Level-set inequalities on fractional maximal distribution functions and applications to regularity theory

Abstract

The aim of this paper is to establish an abstract theory based on the so-called fractional-maximal distribution functions (FMDs). From the basic ideas introduced in [1], we develop and prove some abstract results related to the level-set inequalities and norm-comparisons by using the language of such FMDs. Particularly interesting is the applicability of our approach that has been shown in regularity and Calderón-Zygmund type estimates. In this paper, due to our research experience, we will establish global regularity estimates for two types of general quasilinear problems (problems with divergence form and double obstacles), via fractional-maximal operators and FMDs. The range of applications of these abstract results is large. Apart from these two examples of the regularity theory for elliptic equations discussed, it is also promising to indicate further possible applications of our approach for other special topics.

Introduction

Before embarking on the main objective of this paper, we take into account a class of nonlinear elliptic equations of the type

$$\mathrm{div}(\mathbb{A}(x,\mathrm{\nabla }u))=\mathrm{div}(|\mathbf{F}{|}^{p-2}\mathbf{F}),\text{in}\mathrm{\Omega }$$

as an example and review some recent progresses that have been made in the last several years. As far as we know, researchers have long been interested in regularity theory for linear/nonlinear differential equations and despite plenty of works concerning this kind equations, the picture of regularity for their solutions is somehow incomplete. Let us briefly give a survey of classical and recent regularity results the related to (1.1), in which the main goal is to transfer the regularity of datum F to the gradient of solutions ∇u in the norm of some functional space $\mathbb{X}$. The gradient estimates for solutions to equations (1.1) can be written as

$${\Vert \mathrm{\nabla }u\Vert }_{\mathbb{X}}\le C{\Vert \mathbf{F}\Vert }_{\mathbb{X}},$$

or in terms of the local Calderón-Zygmund estimates

$${\Vert \mathrm{\nabla }u\Vert }_{\mathbb{X}(B_{R/2})}\le C{\Vert \mathbf{F}\Vert }_{\mathbb{X}(B_R)}+\mathtt{\text{lower-order terms on}}\mathrm{\nabla }u,$$

where $B_R$ is a open ball in ${\mathbb{R}}^n$ of radius R such that $B_R\subset \mathrm{\Omega }$. These types of estimates, in a sharp way, are important in studying many problems concerning nonlinear equations or systems.

Classical regularity results when considering p-Laplace equations with $\mathbf{F}\equiv 0$ have been obtained and proved by several authors by years. In a fundamental work by Uralt́zeva in [90], it is possible to obtain the $C^{1,\alpha }$ Hölder regularity for $p\ge 2$; and this result was later established for systems by Uhlenbeck [89]. Afterwards, there has been extensive research on the p-harmonic functions since the 1980s such as [32], [40], [60], [81] and many recent advances. The very first and well-known regularity result related to the p-Laplace equation $\mathrm{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)=\mathrm{div}(|\mathbf{F}{|}^{p-2}\mathbf{F})$, where the estimate (1.2) established by Iwaniec in [48] when $\mathbb{X}=L^q$, for all $p\le q<\infty$. The method of Iwaniec relies on the use of sharp maximal operators in Harmonic Analysis and priori estimates for solution to homogeneous equations $\mathrm{div}(\mathbb{A}(x_0,\mathrm{\nabla }u))=0$. Iwaniec's results were later improved and extended by DiBenedetto and Manfredi in [34] for the systematic elliptic equations when $\mathbb{X}=\text{BMO}$ for $p\ge 2$. During the past several years, there have been extensive studies and promising technical approaches on Calderón-Zygmund and regularity for quasi-linear elliptic equations (1.1). So far many progresses have been made and the literature on regularity theory has been further expanded. In 1993, Caffarelli and Peral in [24] proposed a very important approach to $L^p$ estimates for quasi-linear elliptic equation $\mathrm{div}(\mathbb{A}(x,\mathrm{\nabla }u))=0$. This approach relies on Calderón-Zygmund decomposition and the boundedness of Hardy-Littlewood maximal functions. It is valuable to further develop a roadmap for Calderón-Zygmund gradient estimates and regularity theory for more general class of nonlinear elliptic/parabolic equations over time.

In the past couple of years, inspired by this beautiful idea, there have been several attempts to study regularity theory for solutions to elliptic equations in form $\mathrm{div}(\mathbb{A}(x,\mathrm{\nabla }u))=\mathrm{div}(|\mathbf{F}{|}^{p-2}\mathbf{F})$, in which both local estimates (1.2) and (1.3) obtained in $\mathbb{X}=L^p$ and $W^{1,p}$ spaces, such as [9], [20], [67]. Regarding extendable regularity estimates for solutions to boundary value problems, it enables us to refer [13], [14], [15], [92]- the works by Byun and Wang; [10], [11], [16], [17], [65] - for further studies by others, under various weak assumptions on the boundary of the domain.

Besides, plenty of interesting approaches yield regularity results. It refers to seminal works by Iwaniec and Sbordone in [50] with a method of using Hodge decomposition theorem, Lewis in [61] with method based on the truncation of certain maximal operators. Or technique in [25], [56], is particularly useful to study regularity estimates for equations with VMO-coefficients. It is worth mentioning that classical ingredients mainly based on Calderón-Zygmund theory, Harmonic Analysis, interpolation inequalities, methods of freezing the coefficients, VMO coefficients via commutator theorem, etc.

It is worthy to emphasize that Acerbi and Mingione, in an impressive paper [1], generated a new idea to develop a local Calderón-Zygmund theory for degenerate parabolic systems. Therein the authors first proposed a technique with no use of maximal operators, Harmonic analysis free, the basic analysis is an application of DiBenedetto's intrinsic parabolic geometry estimates [33] (that is a scaling depending on solution itself) and Vitali's covering lemma. Although only dealing with the quasi-linear parabolic systems, the results for elliptic ones are also well understood (intrinsic cylinders are replaced by balls in the elliptic problems). This new approach opened the majority of intensive works for nonlinear elliptic and parabolic problems, which are still being mined for new and interesting results. There have been a large number of studies conducted, such as the terminology ‘large-M-inequality’ principle; geometrical approach by Byun and Wang in [14], [16] adapted to non-smooth domains; types of ‘good-λ bounds’ technique by many others [72], [73], [82], [84], [85], [86] working with balls instead of cubes. Several regularity results have been extensively treated in more general functional spaces $\mathbb{X}$: $L^p\text{log}L$, Lorentz, Morrey, Lorentz-Morrey spaces, or even Orlicz spaces, studied and addressed in a series of papers [5], [6], [21], [26], [41], [75], [76], [87] with related works.

Let us briefly describe the idea underlying this effective approach. We refer to the pioneering works in [1], [59], [67] for further reading. In order to obtain a local Calderón-Zygmund estimates (1.3), in $\mathbb{X}=L^q$ for example, starting with the integral of ∇u, understood in the sense of the Choquet integral as follows:

$$\int |\mathrm{\nabla }u{|}^q=q\underset{0}{\overset{\infty }{\int }}{\lambda }^{q-1}|\{|\mathrm{\nabla }u|>\lambda \}|d\lambda ,$$

and with change of variable yields

$$\int |\mathrm{\nabla }u{|}^q=M^{q}q\underset{0}{\overset{\infty }{\int }}{\lambda }^{q-1}|\{|\mathrm{\nabla }u|>M\lambda \}|d\lambda ,$$

for every $\lambda >0$ suitably large, where $M>0$ is an arbitrary constant (see for e.g. [3]). The key point is that, as in (1.3), we want to find a decay estimate for the level-sets $|\{|\mathrm{\nabla }u|>M\lambda \}|$ in terms of level-sets of the datum $|\{|\mathbf{F}|>\lambda \}|$. As an abstract idea, it states: if the following estimate

$$|\{|\mathrm{\nabla }u|>M\lambda \}|\le M^{-(p+\delta )}|\{|\mathrm{\nabla }u|>\lambda \}|+C_M|\{|\mathbf{F}|>\lambda \}|,$$

holds for some $\delta >0$, then the gradient of solutions $|\mathrm{\nabla }u|$ is controlled by the level-sets of data F. More precisely, for a large $M\gg 1$, the $L^q$ regularity estimate of (1.1) will be obtained for all $q<p+\delta$. Otherwise speaking, locally we have

$$\int |\mathrm{\nabla }u{|}^q\le M^{q-(p+\delta )}\int |\mathrm{\nabla }u{|}^q+C_M\int |\mathbf{F}{|}^q.$$

Here, $C_M$ is a positive constant depends only on M, and for simplicity, we denote the Lebesgue measure of a set $E\subset {\mathbb{R}}^n$ by $|E|$ or by ${\mathcal{L}}^n(E)$ later in our main work. As the reader will see, the proof of level-set inequality (1.4) is a key step to conclude local Calderón-Zygmund type estimates (1.3). Further, it can be seen that the idea expressed here is also valuable to obtain the level sets involving Hardy-Littlewood maximal or fractional maximal operator of ∇u in terms of the level sets of F, see, for e.g. [1], [59], [67], [70], [71] or [72], [82], [83], or more literature related to the subject.

To the best of the authors' knowledge, in general, from the example of level-set decay estimate (1.4), it enables us to state: Given two measurable functions $\mathcal{F},\mathcal{G}$, if there holds

$$|\{{\mathbf{M}}_{\alpha }\mathcal{G}>{\sigma }_{\epsilon }\lambda \}|\le \epsilon |\{{\mathbf{M}}_{\alpha }\mathcal{G}>\lambda \}|+C|\{{\mathbf{M}}_{\alpha }\mathcal{F}>{\kappa }_{\epsilon }\lambda \}|,$$

for any $\epsilon >0$ small enough and ${\sigma }_{\epsilon }$, ${\kappa }_{\epsilon }>0$, then the gradient estimate (1.2) can be obtained in terms of ${\mathbf{M}}_{\alpha }$ as

$${\Vert {\mathbf{M}}_{\alpha }\mathcal{G}\Vert }_{\mathbb{X}}\le C{\Vert {\mathbf{M}}_{\alpha }\mathcal{F}\Vert }_{\mathbb{X}}.$$

A question that arises pretty naturally here concerning some sufficient conditions for the level-set inequality (1.4) or likewise to be valid. What are the main tools behind the proof of (1.5) or how it turns out the idea to construct conditions for $\mathcal{F}$ and $\mathcal{G}$ to sharply achieve (1.5)-type inequality? The primary goal of this paper is to answer these questions. With this as motivation, in this study we discuss on some key ingredients for the proof of such type of level-set estimates.

Inspired by the ideas coming from aforementioned example, if one can find two functions φ and ψ such that: φ belongs to a so-called reverse Hölder class; and function ψ is able to be controlled by $\epsilon \mathcal{G}$, for all $\epsilon >0$, via a local integral estimate (see (2.4) and ingredient (A2) in Section 2 below). To better understand these key ingredients, let us turn back to the abstract theory of nonlinear elliptic equations (1.1). Here, a version of Gehring's lemma is applied to improve the degree of gradient integrability of weak solutions v to homogeneous equations of type

$$\mathrm{div}(\mathbb{A}(x,\mathrm{\nabla }v))=0,\text{in}\mathcal{B}\text{and}v=u,\text{on}\partial \mathcal{B},$$

whenever $\mathcal{B}$ is an open ball in Ω, see [43] and later many different versions have been established (see, for e.g., [44, Theorem 6.7], [49]). As a result of Gehring's lemma, the self-improving property of a well-known inequality, called reverse Hölder integral inequality with increasing supports: if v is the unique solution to reference problem (1.6), then there exists a number $\gamma >1$ depending on n, p and the structure of operator $\mathbb{A}$ such that the following inequality holds

$${(\underset{B_{\rho }}{⨏}|\mathrm{\nabla }v{|}^{\gamma p}dx)}^{\frac{1}{\gamma p}}\le C{(\underset{B_{2\rho }}{⨏}|\mathrm{\nabla }v{|}^{p}dx)}^{\frac{1}{p}},$$

for all $B_{2\rho }\subset \mathcal{B}$. As we shall see, the function φ here plays a role of ∇v. On the other hand, function ψ is in fact the difference between gradients of solutions to equations (1.1) and (1.6), also known as the comparison estimates, must be established in most of research papers. In the context of regularity estimates above-described, these technical ingredients are helpful to recover integrability information of solutions from data, as in (1.4). It is worth noting that in order to measure integrability properties of solutions, the use of level-set decays of various fractional maximal operators has been mentioned the first time by Mingione in [67]. And one of our motivations for this study also stemmed from this celebrated paper.

In accordance with the questions arising before, the discussion leads us to another interesting tool for abstract results. From another point of view, level-set inequality (1.5) might actually work on the idea of fractional-maximal distribution functions (FMD) (we refer to Section 3.1 below for the definition), more precisely as

$$d_{\mathcal{G}}^{\alpha }(\mathcal{B},{\sigma }_{\epsilon }\lambda )\le \epsilon d_{\mathcal{G}}^{\alpha }(\mathcal{B},\lambda )+C{d}_{\mathcal{F}}^{\alpha }(\mathcal{B},{\kappa }_{\epsilon }\lambda ).$$

The construction of such appropriate tool can provide new insights of the technical approach when introducing or discussing on regularity theory and its applications. For the sake of clarity and completeness, this level-set type (1.7) will be explained in Section 3.

The aim of this paper is two-fold. First we discuss the basic ingredients to formulate the level-set estimates in terms of fractional maximal functions. Specifically, by using the language of such fractional-maximal distribution functions, we provide a newer landmark for the ‘good-λ’ type bounds technique, that has important theoretical implications in regularity and Calderón-Zygmund type estimates. On a different direction, researching the regularity theory of nonlinear elliptic equations, that is also linked to the double obstacle problems become a new trend in nonlinear PDEs. The second purpose of this paper is an application of the abstract setting for this technique. We shall prove some global regularity estimates for nonlinear elliptic problems. In particular, there are two separate issues discussed here. On the one hand, we develop the level-set decay estimate (1.4) (in terms of fractional maximal operators ${\mathbf{M}}_{\alpha }$) to establish the global regularity estimates for a wide class of nonhomogeneous quasi-linear elliptic equations as follows

$$\mathrm{div}(\mathbb{A}(x,\mathrm{\nabla }u))=\mathrm{div}(\mathbb{B}(x,\mathbf{F}))\text{in}\mathrm{\Omega },u=\mathsf{g}\text{on}\partial \mathrm{\Omega },$$

where $\mathbf{F}\in L^p(\mathrm{\Omega };{\mathbb{R}}^n)$ with boundary data $\mathsf{g}\in W^{1,p}(\mathrm{\Omega })$ for $p\in (1,n]$. This form of equations appears naturally in many engineering or science problems. Here, we focus our attention on the appearance of the degeneracy parameter $\varsigma \in [0,1]$ in the standard assumptions of $\mathbb{A}$ (growth and ellipticity conditions, see Section 5). Apart from the regularity results described in example above, there have been remarkable contributions pertaining to regularity theory for degenerate problems with $\varsigma =0$, see [5], [6], [13], [35], [36], [59], [67], [72], [73], [82] and many extensive literature so far. In this journey, we confine ourselves with regularity estimates for $\varsigma \ge 0$. On the other hand, as the second application, we want to apply the proposed technique to nonlinear elliptic double obstacle problems, where the solutions are constrained to lie between two fixed obstacle functions: $f_1\le u\le f_2$ (see Section 5 below, for detailed description). This constrained variational problem is an interesting topic that has a wide range of applications in elasto-plasticity, mathematical finance, optimal control problem, groundwater hydrology, the study of a soap film, equilibrium of an elastic membrane, transactions costs and other sciences (see reference books in [42], [53], [78], [88] for further mathematical problems and applications). Significant progress has been made for one-sided obstacle problems in [7], [8], [18], [19], [22], [27], [37], [38], [39], [66] and many references given therein. However, there seems not too much works on the double obstacle case, even though it also arises in many applications. In this paper, along with the works [28], [30], [52], [63], [79], and somewhat extends the results in [12], we prove the global gradient estimates of solutions to double obstacle problems by our technical argument, via the theory of FMD.

One of the new aspects of our work is that we deal with fractional maximal operators. Together with Hardy-Littlewood maximal operators, this is one of important variants in analysis and PDEs to study differentiability properties of functions. Fractional maximal operator, usually denoted by ${\mathbf{M}}_{\alpha }$, whose definition will be essentially given in the next section, is useful tool to acquire gradient estimates for solutions for a large class of quasi-linear elliptic/parabolic equations (see [35], [36], [58], [59] and many research papers so far). In this study, we employ ${\mathbf{M}}_{\alpha }$ to take advantage of the efficiency of the proposed technique. To be more precise, gradient estimates of solutions to the general problems (P) are preserved under fractional maximal operators.

Why fractional maximal operators come into play? - In [2], ${\mathbf{M}}_{\alpha }$ has a connection to the Riesz potential ${\mathbf{I}}_{\alpha }$ (fractional integral operator) in the following point-wise inequality:

$${\mathbf{M}}_{\alpha }f(x)\lesssim {\mathbf{I}}_{\alpha }f(x),\text{for every}x\in {\mathbb{R}}^n,$$

and additionally, the fractional maximal function ${\mathbf{M}}_{\alpha }f$ and Riesz potential ${\mathbf{I}}_{\alpha }f$ are often comparable in norm, [69]. It is observed that fractional maximal operator and the Riesz potential are connected via relation (1.8), allowing both size and oscillations of solutions and their derivatives, including ‘fractional derivatives’ ${\partial }^{\alpha }u$ to be controlled (see [59]). Henceforth, it enables us to exploit the ${\mathbf{M}}_{\alpha }$ to transfer the level-set information from given data F to ∇u.

One more to emphasize in this study, global regularity results in Section 5 will be obtained in the setting of Lorentz and Orlicz spaces, respectively. Moreover, an extra attempt has also been made to study results in the Orlicz-Lorentz setting (a generalization of Orlicz and Lorentz spaces - see [51], [68]) in Section 5 below. As a natural way analogous to this work, we plan to combine these results and proposed technique in order to study the boundary value problems for nonlinear parabolic equations/systems, in a forthcoming paper.

Going far beyond, this work has significance for its genre. By rephrasing idea from the references already given (including the relevant contributions and ours), in view of the generalizations to ‘good-λ’ level-set inequalities, this paper develops a general and robust approach to build on the higher regularity via the use of FMD. Apart from being an interesting in its own, this work reveals a wider perspective of such technique in modern analysis. This paper gives a flavor to the reader of the essence behind the proof of Calderón-Zygmund-type estimates, which attracts a number of interesting works during last decades. We call the attention of the reader for enlightening paper [1] and further papers related to this approach. Being a contribution to the study of regularity theory for nonlinear elliptic/parabolic problems, we believe that this paper can provide an inviting reading on the topic, especially to newcomers.

Let us now state our main results which will be summarized into two following theorems. In Theorem A, we discuss some sufficient conditions for the validity of FMD inequalities. In general, these conditions can be represented by the key ingredients in our statements (see Section 2 for details). Next, arising from what obtained in Theorem A, Theorem B enables us to obtain the norm-comparisons in the setting of several spaces, such as: Lorentz spaces, Orlicz spaces and Orlicz-Lorentz spaces. However, it is a remarkable fact that the proofs of Theorem A, Theorem B above are splitted into separate parts, to be convenient to the readers. These new abstract results in this paper allow the application of any type of regularity theory (Calderón-Zygmund estimates) for partial differential equations. As already said, we here only deal with two applications: for a general nonhomogeneous quasilinear elliptic equations and for quasilinear elliptic double obstacle problems.

Theorem A

Let $\gamma >1$ and two functions $\mathcal{F}$, $\mathcal{G}\in L^1(\mathit{\Omega };{\mathbb{R}}^{+})$ satisfy the global comparison in (A3).

• If $\mathcal{F}$, $\mathcal{G}$ satisfy the local comparison $(\mathbf{A}{\mathbf{2}}_{\mathbf{1}})$ then for every $\alpha \in [0,\frac{n}{\gamma })$ there exists ${\epsilon }_0\in (0,1)$ such that the following fractional-maximal distribution inequality

  $$d_{\mathcal{G}}^{\alpha }(\mathit{\Omega };\sigma \lambda )\le C\epsilon d_{\mathcal{G}}^{\alpha }(\mathit{\Omega };\lambda )+d_{\mathcal{F}}^{\alpha }(\mathit{\Omega };\kappa \lambda ),$$

  holds for all $\lambda >0$ and $\epsilon \in (0,{\epsilon }_0)$, with $\sigma ={\epsilon }^{-\frac{n-\alpha \gamma }{n\gamma }}$ and $\kappa =\epsilon c_{\epsilon }^{-1}$.

• If $\mathcal{F}$, $\mathcal{G}$ satisfy the local comparison $(\mathbf{A}{\mathbf{2}}_{\mathbf{2}})$ then there exists ${\sigma }_0={\sigma }_0(n,\tilde{c})>0$ such that the fractional-maximal distribution inequality (1.9) holds for all $\lambda >0$ and $\epsilon \in (0,1)$ and for some $\kappa \in (0,\epsilon )$.

Theorem B

Let $\gamma >1$ and two functions $\mathcal{F}$, $\mathcal{G}\in L^1(\mathit{\Omega };{\mathbb{R}}^{+})$ satisfy the global one (A3).

• If $\mathcal{F}$, $\mathcal{G}$ satisfy the local comparison $(\mathbf{A}{\mathbf{2}}_{\mathbf{1}})$ then for every $\alpha \in [0,\frac{n}{\gamma })$, $0<q<\frac{n\gamma }{n-\alpha \gamma }$ and $0<s\le \infty$ there exists a constant $C>0$ such that

  $${\Vert {\mathbf{M}}_{\alpha }\mathcal{G}\Vert }_{L^{q,s}(\mathit{\Omega })}\le C{\Vert {\mathbf{M}}_{\alpha }\mathcal{F}\Vert }_{L^{q,s}(\mathit{\Omega })}.$$

  Moreover, given Young function $\mathit{\Phi }\in {\mathit{\Delta }}_2$ then there exists $\tilde{q}>0$ such that the following estimate

  $${\Vert {\mathbf{M}}_{\alpha }\mathcal{G}\Vert }_{L^{\mathit{\Phi }}(q,s)(\mathit{\Omega })}\le C{\Vert {\mathbf{M}}_{\alpha }\mathcal{F}\Vert }_{L^{\mathit{\Phi }}(q,s)(\mathit{\Omega })},$$

  holds for every $\alpha \in [0,\frac{n}{\gamma })$, $0<q<\tilde{q}$ and $0<s\le \infty$.

• If $\mathcal{F}$, $\mathcal{G}$ satisfy the local comparison $(\mathbf{A}{\mathbf{2}}_{\mathbf{2}})$ then both inequalities (1.10) and (1.11) even hold for all $\alpha \in [0,n)$, $0<q<\infty$ and $0<s\le \infty$.

We conclude this section by outlining the content of this paper. In Section 2, we introduce some general notation and basic definitions that will be used throughout the paper. Furthermore, this section is also dedicated to discuss on some crucial ingredients emerged in our approach. Section 3 will establish level-set inequalities by specifying via FMDs; then the definitions of considered functional spaces can be reformulated in terms of such distribution functions. We also state our chief result in Section 3. The next section 4 brings these FMD inequalities back to the norm inequalities. We also state and prove some abstract results related to comparisons for different functional spaces (Lorentz, Orlicz and the Orlicz-Lorentz spaces). At the end, we will present two applications where our results take place. For a wider understanding, Section 5 will detail the global gradient estimates for a general class of quasi-linear elliptic equations via fractional maximal operators based on the idea of FMD established in Section 3 and 4; and further the global regularity results are also driven with elliptic double obstacle problems, thus providing the complete picture for its applications.