Homogenization of oblique boundary value problems
-
Sunhi Choi
Abstract
We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the Hölder continuity of the homogenized boundary data. While we follow the outline of Choi and Kim (Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data, Journal de Mathématiques Pures et Appliquées 102 (2014), no. 2, 419–448), new challenges arise due to the presence of tangential derivatives on the boundary condition in our problem. In addition, we improve and optimize the rate of convergence within our approach. Our results appear to be new even for the linear oblique problem.
1 Introduction
For given
Here,
The examples of boundary conditions we consider include the linear oblique problem:
where the vector field
where
where
We are interested in the behavior of
Note that, as first pointed out by Bensoussan et al. [5], if
Let us state a convergence result on
Theorem 1.1
Let
(here,
The proof of Theorem 1.1 will be given later in this section, based on our main result (Theorem 1.2), which establishes rates of convergence for (approximate) cell problem solutions. Our work extends the previous results in [8] on linear Neumann problems, where
It is unknown whether the form of the boundary condition such as (1) or (2) is preserved in the limit
Literature. Before proceeding further, let us briefly describe some of the relevant literature. In the classical article in [5], the following problem was considered:
For this co-normal boundary value problem, explicit integral formulas have been derived for the limiting operator as well as for the limiting boundary data, under the assumption that
For linear elliptic systems with either Dirichlet or Neumann problem with co-normal derivatives, there has been a recent surge of development in quantitative homogenization relying on the integral representation of solutions: we refer to [2,15,20] and the references therein.
For nonlinear problems, or even for linear problems with non co-normal boundary data, until recently, the focus has been on half-space type domains with rational normal, with the origin on the boundary. In [21], Tanaka considered some model problems in half-space whose boundary is parallel to the axes of the periodicity by purely probabilistic methods. In [1], Arisawa studied specific problems in oscillatory domains near half spaces going through the origin. Generalizing the results of Arisawa [1] for nonlinear boundary conditions, Barles et al. [4] studied the problem for operators with oscillating coefficients, in half-space type domains whose boundary is parallel to the axes of periodicity. We also refer to [14], which adopts an integro-differential approach to study linear scalar problems with the specific Neumann problem
For the linear Neumann problem
Cell problem. By the formal expansion
with
For general
has a solution with sublinear growth at infinity, for any
A discussion on assumptions on
1.1 Assumptions and main results
Let
(Uniform Ellipticity) There exist constants
for all
(1-Homogeneity)
(Lipschitz continuity) There exists
(At most linear growth)
(Lipschitz continuity)
(Oblicity)
A typical example of an operator
where
where
For
and a hyperplane
We will denote
Now we are ready to state the main result.
Theorem 1.2
Let
There exists
Here,
(6)where
When
The proof is given in Theorems 4.1, 5.2, and 5.1.
A discussion on the rate of convergence
If
and thus yields a uniform bound
If
Then
Since
Proof of Theorem 1.1
Once Theorem 1.2 (a) is obtained, one can derive our main theorem by the perturbed test function arguments introduced by Evans [10].
Let
where
where
We claim that
Below we will only show that
First, suppose that
Let
and thus, the choice of
Due to Theorem 1.2, we have
Since
Now suppose that (9) is false, then there exists
This means that the right-hand side of (13) is strictly negative at
Next suppose that
2 Preliminaries
We adopt the following definition of viscosity solutions, which is equivalent to the one given in [9]. Let
where
Definition 2.1
An upper semi-continuous function
A lower semi-continuous function
Existence and uniqueness of viscosity solutions of
Theorem 2.2
Let G and F satisfy the conditions (G1) and (G3)
For a symmetric
and
where
Next we state some regularity results that will be used throughout this article.
Theorem 2.3
[Chapter 8, [6], modified for our setting] Let u be a viscosity solution of
where
As mentioned in Section 1, regularity results for nonlinear Neumann problems are rather limited.
Theorem 2.4
[18,19] Let u be a viscosity solution of
Let u be a viscosity solution of
For F and G satisfying (F1)–(F3) and (G1)–(G3), suppose that either (A)
where C depends on
Our proof extends in general to the cases where estimate (15) holds for some
Below we mention interior homogenization result from [7], which is a modified version of homogenization results such as in [11].
Theorem 2.5
(Theorem 2.14, [7]) Let K be a positive constant and let
Then for any
where
Next we state some consequences of ergodic property of irrational numbers in
Lemma 2.6
[Lemma 2.11 in [13]] For
Finally, we present a lemma that states ergodic property of hyperplanes with irrational normals in
Lemma 2.7
[Lemma 2.7 in [8], Lemma 2.3 in [12]]. For
Suppose that
where
Suppose that
and
We recall that
If
3 Localization lemmas
In this section, we prove several lemmas on perturbing and localizing the solutions, which will be used frequently throughout the article. Below we prove a localization lemma, and as a corollary, we prove existence and uniqueness of solution
First, we state a basic lemma, which will be frequently used. The proof is a direct consequence of the oblicity assumption
Lemma 3.1
There exists
Lemma 3.2
Let
Proof
Without loss of generality, let us set
where
and
Note that in
Also
Hence, to show that
Hence, on
where the last inequality follows from the Lipschitz property of
As a corollary of Lemma 3.2, we prove existence and uniqueness of solutions in strip regions.
Lemma 3.3
There exists a unique solution
Proof
Let
From Lemma 3.1,
Due to Theorem 2.5 and the Arzela-Ascoli Theorem,
To show uniqueness, suppose both
Hence,
The following is immediate from Theorem 2.2 and the construction of
Corollary 3.4
Suppose u,v are bounded and continuous functions in
Then
Lemma 3.5
There exists
where
Let L denote the Lipschitz bound for
Proof
By our assumption,
After a change of coordinates, we may assume
where
Now suppose
On the other hand,
and since
Then using the fact that
Hence, from (19), we obtain a contradiction if
The lower bound can be obtained with the aforementioned argument applied to
4 Homogenization in a strip domain
Let
where
Theorem 4.1
The followings hold for
For irrational directions
where
[Error estimate] There exists a constant
where
(23)In (23),
Let
Let
Let
To prove Theorem 4.1 we begin with a preliminary lemma. The following lemma states that
Lemma 4.2
Away from the Neumann boundary
and
If
(24)where
If
(25)for any
Proof
First, we consider a rational direction
Hence,
where the third inequality follows from Theorem 2.3.
Next, we consider an irrational direction
Observe that
where, from Theorem 2.3,
Next we project
Then by using Theorem 2.3 with (27) once again, we compare
Finally,
where the last inequality follows since
Since
Corollary 4.3
For a solution
and hence,
Due to the uniform interior regularity of
for
Next, we prove that the subsequential limit is unique, i.e.,
Proof of Theorem
4.1
(a) and (b) for irrational directions: Let
for a constant
where we let
For the proof of (28), let
and denote by
Hence, after translations by
respectively, with
and
where
for some
Let
Note that
From (30) and the comparison principle, it follows that
Here, we denote by
whose respective slopes are
Now we define
and
where
on
and also due to (31),
and
in
where
where
Proof of Theorem
4.1
(a) and (b) for rational directions: Let
The proof of (34) is parallel to that of (28). Let
Proof of Theorem
4.1
(c) and (d): Let
and
If we define
Let
Then for
where the first and last inequalities follow from (36) and (35), respectively. Then by (37),
The infimum is taken when
Hence, we can conclude
Next, we consider a rational direction
The following lemma will be used in the next section.
Lemma 4.4
Let
where N and A are constants. Then there is a constant
Proof
For
where the second inequality is from the interior Lipschitz regularity (Theorem 2.3) applied to
5 Continuity over normal directions
In the previous section, we have shown that for an irrational direction
We first show that
Theorem 5.1
For
Proof
For
is, respectively, a super and subsolution of
From here and Theorem 4.1, it follows that
The dependence of
For
Note
Theorem 5.2
With fixed q, let us denote
then we have
Remark 5.3
In the proof, we indeed show that, for any directions
Remark 5.4
For notational simplicity and clarity in the proof, we will assume that
For the rest of the article, we prove (a) of Theorem 5.2. Theorem 5.2 (b) follows from (38), (39), and Theorem 5.2 (a).
5.1 Basic settings and Sketch of the proof
We denote
We also denote
For given
we divide the unit strip
Then
Next we introduce the parameters
and
Without loss of generality, assume
If
On the other hand, since
The two-scale homogenization procedure has been introduced in [7,8]. It allows studying continuity properties of the homogenized boundary data as we approach the rational direction, which might be singular points as described in Section 1. This point of view was also employed in [12,13] to study homogenization for general operators, by studying the singularity of homogenized operator at rational directions. Let us also point out near the boundary the small-scale oscillation of the operator interacts with that of boundary data to create a meso-scale averaging phenomena. Due to this interaction, characterizing the homogenized boundary condition remains a challenging and interesting open problem. After the first homogenization, the boundary data change to periodic data in a meso-scale (which will be
Below we begin the analysis of the two-step homogenization as described earlier. We will work with small
which can be stated as follows:
since
After the near-boundary homogenization,
On the other hand, (39) of Theorem 5.2 can be stated as follows:
It follows then that
which ensures
Next, we define vertical strips
We define
Then we can observe
since the definition of
On the other hand, by the definitions of
Similarly for
Remark 5.5
Observe that (48) implies
and
Our simplification of
and also
for any
Remark 5.6
For
Observe that if Theorem 5.2 holds for the rational direction
Then
since
Remark 5.7
For the dimension
and
Then parallel arguments as in steps 1–9 in the next section would apply to yield the results in
5.2 Proof of Theorem 5.2
In the first three steps, we follow the aforementioned heuristics and replace the Neumann condition with the locally projected boundary data
Step 1. First homogenization near Boundary (
We proceed to discuss the first homogenization. Denote
and
Definition 5.8
For a given function
Note that the Neumann boundary data of
Lemma 5.9
For
Proof
We will prove the lemma for
and
We will compare both of
Here, observe that in the ball
Below we derive some properties of
Then by Theorem 2.4,
where
and hence, the oscillation of
First, we compare
where the second inequality follows from (49) and the construction of
Note that
Observe that (57) and (59) yield
where
Next, we compare
Recall that the oscillation of
which satisfies the assumptions for the
For
Recall that the Neumann boundaries of
and Lemma 3.5 yields that in
This and (57) yield that in
where
Recalling
In the rest of proof, we will show
Then the aforementioned inequalities and (62) would imply
First, observe that
where the last inequality follows from (47).
Next, recall that
for a solution
where
and we can conclude
Step 2. Constructing middle region barrier
In step 1, we showed that
but for computational clarity, we will keep the symbol
We will construct middle region barrier
Consider a ball
By using this fact with Lemma 3.5, we can construct a
Let
Step 3. Homogenization of the operator in the middle region
Next we show, similar to Lemma 5.9, that the second homogenization does not change too much if the domain
Here,
To this end, we will first compare
Lemma 5.10
For any
Proof
The proof follows from Theorem 2.5 applied to
Next we compare
Lemma 5.11
Let
in
Proof
Observe that
and
Hence, one can apply Lemma 2.9 of [7] to
Step 4. Flatness of
Lemma 5.12
[Flatness of
Proof
Due to Lemma 5.10, it is enough to show aforementioned lemma for
We know that
Hence,
which can be written in terms of
in
Now we construct the near-boundary barrier
Step 5. Flatness of
In this step, we compare
Also from Lemma 5.12 and the definition of
where
in
Before we proceed to the next step, observe that the
Step 6. Patching up
Let
Since
Thus, it follows that
Due to the flatness estimates (70) and (71), we can approximate
Define
Then by (72),
where the last inequality follows by choosing
We denote
Step 7. Comparing the solutions
Parallel arguments as in the previous steps apply to the other direction
Then similarly as in the direction
Here, their corresponding Neumann boundary conditions satisfy
where
Thus, to compare
and
Now we prove the following lemma using the estimate (75).
Proof
By the construction of the viscosity supersolution
where
where
Here,
Recall that
for solutions
and
Note that
where the last inequality follows from (47). Then we can conclude from (78) and (79).□
-
Funding information: Research supported by NSF DMS-1566578. I.K. is partially supported by NSF DMS-2153254.
-
Conflict of interest: Authors state no conflict of interest.
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