zd{εk+aεT} where T is a given compact set and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_\varepsilon \ll \varepsilon $$\end{document}aε≪ε is the size of the perforation in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-cell \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0, \varepsilon )^d\subset \mathbb {R}^d$$\end{document}(0,ε)d⊂Rd. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε tends to zero we establish uniform estimates for p-capacity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<d$$\end{document}1<p<d, of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \cap T_\varepsilon $$\end{document}Γ∩Tε. Additionally, we prove that the intersections \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \cap \{\varepsilon k+a_\varepsilon T\}_k$$\end{document}Γ∩{εk+aεT}k are uniformly distributed over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ and the perforations, in a given bounded domain, is homogenizable when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p<1+\frac{d}{4}$$\end{document}p<1+d4. This result is new even for the classical Laplace operator.

The sieve-like configuration with convex with given strictly convex and C 2 smooth surface as the obstacle and p-Laplacian as the governing partial differential equation is homgenizable provided that p < 1 + d 4 . Moreover, the limit problem admits a variational formulation with one extra term involving the mean capacity, see Theorem 3. The configuration of , ε , T ε and is illustrated in Fig. 1.
This result is new even for the classical case p = 2 corresponding to the Laplace operator. Another novelty is contained in the proof of Theorem 2 where we use a version of the method of quasi-uniform continuity developed in [4].

Statement of the problem
We assume that is a strictly convex surface in R d that locally admits the representation where Q ⊂ R d−1 is a cube. For example, may be a compact convex surface, or may be defined globally as a graph of a convex function. Without loss of generality we assume that x d = g(x ) because the interchanging of coordinates preserves the structure of the periodic lattice in the definition of T ε . We will also study homogenization of the thin obstacle problem for the p-Laplacian with an obstacle defined on ε . Our goal is to determine the asymptotic behaviour, as ε → 0, of the problem min ˆ |∇v| p dx +ˆ hvdx : v ∈ W 1, p for given h ∈ L q ( ), 1/ p + 1/q = 1 and φ ∈ W 1, p 0 ( ) ∩ L ∞ ( ).
We make the following assumptions on , T , , d and p: The compact set T from which the holes are constructed must be sufficiently regular in order for the mapping to be continuous, where e is any unit vector. This is satisfied if, for example, T has Lipschitz boundary.
The size of the holes is This is the critical size that gives rise to an interesting effective equation for (2). (2) is in the range This is to ensure that the holes are large enough that we are able to effectively estimate the intersections between the surface and the holes T ε , of size a ε . See the discussion following the estimate (15). In particular, if p = 2 then d > 4.
These are the assumptions required for using the framework from [4], though the (A 4 ) is stricter here.

Main results
The following theorems contain the main results of the present paper.

Theorem 1
Suppose is a C 2 convex surface. Let I ε ⊂ [0, 1) be an interval, let Q ⊂ R d−1 be a cube and let Next we establish an important approximation result. We use the notation T k ε = εk + a ε T and k ε = ∩ T k ε .

Theorem 2
Suppose is a C 2 convex surface and P x a support plane of at the point x ∈ . Then 1 • the p-capacity of P k x = P x ∩ T k ε approximates cap p ( k ε ) as follows 2 • Furthermore, if P 1 and P 2 are two planes that intersect {a ε T + εk} at a point x, with normals ν 1 , ν 2 satisfying |ν 1 − ν 2 | ≤ δ for some small δ > 0, then where lim δ→0 c δ = 0.
As an application of Theorems 1, 2 we have Theorem 3 Let u ε be the solution of (2). Then u ε u in W In (5), ν(x) is the normal of at x ∈ and cap p,ν(x) (T ) is the mean p-capacity of T with respect to the hyperplane P ν(x) = {y ∈ R d : ν(x) · y = 0}, given by where cap p (E) denotes p-capacity of E with respect to R d . Theorem 3 was proved by the authors in [4] under the assumption that is a hyper plane, which was in turn a generalization of the paper [5]. In a larger context, Theorem 3 contributes to the theory of homogenization in non-periodic perforated domains, in that the support of the obstacle, ε , is not periodic. Another class of well-studied non-periodic perforated domains, not including that of the present paper, is the random stationary ergodic domains introduced in [1]. In the case of stationary ergodic domains the perforations are situated on lattice points, which is not the case for the set ε . The perforations, i.e. the components of ε , have desultory (though deterministic by definition) distribution. For the periodic setting [2] is a standard reference.
The proof of Theorem 3 has two fundamental ingredients. First the structure of the set ε is analysed using tools from the theory of uniform distribution, Theorem 1. We prove essentially that the components of ε are uniformly distributed over with a good bound on the discrepancy. This is achieved by studying the distribution of the sequence for g defined by (1) and εk ∈ Q . Second, we construct a family of well-behaved correctors based on the result of Theorem 2. The major difficulty that arises when is a more general surface than a hyperplane is to estimate the discrepancy of the distribution of (the components of) ε over , which is achieved through studying the discrepancy of {ε −1 g(εk )} k . For a definition of discrepancy, see Sect. 2. In the framework of uniform convexity we can apply a theorem of Erdös and Koksma which gives good control of the discrepancy.

Definition 1
The discrepancy of the first N elements of a sequence {s j } ∞ j=1 is given by where I is an interval, |I | is the length of I and A N is the number of 1 ≤ j ≤ N for which s j ∈ I (mod1).
We first recall the Erdös-Turán inequality, see Theorem 2.5 in [7], for the discrepancy of the sequence where n is a parameter to be chosen so that the right hand side has optimal decay as N → ∞.
Observe that s j is the j-th element of the sequence which in our case is s j = f ( j) for a given function f and N = 1 ε . We employ the following estimate of Erdös and Koksma ( [7], Theorem 2.7) in order to estimate the second sum in (8) In order to apply this result to our problem we first need to reduce the dimension of (7) to one. To do so let us assume that the obstacle is given as the graph of a function x d = g(x ) where g is strictly convex C 2 function such that for some positive constants c 0 < C 0 . Next we rescale the ε-cells and consider the normalised problem in the unit cube ε , j ∈ Z d−1 . If d = 2 then we can directly apply (9) to the scaled function f above. Otherwise for d > 2 we need an estimate for the multidimensional discrepancy in terms of D N introduced in Definition 1, a similar idea was used in [4] for the linear obstacle. Suppose for a moment that this is indeed the case. Then we can take F k (t) = k f (t) in (9) and noting one can proceed as follows for some tame constant λ > 0 independent of ε, k. Plugging this into (8) yields for another tame constant λ > 0. Now to get the optimal decay rate we choose 1 n = n N which yields N = n 3 and hence and we arrive at the estimate

Proof of Theorem 1
Proof Suppose Q is a cube of size r . Then there is a cube where (k 1 , k ) = k , a, b are the integer parts of ε −1 α and ε −1 β respectively and |(b − a) − ε −1 r | ≤ 1. We also note that For each k the function h : s → ε −1 g(εs + εk ) satisfies |h (s)| ≤ C 1 and h (s) ≥ ρε for a ≤ s ≤ b. Thus we may apply the Erdös-Koksma Theorem as described above and conclude that It follows that the modulus of the left hand side of (13) is bounded by Cε 1 3 , proving the theorem.

Correctors
The purpose of this section is to construct a sequence of correctors that satisfy the hypotheses given below. Once we have established the existence of these correctors, the proof of the Theorem 3 is identical to the planar case treated in [4].
where cap p,ν(x) is given by (6) and H ∫ is the restriction of s−dimensional Hausdorff measure on .
Then it follows from the definition of cap p [3] that Indeed, we have .

ε). Thus Theorem 1 tells us that the number of components of
Here we need to have ε 1/3 = o(|I ε |), which is equivalent to (A 4 ). Sincê Thus´K |∇w ε | p is uniformly bounded on compact sets K . Since w ε (x) → 0 pointwise for x / ∈ , H1 follows. When verifying H 2 and H 3 we will only prove that Once this has been established the rest of the proof is identical to that given in [4].

Proof of Theorem 2
Proof 1 • Set R ε = ε 2a ε → ∞, then after scaling we have to prove that uniformly in ε where i being a bounded domain with smooth boundary and D t i → S i as t → 0 in Hausdorff distance. Consider such that 0 ≤ η ≤ 1 and η ≡ 1 in B 3 . Using w as a test function we conclude that Since η ≡ 1 in B 3 then applying Hölder inequality we infer that´B where the last bound follows from the estimate v i ≤ W . Combining these estimates we infer for some tame constant K independent of t and ε. Thus, by construction v t Choosing a sequence ψ n such that 1 − ψ m converges to the characteristic function χ D t where Notice that on ∂ D t i we have that ν = − ∇ψ m |∇ψ m | is the unit normal pointing inside D t i . We denote n = −ν and then we have that and returning to (19) we infer Thus we can omit the absolute values of the normal derivatives and obtain Recall that by Lemma 5.7 [6] there is a generic constant M > 0 such that for all ξ, η ∈ R d . First suppose that p > 2 then applying inequality (21) to (20) yields As for the case 1 < p ≤ 2 then from (21) we have But, from Hölder's inequality and (18) we get Therefore, there is a tame constant M 0 such that for any p > 1 we havê .
Letting t → 0 we get with some tame constant M 1 . Since 1 − v t i are nonnegative p-subsolutions in B R ε , from the weak maximum principle, Theorem 3.9 [6] we obtain Take a finite covering of Since v t i W 1, p (B 3 ) ≤ C uniformly for all t > 0 it follows that v t i → v 1 strongly in L p (B 3 ) and v i is quasi-continuous. In other words, for any positive number θ there is a set E θ such that cap p E θ < θ and v i is continuous in B 2 \E θ . Notice that E θ ⊂ S 1 ∪ S 2 and hence H d (E θ ) = 0. This yields where ω i (·) is the modulus of continuity of v i on B 3 modulo the set E θ . Thuŝ Hence (17)  ) the result follows. Observe that L p norm of ∇v t i remains uniformly bounded in B R ε by (18) and hence the moduli of quasi-continuity in, say, B 3 do not depend on the particular choice of k ε or the tangent plane P k x . 2 • We recast the argument above but now for S 1 = 1 a ε P 1 , S 2 = 1 a ε P 2 . Squaring the inequality |ν 1 − ν 2 | ≤ δ we get that 2 sin β 2 ≤ δ where β is the angle between P 1 and P 2 . Since δ now measures the deviation of v t 1 from 1 on D t 2 , (resp. v t 2 on D t 1 ) we conclude that the corresponding moduli of continuity of the limits v 1 , v 2 (as t → 0) modulo a set E θ ⊂ S 1 ∪ S 2 with small p−capacity depend on δ, i.e.
where B r (z i k ) provide a covering of D t i as above but now, say, r = 6δ. Hence we can take c δ = C(ω 1 (12δ) + ω 2 (12δ)).

Proof of Theorem 3
We now formulate our result on the local approximation of total capacity (say in Q ) by tangent planes of and prove (16). where lim δ→0 C δ = 0 and Q = {x ∈ : x ∈ Q }.
Proof Fix x ∈ Q and let P be the plane {y : y · ν x = 0}, where ν x is the normal of at x. Suppose k = (k , k d ) ∈ Z d , εk ∈ Q and let P x k be the tangent plane to at x k = (εk , g(εk )). Then Theorem 2 1 • tells us that ).