An L^p-comparison, p∈ (1,∞), on the finite differences of a discrete harmonic function at the boundary of a discrete box

05/20/2019 ∙ by Tuan-Anh Nguyen, et al. ∙ Universität Duisburg-Essen 0

It is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d≥ 2, the tangential and normal component of the gradient ∇ u on the sphere are comparable by means of the L^p-norms, p∈(1,∞), up to multiplicative constants that depend only on d,p. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the d-dimensional lattice with multiplicative constants that do not depend on the size of the box.

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1 Introduction

This paper formulates and proves a discrete analogue of a classical result in the continuum setting which states that the tangential and normal component of the gradient of a harmonic function on the boundary of a domain are comparable by means of -norms, . For convenience we give a simplified version of this result in Theorem 1.1 below. For complete formulations and proofs we refer the reader to Maergoiz [17] (see, e.g., Theorems 1 and 2), Mikhlin [18] (see § 44 p. 208), and Bella, Fehrman, and Otto [1] (see Lemma 4). This result can be viewed as a

stability estimate for harmonic extensions

of given Dirichlet or Neumann boundary conditions. Thus, it plays an important role in the proof of a Liouville theorem for a class of elliptic equations with degenerate random coefficient fields (see formulas (40) and (41) in [1]) where the so-called idea perturbing around the homogenized coefficients is realized by harmonic extensions of given boundary conditions. The discrete analogue that we want to show here can be applied to prove a Liouville theorem for the random conductance model under degenerate conditions, which is the discrete analogue of [1] (see the paragraph below Lemma 4 in [1] and the PhD thesis of the author [20] (e.g., Section III.2.3 for an outline).

Theorem 1.1.

For every , denote by the Euclidean norm of . Denote by the usual surface integral. For every , let be the sets given by

(1.1)

For every , , let be the gradient of , let be the tangential component of , and let be the normal component of , i.e., it holds for all that is the orthogonal projection of onto the tangential space of at and . Then there exists a function such that for all , , , with and it holds that

(1.2)
Figure 1: Tangential and normal edges of a two dimensional box

In order to formulate the discrete analogue of Theorem 1.1 let us introduce our notation. For the rest of this paper we always use the notation given in Setting 1.2 below.

Setting 1.2 (Notation for the whole paper).

Let . For every let , , be the standard

-dimensional basis vectors and let

be the set of all oriented nearest neighbour edges (for short: edges) on the -dimensional lattice, i.e., the set given by . For every , , let be the discrete Laplacian of , i.e., the function which satisfies for all with that

(1.3)

and let be the function which satisfies that for all with it holds that . For every , , every finite set , and every function let be the real number given by

(1.4)

Note that in Setting 1.2 above the arguments of are edges. We will also introduce another notation for discrete derivatives which are functions of vertices (see Section 3). However, to formulate the main result let us temporarily use the notation in Setting 1.3 below.

Setting 1.3.

For every let be the sets of edges which satisfy that

(1.5)
(1.6)

let be the set of vertices given by , let be the set of functions , let be the set of functions with , and let be the set of functions which satisfy that

In Setting 1.3 above for we consider boxes on the -dimensional lattice instead of balls on the -dimensional Euclidean space. Fig. 1 illustrates the sets , , for , : contains all red points, contains all red edges, and contains all blue edges. For fixed the set can be viewed as a discrete boundary of the box , a function is thus a discrete Dirichlet condition, an edge can be viewed as a tangential vector, an edge can be viewed as a normal vector, and a function with is thus a discrete Neumann condition. Here, the vanishing mean is a necessary condition for the Neumann problem to have a solution, which also holds in the continuum setting. Finally, the set is the set of functions which is defined on the box and harmonic in the interior of the box.

Theorem 1.4 (Main result).

Assume Setting 1.3. Then there exists a function , there exists a unique family of linear operators

(1.7)

and there exists a family of linear operators

(1.8)

such that

  1. it holds for all , , that ,

  2. it holds for all , , that

    (1.9)
  3. it holds for all , , that , and

  4. it holds for all , , that

    (1.10)

Item i in Theorem 1.4 above implies that for every , the function is the solution to the discrete Dirichlet problem

(1.11)

and the family therefore exists uniquely as in the statement of Theorem 1.4. Next, Item iii in Theorem 1.4 above implies that for every , the function is a solution to the discrete Neumann problem

(1.12)

Note that there is no full statement on the uniqueness of the Neumann problem (1.12). More precisely, the uniqueness of the Neumann problem (1.12) only holds up to a constant on , i.e., if and if is a solution to (1.12), then the restriction of on is a constant function. In addition, note that for fixed , there exists no real numbers such that for every solution to (1.12) it holds that . Indeed, e.g., in the case we can freely change the value of at the four corners of the rectangle in Fig. 1 to make arbitrary large without damaging the fact that is a solution to (1.12). Consequently, it is impossible to make any claims on the uniqueness of the family in the statement of Theorem 1.4.

Next, let us give a brief and rough explanation why Theorem 1.4 is useful for the idea of using harmonic extensions in the proof of the Liouville theorem in [20]. Let be a function defined on the box in Fig. 1. We keep the Dirichlet condition of at red points and replace the values of at other points by an extension that is harmonic in the interior of the box. This will clearly erase the Neumann condition of . However, Theorem 1.4 claims that the new Neumann condition can still be bounded by the remaining Dirichlet condition.

Discrete Laplacian and discrete harmonic functions are interesting topics that date back to 1920s (see, e.g., the fundamental works by Lewy, Friedrichs, and Courant [16], Heilbronn [11], Duffin [5]). Discrete boundary problems have been widely studied in numerical analysis, e.g., to approximate the continuum solutions (see, e.g., the classical work by Stummel [21] and for further references see, e.g., Gürlebeck and Hommel [12], [9], [10], who studied Dirichlet and Neumann boundary problems on general two-dimensional discretized domains using difference potentials, and the references therein).

Although discrete and continuum objects often have many similar properties, it is not always trivial to adapt things from the continuum case to the discrete case and vice verse. To the best of the author’s knowledge, there exists no result in the discrete case which deals with the bounds (1.9) and (1.10), while -comparisons, , between the tangential and non-tangential components of harmonic functions on Lipschitz and -domains and related topics have been studied by several papers, e.g., in chronological order: Mikhlin [18], Maergoiz [17], Calderon, Calderon, Fabes, Jodeit, and Rivièrie [2], Fabes, Jodeit, and Rivièrie [6], Jerison and Kenig [13], Verchota [22], Dahlberg and Kenig [3], Mitrea and Mitrea [19]. The main issue in the discrete case is to show that the functions in (1.9) and (1.10) do not depend on the size of the discrete box while in the continuum case this is not an issue due to a simple scaling argument. In fact, for (1.2) we only need to consider .

The proof of Theorem 1.4 that we represent here essentially mimics the proof of Lemma 4 in Bella, Fehrman, and Otto [1] who formulate and prove Theorem 1.1 with balls replaced by boxes in the continuum case. We separate the proof into several steps and organize the paper as follows. Section 2 formulates and proves a discrete counterpart of inequality (88) in [1], which was shown by using the continuum Poison kernels. In order to adapt this idea to the discrete case we use a result in Lawler and Limic [15] to approximate the discrete Poison kernels by the continuum Poison kernels. Estimates by means of the Marcinkiewicz multiplier theorem, e.g., inequalities (78), (79), (82), and (99) in [1] are adapted in Section 3 which focuses on discrete harmonic functions on haft spaces with periodic boundary conditions. In order to avoid many tedious calculations with higher derivatives of the multipliers we apply Cauchy’s integral formula. In addition, with some elementary arguments, Section 3.4 provides a result of independent interest that the author has not found in the literature. Finally, Section 4 applies the results obtained in Sections 3 and 2 to prove the main result, Theorem 1.4. As Bella, Fehrman, and Otto [1] we call estimate (1.9) the Dirichlet case and estimate (1.10) the Neumann case and prove them separately. The main techniques here are basically to adapt two ideas learnt from [1] to the discrete case: i) returning to the case of periodic boundary conditions by using even and odd reflections and ii) reducing to the case of haft spaces. Concerning the idea of using reflections, Section IV.2.1 in the author’s dissertation [20]

may provide a simple illustration with figures in the two-dimensional case that may help to understand the general case. Another interesting application of even and odd reflections and the discrete Marcinkiewicz multiplier theorem is to prove

-estimates for discrete Poisson equations (see Section 2.5.2 in Jovanović and Süli [14]).

For convenience, throughout this paper, the arguments here are often compared with that in the continuum case in [1]. However, since there are several differences between the discrete case and the continuum case, this paper is organized so that the reader can easily start from scratch.

Finally, the proof shows that the functions in Theorem 1.4 may depend exponentially on the dimension: this result, as finite difference method in general, may not be quite useful for high-dimensional applications (the so-called curse of dimensionality).


Our notation will be defined clearly in the formulation of each result. In addition, remember that throughout this paper we always use the notation in Setting 1.2 above and the usual conventions in Setting 1.5 below.

Setting 1.5 (Conventions).

Denote by the imaginary unit. Denote by and the real and imaginary part of , respectively, where . Write and . For , , denote by the -th coordinate of (if no confusion can arise), denote by the standard scalar product of and , i.e., , and denote by the maximum norm of , i.e., . For every set denote by the cardinality of . Partial derivatives will be denoted by , , . When applying a result we often use a phrase like ’Lemma 3.8 with ’ that should be read as ’Lemma 3.8 applied with (in the notation of Lemma 3.8) replaced by (in the current notation)’ and we often omit a trivial replacement to lighten the notation, e.g., we rarely write, e.g., ’Lemma 3.28 with ’.

Acknowledgement

This paper is based on a part of the author’s dissertation [20] written under supervision of Jean-Dominique Deuschel at Technische Universität Berlin. The author thanks Benjamin Fehrman and Felix Otto for useful discussions and for sending him the manuscript of [1]. The author gratefully acknowledges financial support of the DFG Research Training Group (RTG 1845) ”Stochastic Analysis with Applications in Biology, Finance and Physics” and the Berlin Mathematical School (BMS).

2 Potential-theoretic results for harmonic functions on haft spaces

2.1 Main result

In this section we essentially prove Corollary 2.2 below, which formulates a discrete analogue of inequality (88) in Bella, Fehrman, and Otto [1]. We basically follow the proof in [1]. However, to make the argument more illustrative we introduce a simple random walk in Setting 2.3. Lemmas 2.11 and 2.7 are discrete counterparts of inequality (92) and (93) in [1]. Combining Lemmas 2.11 and 2.7

with a Marcinkiewicz-type interpolation argument we obtain

Corollary 2.12. Approximating the discrete Poisson kernels by the continuum counterparts we obtain Lemma 2.10. This and Corollary 2.12 imply Corollary 2.2.

Setting 2.1.

For every , let be the set given by and let be the set of all bounded functions with the properties that

  1. it holds for all , that and

  2. it holds for all that .

Corollary 2.2.

Assume Setting 2.1. Then there exists such that for all , , , , , it holds that

2.2 Results which directly follow from the simple random walk representation

Throughout this section we use the notation given in Setting 2.3 below. Due to the Riesz-Thorin interpolation argument for Corollary 2.6 we have to consider the function in Setting 2.3 as a complex-valued function. For other results we only need to replace by .

Setting 2.3 (Simple random walks).

Let be fixed, let

be a probability space with expectation denoted by

, let ,

, be independent random variables which satisfy for all

, that and let , , be the random variables which satisfy for all that

(2.1)
Lemma 2.4.

Assume Settings 2.3 and 2.1 and let , . Then

  1. it holds for all that and

  2. it holds for all that .

The following proof relies on martingale theory. For an elementary proof see Section A.2.

Proof of Lemma 2.4.

The assumption that and the assumption that is bounded demonstrate for all that is a bounded martingale. The optional stopping theorem proves that

(2.2)

This shows Item i. Furthermore, (2.2), linearity, and periodicity imply for all that

(2.3)

The proof of Lemma 2.4 is thus completed. ∎

Lemma 2.5.

Assume Setting 2.1 and let , , . Then it holds that

(2.4)
Proof of Lemma 2.5.

Recall that we use the notation in Setting 2.3. The fact that -almost surely it holds that and the assumption on periodicity, i.e., imply that -almost surely it holds that

(2.5)

Furthermore, Lemma 2.4 shows for all that

(2.6)

This and (2.5) establish that

(2.7)

and

(2.8)

This completes the proof of Lemma 2.5. ∎

Combining Lemma 2.5 with a Riesz-Thorin interpolation we obtain Corollary 2.6 below.

Corollary 2.6.

Assume Setting 2.1 and let , , , , . Then it holds that

Lemma 2.7 (-estimate).

Assume Setting 2.1 and let , , , . Then it holds that

(2.9)
Proof of Lemma 2.7.

Recall that we use the notation in Setting 2.3. First, observe that Lemma 2.4, Jensen’s inequality, and linearity of show that

(2.10)

Next, Jensen’s inequality and Corollary 2.6 ensure that

(2.11)

Combining this, (2.10), and the triangle inequality completes the proof of Lemma 2.7. ∎

Lemma 2.8.

Assume Setting 2.1 and let , , , , , satisfy that . Then it holds that

(2.12)
Proof of Lemma 2.8.

Jensen’s inequality and the assumption ensure that

(2.13)

This completes the proof of Lemma 2.8. ∎

2.3 The Poisson kernel revisited

Setting 2.9.

Assume Setting 2.3, let be the function which satisfies for all that , and let be the real extended number given by

(2.14)
Lemma 2.10.

Assume Setting 2.9. Then .

Proof of Lemma 2.10.

Throughout this proof let be the surface area of the -dimensional unit sphere and denote by the Euclidean norm. The definition of and Theorem 8.1.2 in Lawler and Limic [15] (applied with for and combined with the definition of the Poisson kernel at the beginning of Section 8.1.1 in [15]) shows that there exist , which satisfy for all that

(2.15)
(2.16)

The triangle inequality then implies for all that

(2.17)

Next, (2.16) implies that for all it holds that