Homogenization of the boundary value for the Dirichlet Problem

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a sharp geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in $C^1$-domains.

H. Shahgholian has been supported in part by Swedish Research Council. K. Lee has been supported by Korea-Sweden Research Cooperation Program. This project is part of an STINT (Sweden)-NRF (Korea) research cooperation program. 1 1. Introduction 1.1. Background. In this paper we consider Dirichlet and related problems, with rapidly oscillating data. Although we treat the case of Laplace equation, our analysis straightforwardly extends to general cases for equations that admit a Poisson/Green representation. Such integral representation, in turn, reduces the study of the problem to the corresponding integral equation, and hence the analysis of the integral of rapidly oscillating functions becomes central.
To fix ideas, let Γ be a C 1 surface in R n (n ≥ 2), not necessarily bounded. We assume g(x, y) is integrable in both variables over Γ, and 1-periodic in y-variable i.e., g(x, y + k) = g(x, y) for k ∈ Z n . In this paper, we shall study possible limit behaviors of the integral (1) lim ε→0 Γ g y, y ε dσ y , and we shall prove that under mild conditions on the surface Γ there is an effective limit as ε tends to zero. In general we show that the above integrals stay within the bounds of the interval Γ g * (y, ν y ), Γ g * (y, ν y ) where g * , g * are defined below, as the infimum respectively supremum of the average-integrals of g(y, ·) over closed loops of the plane {x : x · ν y = 0} on the torus; here ν y denotes the normal vector of Γ at y.
To illuminate the application of this to the Dirichlet problem, let us consider a bounded C 1 -domain Ω ⊂ R n (n ≥ 2). Let further g(x, y) and f (x, y) be continuous in both variables, and periodic in y. Let u ε be a solution to the Dirichlet problem (P ε ) u ε = −µ ε in Ω, u = g(x, x/ε) on ∂Ω, where µ ε = f (x, x/ε)χ Γ 0 dσ x , and Γ 0 is a C 1 surface compactly in Ω.
Since solutions to this problem can be represented by surface integrals of g and f (through Poisson and Green functions), we may directly apply our results. Hence, a by-product of our surface integral homogenization, is nontrivial limit scenarios, as ε tends to zero, of equation (P ε ) and its solution.
It is noteworthy that, through integral equations of Fredholm type, or just standard functional minimization, our technique applies to homogenizations with oscillating Neumann data (see Section 6.3). For fully nonlinear operator the Neumann problem has been treated in [CKL], see also [BDLS].
Another issue that occurs along such analysis is the study of the speed of convergence. In problems, where governing partial differential equations have oscillating coefficients, one uses the standard method of expansion in an appropriate space. Here u bl, is the so-called boundary layer term, and finding it is part of the problem. The function u bl, will then solve a Dirichlet problem with oscillating boundary data. We refer to [AA] for some backgrounds and to [GM] for recent developments in the topic.
Remark 1.1. It is noteworthy that our approach applies to general equations of the type div(A(x/ε)∇u) = f (x/ε) with oscillating Dirichlet/Neumann data. This, even though straightforward, becomes quite technical and is therefore outside the scope of this paper. Thus for clarity of the exposition we shall treat the Laplacian case, only.

Heuristics. Let us set
, and y − x ∈ Z n , and adopt the standard notation from homogenization, as well as that of ergodic theory.
For clarity of the ideas, let us also deal with problem (P ε ) in the case f ≡ 0, and see how it reduces to the integral averaging, and then how the limit integral is obtained.
Since harmonic functions can be represented using Poisson kernel P(x, y), u (x) = ∂Ω P(x, y)g(y, y/ ) dσ y , we must then analyze the behavior of this integrals as ε tends to zero. For x ∂Ω, P(x, y) is continuous and we can rewrite the integral where r j is small enough, independent of ε, and Q r j (y j ) is a cube with size r j and center y j . This obviously brings us to integral (1). Observe that at this stage we cannot replace the second variable of g with y j , due to rapid oscillation for ≈ 0. Further assume that ∂Ω ∩ Q r j (y j ) is so flat that at -scale it is approximately 2 -away from its tangent plane; this requires a C 2 -graph locally, but our techniques/proofs work for C 1 -surfaces.
The idea now is to cover the boundary ∂Ω ∩ Q r j (y j ) with finite many, and small enough cubes, so that each part of the surfaces is as flat as we want to. In particular we have where ε i is to be chosen small enough. This part is slightly more delicate, and needs extra care. We would also prefer to take ε i = √ ε. The trouble is that this might not work, as the points are changing, and we may loose lots of information about the behavior of the surface ∂Ω ∩ Q ε i r j (y i,j ). Indeed, an scaling of the integral gives ∂Ω∩Q (ε i /ε)r j (y i,j ) g(y j , y) dσ y , and the question is whether this integral will converge to the mean Q + 1 g(y j , y)dy. This would happen exactly when the normal vector of the surface ∂Ω at y i,j is irrational. In other words the surface ∂Ω ∩ Q (ε i /ε)r j (y i, j ) (which is almost a plane) will foliate the n-dimensional torus, as ε tends to zero, provided we have chosen ε i /ε → ∞, and the normal vector of the surface ∂Ω at y i, j is irrational. In particular g(y j , y) dσ y = g(y j ).
If the set of points on ∂Ω with rational direction has zero surface measure (this may fail if there are so-called flat spots with rational directions) then we could cover the boundary with small cubes centered at points with irrational normals. From here combining (2)- (3) we obtain It is apparent that if there is a flat portion of the boundary with rational normal, then a full foliation cannot take place at such portions. Consequently, the resulting (mod 1) surface will be a close simple curve over the n-dimensional torus. Hence different sequence of may give different shifts of this close loop, and hence the possibility of a parameter family of values, in between [g * (y, ν y ), g * (y, ν y )].
Remark 1.2. A word of caution: As ε tends to zero, one may obviously rescale the integral, by a change of variables, as we did in our heuristic explanations above. This scaling makes the surface to be scaled (we assume the origin is NOT on the surface) so as to disappear in the limit. This naturally would make it impossible to compute the limit integral. However, the periodicity of the function g in its second variable, implies that we can bring back the surface so as it passes through the fundamental cube Q + 1 using (mod1) argument. In particular this means that despite the integration is on a fixed surface, the surface itself will start jumping forth and back due to the variable ε.
It would be a good idea for the reader to consider simple examples such as integration over a line segment in the plane, by varying the normal direction of the plan from rational to irrational.
1.3. Plan of the paper. In the next section we shall introduce all definitions and notations. We take care of some technicalities in Section 3. We shall formulate our main results concerning surface integration in Section 4, and its generalization to other type of functions, such as layered-densities, almost periodic functions will appear in Section 5. Several interesting applications to PDE are mentioned in Section 6, and several Examples are also given in Section 7.

Averaging and Ergodic theory.
Definition 2.1. Let ν be a vector in R n , z a fixed given point, and Q r (z) = {x : |x i − z i | < r}. Let g(x, y), be integrable in both variables over the plain {(y − z) · ν = 0}, and periodic in y-variable. We define Later we shall consider cases where ν = ν z , is the normal vector at z on a given surface Γ. We also define the average of g as Henceforth we shall assume all vectors have length one, unless otherwise stated. We shall also without loss of generality assume that the surface is orientable, and fix a consistent choice of normal in clockwise direction (this choice is obvious if Γ is the boundary of a domain).
It should be noted that if the set {(y − z) · ν = 0}(mod 1) foliates the cube Q + 1 then the integral converges to the average g(z), and this happens exactly when ν is irrational direction (see Lemma 3.2). When the set {(y − z) · ν = 0}(mod 1) does not foliate the cube Q + 1 (or the n-dimensional torus) then we shall get the limit as an integral over a closed loop flow on the torus. In particular the value of the integral exists, and depends on ν and . For different values of this loop translates over the torus and will give rise to supremum respectively infimum values of the integrals, as defined above by g * , g * , respectively.
Remark 2.1. It should be remarked that in the definitions of the above averages g * , g * we could have replaced for any domain D containing the point z. The reason for this is that the set T = {(y − z) · ν = 0}(mod 1) either foliates the whole n-dimensional torus, or it is a closed simple curve on the torus. In either case, when tends to zero, the piece of plane {(y − z) · ν = 0} ∩ D will have the same effect as T.
Since the direction of the normal of the plane {(y − z) · ν = 0} will play a crucial role in our analysis, we introduce proper definitions well-known in Ergodic theory. For readers' convenience we also prove some of these well-known results here.
for the surface measure σ Γ . Here ν x denotes the normal to Γ at x.

Technical preliminaries
In this section we shall recall some standard facts from Ergodic theory, and also state and prove some averaging results that will be needed for the proof of the main theorems.
The first lemma is a version of Weyl's Lemma and is well-known fact about uniform distribution.
That (1, ν 1 , · · · , ν d ) is an irrational direction, means that at least one of the numbers ν j is irrational. In this paper d = n − 1 is the only case that is used.
to be the sum of Dirac delta functions. First, let us consider the case h = h m = e 2iπmt and m 0. By the assumption there is ν l , which is an irrational number, and hence since mν l cannot be an integer i.e. 1 − e 2iπmν l 0. Now by Fourier expansion (extending h as 1-periodic function), for some b m ∈ R, which in combination with (4) results in Lemma 3.2. Let g, g * , g * , and g be as in Definition 2.1. Then the following hold: (ii) The following inequalities always hold Proof. First note that we can replace the integral in the definition of g * , and g * by with N positive integers, and N → ∞. This simplifies the matter slightly.
We may without loss of generality assume ν = (1, ν ). Since ν is irrational direction, Π(mod 1) will foliate the unit cell, according to standard foliation theory and results in integral average over Q + 1 . To see this we assume (by using periodicity) that x ∈ Q + 1 and that the plane Π cuts the x 1 -axis (or some other axis). let a 0 be the point of intersection between the x 1 -axis and this plane, so that Π(ν, x) = {x : x 1 = a 0 − ν · x }, and (1, ν ) is irrational (by the assumption), and 0 ≤ a 0 < 1 due to periodicity of g(x, y) in y. Having fixed ν, we shall now use the notation Π t = Π(ν, te 1 ), the plane with normal ν through the point (t, 0 ).
Let N > 0 be a large number, and I N = {k ∈ Z n−1 : |k i | ≤ N}. Then by periodicity of g(x, ·) Set now w(a k ) = w(x, a k ) := Π a k g x, y dσ y . Then Since ν is irrational, from Lemma 3.1 we conclude whereΠ t = Π t ∩S 0 (mod 1). Therefore lim r→∞ {(y−x)·ν=0}∩Q r g x, y ε dσ y = g(x) which is independent of ε. Now we have conclusion (i) from the definition of g * (x, ν) and g * (x, ν). To prove (ii) let us suppose that ν is a rational direction, otherwise the conclusion follows by the equality in (i). Since ν is rational, it is not hard to see that the restriction of g(x, y) on this hyperplane will be periodic with period T, say; for g(x, y/ε) the period will be εT.
In particular the integral can be seen as integration over a spiral-like plane on the torus (a closed loop). The two dimensional case can be illustrated by a curve on the torus that loops over itself. In particular the limit integral (w.r.t. r) in the definition of g * , g * can be replaced by An observation here is that the supremum value of the mean integral w.r.t. actually is taken for some value 0 , and naturally for many other values, due to periodicity in . One can think of the situation as parallel planes in R n with fixed distance 1 from each other are moving, simultaneously by keeping a fixed distance between them, in the orthogonal direction of the plane (mod 1), when changes. Obviously, these planes foliate Q + 1 once ranges [0, 1).
From here one deduces In our analysis of the limit behavior of the integral (1) we will use Definition 2.1 in a slightly different way. Next lemma will give us a hint in that direction. Lemma 3.3. Let g be as before, Π(ν, z) = {(y − z) · ν = 0}, and z ∈ Γ. Then for any R ε ∞ (as ε 0) we have and lim inf ε→0 Π(ν,z)∩Q εRε (z) g(z, y/ε)dσ y ≥ g * (z, ν).
We leave the the reader to verify this simple fact. We shall next make the previous lemma even more general by letting the plane be replaced by very smooth surfaces. Let Γ be a smooth surface, with module of continuity τ = τ Γ for its C 1 norm. Define Lemma 3.4. Let g be as before, and Γ a smooth C 1 surface, with module of continuity τ = τ Γ for its C 1 norm. Let further ρ ε = M where M is as in (6). Then, for z ∈ Γ, and η > 0 there exists ε ν z ,η such that for all ε ≤ ε ν z ,η we have Proof. Set z ε = z ε (mod 1), and Γ z := {x : (x + z) ∈ Γ}. Then we have If we set Π ν z = {x : x · ν z = 0}, then by continuity of g(z, .), and that for g z, z ε +ỹ − g z, z ε + y dσ y .
Hereỹ = y + ν z |y|s ε with s ε = τ(εM ε ). Estimating I we obtain |I| ≤ Cτ g (|y|s ε ) ≤ Cτ g (M ε s ε ) < η/2, for ε small enough. Here τ g is the module of continuity for g From here, and Lemma 3.3 the statements in the lemma follows, provided we have taken ε small enough depending on ν z .

Surface integrals of oscillating functions
Theorem 4.1. Let Γ be a C 1 surface in R n , and g(x, y) be integrable in x-variable over Γ, and continuos and 1-periodic in y in R n . Then Moreover if Γ satisfies IDDC then an effective limit exists and we have lim ε→0 Γ g y, y ε dσ y = Γ g(y)dσ y .
Proof. We shall prove the limit superior estimate only. The limit inferior estimates follows in a similar way. The last statement follows in an obvious manner. Let us fix a small positive constant η > 0, to be decided later. Without loss of generality, we may assume Γ is bounded, and g(x, y) is uniformly continuous function on Γ × R n .
As η 0 was arbitrary we have the two main estimates in the statement of the theorem.
Putting these together along with the IDDC we shall have the third statement. Indeed, due to IDDC we have that the set {x ∈ Γ : ν x rational } has zero surface measure, and the integral over this set is zero. For the rest of Γ we have irrational normals only, and hence the full averaging (Lemma 3.2) takes place and we obtain g * = g * .

The case of Layered densities, almost periodicity, and ergodicity
In this section we shall deduce, similar results of that in Theorem 4.1 while replacing the periodicity assumption with layered materials/densities, almost-periodic and ergodic case. Nevertheless we shall only mention the results without deepening much into the analysis. The reader may easily verify the statements. 5.1. Layered Materials. If we assume the function g(x, y) is independent of (y k+1 , · · · , y n ) and is 1-periodic in (y 1 , · · · , y k ) then one may naturally obtain results reminiscent of that of layered materials in homogenizations for PDE. Indeed one can obtain the following obvious result: If the surface Γ does not have any flat parts in directions e i (i = 1, · · · , k), then the averaging takes place.

Almost Periodic Case.
In the case of almost periodic functions (see [Bohr]) one obtains similar results by replacing the average integral g(x) withĝ (y) = lim r→∞ Q + r g(x, y)dy.
The obvious details are left to the reader.

Ergodic Case.
The results of our main theorem can be generalized further to the case of functions with ergodic properties (see [JKO]). Indeed, if we assume g(x, y, ω) is defined on Γ × R n × D, for some D ⊂ R n and that g(x, y, ω) is statistically homogeneous field in (y, ω)-variable, i.e., g(x, y, ω) = h(x, T y ω) for some random variable h(x, z) (random w.r.t. second variable) with underlying probability space and an ergodic n-dimensional dynamical system T y . Hence h(x, T y ω) admits an averaging One may now deduce similar results as that in Theorem 4.1 with g(z) = Exp(g(z, ·, ·)).
6. Applications to Partial Differential Equations 6.1. Dirichlet problem: Elliptic Case. Let Ω be a bounded domain in R n , with piecewise smooth boundary. Let g(x, y) be as before, and f (x, y) have the same property as g. For simplicity, we shall assume f, g are continuous in both variables (actually L 2 (∂Ω) would suffice). Let further Γ 0 ⊂ Ω be a piecewise C 1 -curve, and define Then the following result hold.
Theorem 6.1. For a solution u ε of (P ε ), we define (in Ω) Then the following hold (in the weak sense): Moreover for any sequence of u ε , there is a subsequence converging to a function u in Ω, satisfying −µ * ≤ ∆u ≤ −µ * in the weak sense in Ω, Proof. By Green's and Poisson's representations we have where P and G are Poisson respectively the Green functions of the domain. By Theorem 4.1 which implies the first two statements in the theorem. Now the last statement follows from the above inequalities, in an obvious way.
To obtain an effective limit, in the above theorem, one needs to consider domains satisfying IDDC. In particular, using the above theorem and Lemma 3.2 (ii) we have the following result. Corollary 6.2. Let ∂Ω and Γ 0 satisfy IDDC, and u be a solution to problem (P ε ). Then, u ε converges to a function u in Ω, satisfying The general nature of the method employed here suggests that we can apply this to situations where integral representations are possible. This can naturally go beyond the Dirichlet problem, or the Laplace operator, and as general as to systems, and equations of higher orders. We state explicitly that once one has an integral representation of any function, through kernel functions, then one may conclude similar statement as that of Theorem 6.1 and Corollary 6.2. We leave it to the reader to apply this to their favorite scenarios.
Here one may consider different cases, such as Ω = D × (0, T), with D a domain in R n , or Ω ⊂ R n × R and time varying. Also Γ 0 can be take to be either time independent or varying in time.
Now a similar argument, using Poisson and Green representations, as in Section 6.1 can give us various type of results. In the case Ω = D × (0, T) one obtains same type of results as that of Theorem 6.1. To obtain results of the nature of Corollary 6.2 one needs • either to assume Ω = D × (0, T), and D has IDDC condition along with f , and g being independent of their fourth variable, i.e. f ε (x, t) = f (x, x/ε, t), g ε (x, t) = g(x, x/ε, t) • or Ω and Γ 0 , both have IDDC condition in R n+1 .
6.3. Neumann problem. As mentioned at the end of the previous section, one can apply the technique of this paper to far reaching scenarios and problems, involving integral representations. The Neumann problem naturally fits into this category through Fredholm's alternative, and an integral representation. Indeed, let u ε be a solution to the problem with g satisfying compatibility condition ∂Ω g(y, y/ε)dσ y = 0 for all ε > 0. Then it is well-known that where F is the Fundamental solution for Laplace operator (or the corresponding operator), and φ solves the Voltera integral equation of second kind, i.e., As ε tends to zero, φ ε tends (weakly in L 1 (∂Ω)) to a limit φ 0 solving in a weak sense over the boundary of Ω. This happens exactly when the boundary of Ω has IDDC. More accurately, the kernel of the bounded operator acting on L 1 (∂Ω) space, is upper semi-continuous and has a unique element. In particular lim ε ker(T ε ) ⊂ ker(T 0 ), where By uniqueness of the solutions to the Fredholm operator this kernel must have only one element, and hence φ ε → φ 0 , with φ 0 ∈ Ker(T 0 ). In other words φ 0 solves the Voltera equation above for the function g(x). From here it follows that u ε converges to u 0 = ∂Ω F(x, y)φ 0 (y)dσ y , with φ 0 solving φ 0 (x) = ∂Ω ∂ ν F(x, y)φ 0 (y)dσ y + g(x).
Hence u 0 solves the averaged/effective Neumann problem A different way of analyzing this is to consider the solution of the Neumann problem in the weak form Ω ∇u ε (y) · ∇φ(y)dy = ∂Ω g(y, y/ε)φ(y)dσ y , where φ is a test function in a reasonable class. Letting ε → 0 we see that the integrals converge to Ω ∇u 0 (y) · ∇φ(y)dy = ∂Ω g(y)φ(y)dσ y .
The latter in turn solves the Neuman problem with g(y) as the amount of flux at each boundary point.

Examples and illustrations
The behavior of the limit integrals in 2.1 are directly related to foliation of the fundamental cell Q + 1 . To illustrate this (in R 2 ) consider a sequence p j = (− √ 2/j, 1). For a ∈ [0, 1), let l a j be the line through the point (0, a) and orthogonal to p j . Then, due to the fact that p j is rationally independent Lemma 3.2 implies lim ε→0 l j ∩Q ρε g(x, y/ε)dσ y = g(x).
On the other hand l j → l 0 which is a line through (0, a) and parallel to the x 1 -axis. For the limit of the average for this line we then have lim ε→0 l 0 ∩Q ρε g(x, y/ε)dσ y = g(x, ·, a).
Let us set L a j = l a j ∩ Q ρ ε 0 /ε 0 (mod 1). Then one readily verifies that for ε 0 fixed, and very small where M ε 0 = ρ ε 0 /ε 0 . In particular, for j large L j will never foliate the unit cell, and hence it is impossible to approximate the integral over ∂Ω by any covering, however small. 7.1. Example 1. We consider the case when g is periodic only in x 1direction and when the domain is a slab with a unit normal direction ν.
For a ν ∈ S n−1 , set Ω = {x : −R 1 < x · ν < R 2 }. Let g(x) = g(x 1 ) be independent of (x 2 , · · · , x n ) and 1-periodic, i.e. g(x + k) = g(x 1 + k 1 ) = g(x 1 ) = g(x) for k ∈ Z n . Now let u ε be a solution of the following equation We discuss three possible limits of u ε whose homogenized equation can be found. Namely, Figure 2). It turns out that u * , u * , and u don't follow simple homogenization whose boundary data is a simple average g. It means there are nontrivial homogenization processes for each different limits. For R 0, set g * (R, e 1 ) = max g, g * (R, e 1 ) = min g, If R = 0, let g * (0, e 1 ) = g * (0, e 1 ) = g(0, e 1 ) = g(0). Proposition 7.1. For the particular choice ν = e 1 , in equation (10), the limit functions u * , u * , and u will satisfy Proof. Select ε i such that g x 1 ε i = max g and then u ε i = u * since all u ε i have the same boundary values. Hence it is clear that lim sup ε→0 u ε = u * . Similar argument can be applied to u * and u to have the conclusion.
Letting tend to zero and using first Lemma 3.3, and then Lemma 3.2 we shall have lim →0 x·ν=R 2 P(x − y)g y 1 ε dσ y = x·ν=R 2 P(x − y)gdσ y , which implies the conclusion. The next interesting question is to find the limit equation for the general converging sequence. In the following lemma, we will show there is a converging subsequence whose limit takes any value between the supremum of g and its infimum on x · e 1 = R 2 . Proposition 7.3. For any A such that min g ≤ A ≤ max g, there is a sequence {u ε i } converging uniformly to u such that its limit u satisfies in Ω, u(x) = M on x · e 1 = −R 1 , u(x) = A on x · e 1 = R 2 .
Proof. There are ε i → 0 such that g x 1 ε = A since g is 1-dimensional. Therefore all u ε i have the same boundary values, implying u ε i = u(x). 7.2. Example 2. In this example we confine ourselves to R 2 . We consider the case when g is periodic only in x 1 -direction and the domain Ω is convex with two parallel flat parts of boundaries, orthogonal to ν = e 1 . For exactness we consider the following stadium like domain Ω = x : |x 1 | < R, |x 2 | ≤ 1 + R 2 − x 2 1 .
Let g(x) = g(x 1 ) be 1-periodic, and independent of x 2 -direction. Now let u ε be a solution of the following equation then u * and u * (see (11)) are sub-and super-solutions of (18) respectively. In general, they are not solutions.
In the next result we state that the homogenized boundary data may not be continuous even though g(x) is smooth.
Proposition 7.4. There is a a smooth 1-periodic function g(x 1 ) of one variable x 1 and a subsequence of solutions {u ε i } to equation (18), converging to u such that u is not continuous on ∂Ω.
Proof. If x 1 ±R, then x = (x 1 , x 2 ) ∈ ∂Ω satisfies IDD condition and then any limit will satisfy the boundary condition u(x) = g. Now we select ε i so that g x 1 ε i = max g g. Then u ε i has a converging subsequence to u which will be discontinuous at x = (R, 1).