Besov regularity of solutions to the p-Poisson equation 1footnote 11footnote 1This work has been supported by Deutsche Forschungsgemeinschaft DFG (DA 360/18-1, DA 360/19-1) and European Research Council ERC (Starting Grant HDSP-CONTR-306274).

Besov regularity of solutions to the -Poisson equation 111This work has been supported by Deutsche Forschungsgemeinschaft DFG (DA 360/18-1, DA 360/19-1) and European Research Council ERC (Starting Grant HDSP-CONTR-306274).

Stephan Dahlke    Lars Diening    Christoph Hartmann222Corresponding author.    Benjamin Scharf    Markus Weimar
July 17, 2019
Abstract

In this paper, we study the regularity of solutions to the -Poisson equation for all . In particular, we are interested in smoothness estimates in the adaptivity scale , , of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to -Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local Hölder with global Sobolev estimates. In particular, we prove that intersections of locally weighted Hölder spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.

Keywords: -Poisson equation, regularity of solutions, Hölder spaces, Besov spaces, nonlinear and adaptive approximation, wavelets.

Subject Classification: 35B35, 35J92, 41A25, 41A46, 46E35, 65M99, 65T60.

1 Introduction

This paper is concerned with regularity estimates of the solutions to the -Poisson equation

(1)

where and denotes some bounded Lipschitz domain. The corresponding variational formulation is given by

(2)

Problems of this type arise in many applications, e.g., in non-Newtonian fluid theory, non-Newtonian filtering, turbulent flows of a gas in porous media, rheology, radiation of heat and many others. Moreover, the -Laplacian has a similar model character for nonlinear problems as the ordinary Laplace equation for linear problems. We refer to [36] for an introduction. By now, many results concerning existence and uniqueness of solution are known, we refer again to [36] and the references therein. However, in many cases, the concrete shape of the solutions is unknown, so that efficient numerical schemes for the constructive approximation are needed. In practice, e.g., for problems in three and more space dimensions, this might lead to systems with hundreds of thousands or even millions of unknown. Therefore, a quite natural idea would be to use adaptive strategies to increase efficiency. Essentially, an adaptive algorithm is an updating strategy where additional degrees of freedom are only spent in regions where the numerical approximation is still “far away” from the exact solution. Nevertheless, although the idea of adaptivity is quite convincing, these schemes are hard to analyze and to implement, so that some theoretical foundations that justify the use of adaptive strategies are highly desirable.

The analysis in this paper is motivated by this problem, in particular in connection with adaptive wavelet algorithms. In the wavelet case, there is a natural benchmark scheme for adaptivity, and that is best -term wavelet approximation. In best -term approximation, one does not approximate by linear spaces but by nonlinear manifolds , consisting of functions of the form

(3)

where denotes a given wavelet basis and with . We refer to Section 2 and to the textbooks [13, 39, 50] for further information concerning the construction and the basic properties of wavelets. In the wavelet setting, a best -term approximation can be realized by extracting the biggest wavelet coefficients from the wavelet expansion of the (unknown) function one wants to approximate. Clearly, on the one hand, such a scheme can never be realized numerically, because this would require to compute all wavelet coefficients and to select the biggest. On the other hand, the best we can expect for an adaptive wavelet algorithm would be that it (asymptotically) realizes the approximation order of the best -term approximation. In this sense, the use of adaptive schemes is justified if best -term wavelet approximation realizes a significantly higher convergence order when compared to more conventional, uniform approximation schemes. In the wavelet setting, it is known that the convergence order of uniform schemes with respect to depends on the regularity of the object one wants to approximate in the scale of -Sobolev spaces, whereas the order of best -term wavelet approximation in depends on the regularity in the adaptivity scale , of Besov spaces. We refer to [7, 14, 26] for further information. Therefore, the use of adaptive (wavelet) algorithms for (1) would be justified if the Besov smoothness of the solution in the adaptivity scale of Besov spaces is higher than its Sobolev regularity .

For linear second order elliptic equations, a lot of positive results in this direction already exist; see, e.g., [6, 8, 10]. In contrast, it seems that not too much is known for nonlinear equations. The only contribution we are aware of is the paper [11] which is concerned with semilinear equations. In the present paper, we show a first positive result for quasilinear elliptic equations, i.e., for the -Poisson equation (1). Results of Savaré [41] indicate that, on general Lipschitz domains, the Sobolev smoothness of the solutions to (1) is given by if , and by if . However, under certain conditions, the solutions possess higher regularity away from the boundary, in the sense that they are locally Hölder continuous; see, e.g., [18, 24, 45, 48, 49]. The local Hölder semi-norms may explode as one approaches the boundary, but this singular behaviour can be controlled by some power of the distance to the boundary as shown, e.g., in [19, 32, 34, 35]. We refer to Section 4 for a detailed exposition. (Properties like this very often hold in the context of elliptic boundary problems on nonsmooth domains, we refer, e.g., to [38] and the references therein for details). It turns out that the combination of the global Sobolev smoothness and the local Hölder regularity can be used to establish Besov smoothness for the solutions to (1). In many cases, the Besov smoothness is much higher than the Sobolev smoothness or respectively, so that the use of adaptive schemes is completely justified.

We state our findings in two steps. First of all, we prove a general embedding theorem which says that the intersection of a classical Sobolev space and a Hölder space with the properties outlined above can be embedded into Besov spaces in the adaptivity scale . It turns out that for a large range of parameters, the Besov smoothness is significantly higher compared to the Sobolev smoothness. The proof of this embedding theorem is performed by exploiting the characterizations of Besov spaces by means of wavelet expansion coefficients. Then we verify that under certain natural conditions the solutions to (1) indeed satisfy the assumptions of the embedding theorem, so that its application yields the desired result.

This paper is organized as follows: In Section 2, we introduce all the function spaces that will be used in the paper, including their wavelet characterizations, if possible. Afterwards, in Section 3 and Section 4, we state and prove our main results: Our general embedding (Theorem 3.1) can be found in Section 3. Its application to the case of the solutions to (1) which yields new, generic Besov regularity results (see Theorem 4.1 and Theorem 4.2) is performed in Subsection 4.1 and 4.2, respectively. Moreover, here we give explicit bounds on the Besov regularity of the unique solution to the -Poisson equation with homogeneous Dirichlet boundary conditions in two dimensions; see Theorem 4.3 and Theorem 4.4. The paper is concluded with an Appendix (Section 5) which contains a couple of auxiliary lemmata and propositions which are needed in our proofs.

Notation: For families and of non-negative real numbers over a common index set we write if there exists a constant (independent of the context-dependent parameters ) such that

holds uniformly in . Consequently, means and .

2 Function spaces and wavelet decompositions

In this section we recall the definitions of several types of function spaces that will be needed in the sequel. Moreover, we collect some well-known assertions such as, e.g., the characterization of Besov spaces in terms of wavelet coefficients.

2.1 Strongly differentiable functions: (weighted) Hölder spaces

Let be some bounded domain, i.e., an open and connected set. Then, for , furnished with the norm

denotes the space of all real-valued functions on such that is uniformly continuous and bounded on for every multi-index with . Therein denote the -th order strong derivatives. If is a compact subset of  (denoted by ), the spaces are defined likewise. Unless otherwise stated we restrict ourselves to those which can be described as the closure of some open and simply connected set. Next let us recall that for the -th order Hölder semi-norm with exponent is given by

(4)

Consequently, for and ,

denote the (classical) Hölder spaces on . Again we can replace by at every occurrence to define the Hölder spaces also for compact subsets . Standard proofs yield that all the spaces we defined so far are actually Banach spaces; see, e.g., [22, 31].

Furthermore, let us introduce the collection of all functions on which are locally Hölder continuous (of order with exponent ). This set will be denoted by

where we simplified the notation by denoting the restrictions of functions from to compact subsets by again. Since the latter collection of functions does not perfectly fit for our purposes, in the sequel the following closely related (non-standard) function spaces will be used instead. Let denote an arbitrary but non-trivial family of compact subsets . Then for every the quantity

(5)

i.e., the distance of to the boundary of , is strictly positive. Thus, for each , all , and every , the space

is well-defined and it is easily verified that provides a semi-norm for this space. In our applications below will be the set of all closed balls (with center and radius ) such that the (open) ball is still contained in . Here denotes a constant which we assume to be given fixed in advance. Actually, it is not hard to see that the space is independent of . Consequently, we simply write for . Those spaces are then referred to as locally weighted Hölder spaces.

Remark 2.0.

Obviously, for every choice of the parameters, contains as a linear subspace, but it also contains functions whose local Hölder semi-norms grow to infinity as the distance of to the boundary tends to zero. However, this possible blow-up is controlled by the parameter . Moreover, in the Appendix we show that the intersection of with some Besov space is a Banach space with respect to the canonical norm; see Section 5. Finally, we want to mention that the spaces are monotone in , meaning that for . This can be seen by checking that for some universal constant (e.g., ), thus .

For the sake of completeness, we mention here that (as usual) the set of all infinitely often (strongly) differentiable functions with compact support in will be denoted by or . For its dual space we write . Once more, these definitions apply likewise when is replaced by some compact set .

2.2 Weakly differentiable functions: Sobolev spaces

Assume to be either itself, or some bounded domain. Given the Lebesgue spaces consist of all (equivalence classes of real-valued) measurable functions  on for which the (quasi-)norm

is finite.

Moreover, for and , let

denote the classical Sobolev spaces on , where are the weak partial derivatives of order . For fractional smoothness parameters (with and ) we extend the definition in the usual way by setting

where here the norm is given by and

denotes the common Sobolev semi-norm on .

Furthermore, for and , let us denote the closure of in the norm of by . Then we define to be the dual space of , where is determined by the relation .

For a detailed discussion of the scale of Banach spaces , , we refer to standard textbooks such as [1, 46] and the references given therein.

2.3 Generalized smoothness: Besov spaces

A more advanced way to measure the smoothness of functions is provided by the framework of Besov spaces which essentially generalizes the concept of Sobolev spaces introduced above. Besov spaces can be defined in various ways which (for a large range of the parameters involved) lead to equivalent descriptions; cf. [3, 9, 46, 47]. For our purposes the following approach based on iterated differences seems to be the most reasonable one, since it provides an entirely intrinsic definition when dealing with Lipschitz domains (i.e., domains which possess a Lipschitz boundary; cf. [47, Def. 1.103]). We refer, e.g., to [4, 14, 15, 16, 17].

In the following let be either itself, or some bounded Lipschitz domain. Moreover, let and . Then denotes the set of all such that the line segment belongs to . Moreover, for functions on the iterated difference of order with step size is recursively given by

for every . It is easily verified that

Those differences can be used to quantify smoothness: For and every let

denote the modulus of smoothness of order . It is well-known that monotonically as tends to zero and the faster this convergence the smoother is .

Now let with and . Then, for , the Besov space is defined as the collection of all for which the semi-norm

(6)

with is finite. Endowed with the canonical (quasi-)norm

these spaces turn out to be quasi-Banach spaces (and Banach spaces if ). Roughly speaking, with we can control all (weak) partial derivatives up to the order , measured in . Since the influence of the additional fine index is neglectable for many applications, we will mainly focus on the smoothness parameter , as well as on the integrability index , and simply set in what follows.

Remark 2.0.

Some comments are in order:

  • We note that different choices of in (6) lead to equivalent (quasi-)norms. The same is true when we restrict the range for in (6) to the interval .

  • The scale of Besov spaces as defined above is well-studied. In particular, sharp assertions on embeddings, interpolation and duality properties, characterizations in terms of various building blocks (e.g., atoms, local means, quarks, or wavelets) and best -term approximation results are known; see, e.g., [9, 14, 17, 27]. Many of them can also be shown using the Fourier analytic definition of as spaces of (restrictions of) tempered distributions [25, 46, 47]. It is known [20, 42, 47] that both definitions coincide in the sense of equivalent (quasi-)norms if

    (7)
  • The demarcation line for embeddings of Besov spaces into , , is given by

    (8)

    Every Besov space with smoothness and integrability indices corresponding to a point above that line is continuously embedded into (regardless of the fine index ). The points below this line never embed into . For spaces with that satisfy (8) some care is needed. However, if , then the embedding still holds. Observe that (8) exactly coincides with the adaptivity scale of Besov spaces we are interested in.

  • Besov spaces are closely related to Sobolev spaces. Indeed, it has been shown that for bounded Lipschitz domains , , and the space coincides with in the sense of equivalent norms; see, e.g., [17, Theorem 6.7]. Using the fact that for and arbitrary small we thus have

    for all and every .

  • For every bounded Lipschitz domain there exists a linear extension operator

    which is simultaneously bounded for all parameters that satisfy (7); cf. [40]. Moreover, is local in the sense that is contained in some bounded neighborhood of ; see [9].

2.4 Wavelet characterization of Besov spaces

Under suitable conditions on the parameters involved it is possible to characterize Besov spaces by means of wavelet decompositions [13, 28, 39, 47]. These characterizations are one of the most important ingredients of wavelet analysis. In particular, they provide the basis for several numerical applications such as preconditioning and the design of adaptive algorithms. We refer to [4, 5, 7] for details. Moreover, the resulting (quasi-)norm equivalences provide a powerful tool which allows to prove continuous embeddings such as the one stated in Theorem 3.1 in Section 3 below.

To start with, we recall some basic assertions related to expansions w.r.t. Daubechies wavelets. We essentially follow the lines of [8]: Let denote the univariate family of compactly supported Daubechies wavelets [12, 13]. We remind the reader that has  vanishing moments and the smoothness of these functions increases without bound as tends to infinity. So, let us fix an arbitrary value of and let denote the univariate scaling function which generates the wavelet . Furthermore, by we denote the non-zero vertices of the unit cube . Then, in dimension , the set

of (tensor product) functions generates (by shifts and dilates) an orthonormal wavelet basis for as follows: If

denotes the set of all dyadic intervals in , then the basis consists of all functions of the form

(9)

In view of our application below, we remark that there exists some open cube , centered at the origin with sides parallel to the coordinate axes, such that for all . Accordingly, all basis functions (9) satisfy , where

(10)

For every the system defined in (9) also forms an unconditional basis for . Hence, for those each possesses a wavelet expansion

(11)

which converges in .

For our purposes it is convenient to slightly modify this decomposition. Therefore let  be the closure of all finite linear combinations of integer shifts of in and let  denote the orthogonal projector which maps onto . Then, for every , the operator can be extended to a projector on and in (11) we can restrict ourselves to those for which

i.e., to wavelets corresponding to levels . Moreover, we shall renormalize our wavelets and set

such that does not depend on . Incorporating these conventions, from (11) we conclude that every , , can be expanded as

(12)

where satisfies .

Lemma 2.0.

Let , , and . Moreover, choose such that . Then a function belongs to the Besov space if and only if (12) holds with

(13)

Furthermore, (13) provides an equivalent (quasi-)norm for .

The proof of this assertion is quite standard. For the case of Banach spaces () it can be found, e.g., in [39]. For the quasi-Banach case we refer to [33]. Similar assertions can also be found in [47].

Remark 2.0.

We stress the point that due to every belongs to some , , such that (12) is well-defined; see Subsection 2.3(iii). Moreover, we can use the extension operator described in Subsection 2.3(v) to obtain similar norm equivalences for functions in , where is a bounded Lipschitz domain.

As mentioned already in the introduction, we are particularly interested in Besov spaces within the adaptivity scale of , , i.e., spaces with parameters that satisfy (8). Therefore, we specialize Subsection 2.4 for the corresponding spaces on :

Proposition 2.0.

Let , , as well as , and . Moreover, choose such that . Then a function belongs to the Besov space if and only if

with

(14)

and (14) provides an equivalent (quasi-)norm for .

Proof.

Observe that implies . Then the proof easily follows from Subsection 2.4. ∎

3 A general embedding

In this section we prove that, under some growth conditions on the local Hölder semi-norm, the intersection is continuously embedded into certain Besov spaces .

Theorem 3.1.

For with , let denote some bounded Lipschitz domain. Moreover, let and , as well as , , and . If we define

(15)

then for all

(16)

we have the continuous embedding

i.e., for all it holds

(17)

Let us briefly comment on Theorem 3.1 before we give its proof: From the theory of function spaces it is well-known that (standard) embeddings between Besov spaces, e.g.,

are valid only if the regularity of the target space is at most as large as the smoothness of the space we start from, i.e., only if . Theorem 3.1 now states that, under suitable assumptions on the parameters involved, exploiting the additional information on locally weighted Hölder regularity (encoded by the membership of in ) enables us to prove that functions from indeed possess a higher-order Besov regularity measured in the adaptivity scale corresponding to . Since almost equals the Sobolev space (cf. Subsection 2.3(iv)) this shows that approximating in an adaptive way is justified whenever defined by (15) is larger than . At this point we remark that is a continuous piecewise linear function of which decreases to zero when approaches its upper bound. Hence, in any case . Thus, for a fixed value of , the maximal regularity is achieved if is sufficiently large and is small enough.

The proof of Theorem 3.1 given below is inspired by ideas first given in [8]. Due to extension arguments in conjunction with the wavelet characterization of Besov spaces on (see Subsection 2.4) it suffices to find suitable estimates for the wavelet coefficients , , , which then imply (17). The contribution of (the relatively small number of) wavelets supported in the vicinity of the boundary of (boundary wavelets) can be bounded in terms of the norm of in . Here the restriction comes in. The coefficients corresponding to the remaining interior wavelets can be upper bounded by the semi-norm of in using a Whitney-type argument which then gives rise to the restriction . The detailed proof reads as follows:

Proof (of Theorem 3.1).

Step 1. Let . Since for it is and , every such can be extended to some ; see Subsection 2.3(v). In particular, such that it can be written as

Here the form a system of Daubechies wavelets (9), where is chosen such that and for some with ; see Subsection 2.4 for details. We restrict the latter expansion and consider only those wavelets for which belongs to

Therein denotes the ball (see (10)) concentrically expanded by the factor which we used to define the class ; cf. Subsection 2.1. Note that thus for all and . Next we split up the index sets once more and write

for every dyadic level . Note that, due to the boundedness of , there exists an absolute constant such that for all and . For example, we may take . Moreover, our assumption that is a bounded Lipschitz domain ensures that all remaining index sets satisfy at least . Finally, we note that all balls corresponding to with and strictly larger than are completely contained in . These considerations justify the disjoint splitting , where

correspond to the sets of boundary and interior wavelets at level , respectively. Observe that then , defined by

is an extension of as well, i.e., it satisfies . In Step 2–4 below we will show that for the adaptivity scale it holds

if (18)
if (19)
if (20)

Suppose we already know that those relations hold for all and that satisfy (16). Then we can extend the index set in (19) from to and the wavelet characterization of (cf. Subsection 2.4) together with the continuity of implies

(21)

which is finite due to our assumptions. Therefore, the special choice , in conjunction with (20) and (21), yields the desired estimate

This proves Theorem 3.1 since with particularly implies that , due to and the boundedness of . Hence, .

Step 2 (Estimate for ). To show the bound on the projection onto the coarse levels let and . We note that for all and , i.e., . Moreover, by definition, this equals which has compact support in  since is local; see Subsection 2.3(v). Subsection 2.4, i.e., the wavelet characterization of , therefore gives