Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity

In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use standard methods including De Giorgi's truncation and freezing coefficient methods.

For the -Poisson equations, that is, in the case = 0, Hölder continuity of a gradient was already established by many experts ( [15], [20], [31], [32], [40], [43], [44]). These regularity results should not hold for the one-Poisson equations, even if is sufficiently smooth. In fact, in the one-dimensional case, a non-smooth function = ( 1 ) may solve −Δ 1 = 0, as long as it is absolutely continuous and non-decreasing. Hence, it seems impossible to show continuous differentiability of weak solutions, even for the one-Laplace equation. This problem is due to the fact that the ellipticity of one-Laplacian degenerates in the direction ∇ , which is substantially different from -Laplacian. On the other hand, in the multi-dimensional case, the ellipticity of one-Laplacian is non-degenerate in directions that are orthogonal to ∇ . It should be mentioned that this ellipticity becomes singular in a facet, the place where a gradient vanishes. By anisotropic diffusivity of one-Laplace operator Δ 1 , which consists of both degenerate and singular ellipticity, Δ 1 seems, unlike Δ , impossible to deal with in existing elliptic regularity theory.
For this reason, it will be interesting to consider a question whether a solution to (1.1), where , contains both Δ 1 and Δ , is continuously differentiable ( 1 ). In the special case where the solution is convex, this 1 -regularity problem is already answered affirmatively [24,Theorem 1]. The aim of this paper is to establish continuity of derivatives without the convexity assumption.
The equation (1.1) is concerned with a minimizing problem of the energy functional Here, as the principal space, we often choose a closed affine space 0 + 1, 0 (Ω) for some fixed 0 ∈ 1, (Ω). Formally considering the Euler-Lagrange equation corresponding to this energy minimizing problem, we can obtain (1.1). However, since the mapping R ∋ ↦ → | | ∈ [0, ∞) is no longer differentiable at the origin, the term ∇ /|∇ | should be treated not in the classical sense on a place {∇ = 0}, often called a facet of . This problem can be overcome by introducing a subdifferenital, which is often useful for analysis of non-smooth convex functions. Also, since a solution is assumed to be in the first order Sobolev space, we should treat a weak solution in a distributional sense. Our definition of a weak solution is described in Section 1.3.
The energy density can be found in fields of materials science and fluid mechanics. We would like to mention some mathematical models for each field briefly.
A typical example of the former is the relaxation dynamics of a crystal surface below the roughening temperature, which was modeled by Spohn in [39]. There, evolution of = ( , ), a height function of the crystal in a two-dimensional domain Ω, is modeled to satisfy = −Δ and = − Φ from a thermodynamic viewpoint (see [29] and the references therein). Here is often called the chemical potential, and Φ denotes the crystal surface energy, given by Finally, evolution of a crystal surface results in a fourth order parabolic equation = Δ , 3 with = 1/(3 ) > 0. When is stationary, this model becomes a second order elliptic equation , 3 = .
Here is sufficiently smooth since it is harmonic. For the latter model, the density often appears with ( , ) = (2, 2) when modeling an incompressible laminar flow of a Bingham fluid in a cylinder Ω × R ⊂ R 3 . To be precise, we assume that the velocity field is of the form = (0, 0, ( 1 , 2 )), and hence the incompressible condition div = 0 clearly holds. It is also assumed that its speed is sufficiently slow enough to discard convection effects. Under these settings, the Euler-Lagrange equation for a Bingham fluid is given by Here and are positive constants, and = ( 3 ) denotes the pressure function, whose slope becomes constant under the laminar setting. Mathematical formulations on motion of Bingham fluids are found in [18, Chapter VI], [26]. Bingham fluids have two different aspects of plasticity and viscosity, which are respectively reflected by the diffusion operators Δ 1 and Δ 2 = Δ in the equation (1.1). It is worth mentioning that when = 0, this problem models motion of incompressible Newtonian fluids. The density also arises in an optimal transport problem for a congested traffic dynamics ( [11], [13]). There the optimal traffic flow is connected with a certain very degenerate elliptic problem, and higher regularity results related to this have been well-established. Our 1 -regularity result is inspired by a recent work by Bögelein-Duzaar-Giova-Passarelli di Napoli [10], which is also concerned with continuity of gradients on the very degenerate problem related to the optimal traffic flow. For further explanations and comparisons, see Section 1.2.
More generally, we will also consider where the partial differential operators L 1 and L generalize −Δ 1 and −Δ respectively. The detailed conditions of L 1 , L will be described later in Section 1.3.

Our main results and strategy
For simplicity, we first consider the model case (1.1). Our result is Theorem 1.1. Let be a weak solution to (1.1) with ∈ (Ω) for some ∈ ( , ∞]. Then, is in 1 (Ω).
By Giga and the author [24], this type of 1 -regularity result was already established in the special case where is convex. The proof therein is substantially based on convex analysis and a strong maximum principle. In particular, the gradient ∇ is often considered not only as a distribution, but also as a subgradient, which made it successful to show 1 -regularity rather elementarily.
After that work was completed, the author has found it possible to prove continuity of derivatives of non-convex weak solutions. Instead of the strong maximum principle, we use standard methods from the Schauder theory and the De Giorgi-Nash-Moser theory to justify continuity of a gradient ∇ when it is truncated near a facet. Precisely speaking, we would like to show that for each small ∈ (0, 1), the truncated gradient is locally Hölder continuous. This result can be explained by computing eigenvalues of the Hessian matrix ∇ 2 (∇ ) = ( (∇ )) , . In fact, ∇ 2 satisfies (ellipticity ratio of ∇ 2 ( 0 )) = (the largest eigenvalue of ∇ 2 ( 0 )) (the lowest eigenvalue of ∇ 2 ( 0 )) ≤ ( ) 1 + | 0 | 1− for all 0 ∈ R \ {0}. In particular, for each fixed > 0, the operator , can be said to be uniformly elliptic over a place {|∇ | > }. It should be emphasized that G (∇ ) is supported in this place, and therefore this vector field will be Hölder continuous. Since it is easy to check that the continuous vector field G (∇ ) uniformly converges to ∇ , this implies that ∇ is also continuous. We should note that our analysis for G (∇ ) substantially depends on the truncation parameter > 0. Thus, our Hölder continuity estimates may blow up as tends to 0. In particular, on quantitative growth estimates of a gradient near a facet, less is known. This problem is essentially because that the equation (1.1) becomes non-uniformly elliptic near a facet. It should be emphasized that our problem substantially differs from what is often called the ( , )-growth problem, where non-uniform ellipticity appears as a gradient blows up [34], [35].
The proof of Hölder continuity of G 2 (∇ ), where we have replaced by 2 , broadly consists of two parts; relaxing the singular operator , , and establishing a priori Hölder estimates on relaxed vector fields. For relaxations, we will have to consider relaxed problems that are uniformly elliptic. In the special case (1.1), for an approximating parameter ∈ (0, 1), the regularized equation is given by , = . (1.4) Here converges to in a suitable sense, and the relaxed operator , is given by for each ∈ (0, 1). This operator naturally appears when one approximates the density by In particular, we regularize | | by 2 + | | 2 /2 for each ∈ { 1, }, which allows for linearization. It should be noted that for each fixed ∈ (0, 1), the equation (1.4) is uniformly elliptic, in the sense that there holds for all 0 ∈ R . Moreover, sufficient smoothness for coefficients of the operator , is also guaranteed. Therefore, from existing results on elliptic regularity theory, it is not restrictive to assume that a weak solution to (1.4) admits higher regularity ∈ 1, ∞ loc ∩ 2, 2 loc . Moreover, when the external force term is in ∞ , it is possible to let be even in ∞ with the aid of bootstrap arguments [30,Chapter V]. For each fixed ∈ (0, 1), we would like to establish an a priori Hölder continuity of a truncated gradient whose estimate is independent of the approximation parameter that is sufficiently smaller than . Since the singular operator , itself has to be relaxed by a non-degenerate one , , the truncating mapping G 2 should also be replaced by another regularized one G 2 , . For this reason, we have to consider another vector field given by (1.5), which clearly differs from (1.3). The a priori Hölder estimate, which plays as a key lemma in this paper, enables us to apply the Arzelà-Ascoli theorem and to conclude that G 2 (∇ ) is also Hölder continuous. We will obtain the desired Hölder continuity estimates by standard methods, including De Giorgi's truncation and a freezing coefficient argument. Our proofs of a priori Hölder continuity estimates are highly inspired by [10, §4-7]. For a general problem (1.2), we need to establish suitable approximation of the singular operator L. There, for an approximation parameter ∈ (0, 1), we have to introduce a uniformly elliptic operator L that is neither degenerate nor singular. A main problem herein is that we have to justify that the regularized solution converges to the original solution in the distributional sense. In the special case L = , , L = , , and ≡ , this justification is already discussed in [41,Lemma 4], which is based on arguments from [33,Theorem 6.1]. In this paper, we would like to give a more general approximating operator L that is based on the convolution by the Friedrichs mollifier. The novelty of our new approximation argument is that this works for a very singular operator L 1 that differs from Δ 1 . In particular, we are able to generalize the total variation energy functional in F, as long as its corresponding density 1 is convex and positively one-homogeneous. Our setting concerning 1 will be substantially different from previous works [5], [6], [8], [27], which are concerned with energy minimizers of purely linear growth, since ellipticity conditions therein appear to differ from ours.
It is worth mentioning that our strategy works even in the system case, and 1 -regularity on vectorvalued problems has been established in another recent paper [42] by the author. It should be emphasized that when one considers everywhere regularity for the system problem, it would be natural to let the divergence operator L have a symmetric structure, often called the Uhlenbeck structure (see [1], [9], [43]). In particular, when it comes to everywhere 1 -regularity for vector-valued solutions, assumptions related to the Uhlenbeck structure may force us to restrict L 1 = Δ 1 (see Section 2.4 for details). In absence of this assumption, it is well-known that energy minimizers has partial regularity, and often lacks everywhere regularity. Both classical partial regularity results and counterexamples to full regularity are found in [3,Chapter 4], [25,Chapter 9]. On partial regularity of local minimizers in BV spaces, both -growth problems and purely linear growth problems are well-discussed in [5], [6], [27], [38]. Although it will be interesting to consider partial regularity on non-Uhlenbeck-type problems instead of full regularity, we will not discuss this problem in the paper.

Comparisons to previous results on a very degenerate equation
Our proofs on continuity of derivatives are inspired by [10], which is concerned with everywhere regularity for gradients of solutions to very degenerate equations, or even systems, of the form Here ′ denotes the Hölder conjugate of ∈ (1, ∞), i.e., ′ = /( − 1). This equation is motivated by a mathematical model on congested traffic dynamics [13]. There this model results in a minimizing problem of the form opt ∈ arg min (1.7) The equation (1.6) is connected with the minimizing problem (1.7). In fact, in a paper [11], it is proved that the optimal traffic flow opt of the problem (1.7) is uniquely given by ∇ * (∇ ), where solves (1.6) under a Neumann boundary condition. Also, it is worth mentioning that (1.6) can be written by − div(∇ * (∇ )) = , where The question whether the flow opt = ∇ * (∇ ) is continuous has been resolved affirmatively. There it has been to establish continuity of the vector filed G + (∇ ) with > 0. This expectation on continuity can be explained, similarly to (1.1), by calculating the ellipticity ratio of the Hessian matrix ∇ 2 * (∇ ). In fact, for every 0 ∈ R with | 0 | ≥ + , there holds (ellipticity ratio of ∇ 2 * ( 0 )) = (the largest eigenvalue of ∇ 2 * ( 0 )) (the lowest eigenvalue of ∇ 2 * ( 0 )) ≤ ( − 1) 1 + −1 as → 0. This estimate suggests that for each fixed > 0, the truncated gradient G + (∇ ) should be Hölder continuous. We note that it is possible to show G + (∇ ) uniformly converges to G (∇ ) as → 0, and thus G (∇ ) will be also continuous.
When is a scalar function, this expectation on continuity of G + (∇ ) with > 0 was first mathematically justified by Santambrogio-Vespri [37] in 2010 for the special case = 2 with = 1. The proof therein is based on oscillation estimates related to Dirichlet energy. For technical reasons related to this method, the condition = 2 is essentially required. Later in 2014,  established a more general proof that works for any ≥ 2 and any function * whose zero-levelset is sufficiently large enough to define a real-valued Minkowski gauge. This Minkowski gauge plays as a basic modulus to estimate ellipticity ratios on equations they treated. It should be mentioned that their strategy will not work for our problem (1.1), since some structures of the density functions therein seems substantially different from ours. In fact, in our problem, the zero-levelset of is only a singleton, and thus it is impossible to define the Minkowski gauge as a real-valued function. The recent work by Bögelein-Duzaar-Giova-Passarelli di Napoli [10] also focuses on proving Hölder continuity of G + (∇ ) for each > 0 with = 1. A novelty of this paper is that their strategy works even when is a vector-valued function. There they considered an approximating problem of the form The paper [10] establishes a priori Hölder continuity of G 1+2 (∇ ) for each fixed ∈ (0, 1), whose estimate is independent of an approximation parameter ∈ (0, 1). Our proofs on a priori Hölder estimates are highly inspired by [10, §4-7]. There are three significant differences between their proofs and ours. The first is that we need to treat another relaxed vector field G 2 , (∇ ) as in (1.5), different from G 2 (∇ ). This is essentially because our regularized problems as in (1.4) are based on relaxing the very singular operator , (or the general operator L) itself, whereas their approximation given in (1.8) does not change principal part at all. Hence, in the relaxed problem (1.8), it is not necessary to adapt another mapping than G 2 , . The second is some structural differences between density functions and * , which make our energy estimates in Section 3.5 different from [10, §4-5]. The third is that we have to make the approximation parameter sufficiently smaller than the truncation parameter . To be precise, when we prove a priori Hölder continuity of G 2 , (∇ ), we carefully utilize the setting 0 < < /8. This condition will be required especially when two different moduli |∇ | and 2 + |∇ | 2 are both used. In the paper [10], no restrictions on ∈ (0, 1) are made, since their analysis for (1.8) is based on a single modulus |∇ | only.

Some notations and our general results
In Section 1.3, we describe our settings on the equation (1.2) and describe the main result. Before stating our general result, we fix some notations.
We give the definition of weak solutions to (1.2). Here 1 denotes the subdifferential of 1 , defined by It should be noted that (1.15) enables us to apply the compact embedding 1,  [24,Theorem 4], where some tools from convex analysis are basically used, works under weaker assumptions that 1 , ∈ 2 (R \ {0}). Local 2, 0regularity of = 1 + outside the origin is used for technical reasons to make our perturbation arguments successful (see Remark 3.14 in Section 3.6).

Outlines of the paper
We conclude Section 1 by briefly describing outlines of this paper.
Section 2 aims to give the proof of Theorem 1.3 by approximation arguments. For preliminaries, in Section 2.1, we give basic estimates for gradients and Hessian matrices of relaxed density functions. After stating some basic facts on the density 1 in Section 2.2, we would like to check that the relaxation of the density = 1 + convoluted by the Friedrichs mollifier satisfies good properties in Sections 2.3-2.4. Next in Sections 2.5-2.6, we consider an approximating problem for the equation (1.2). There we prove a convergence result on variational inequality problems (Proposition 2.11). As a corollary, we are able to deduce that a solution to an approximated equation surely converges to a weak solution to (1.2), under suitable Dirichlet boundary conditions (Corollary 2.12). Finally in Section 2.7, we would like to prove Theorem 1.3. There Theorem 2.14 is used without proofs, which states that a truncated gradient of an approximated solution, given by (1.5), is locally Hölder continuous, uniformly for an approximation parameter ∈ (0, /8).
Section 3 is focused on giving the proof of Theorem 2.14. First in Section 3.1, we briefly describe inner regularity on generalized solutions, including a priori Lipschitz bounds (Proposition 3.1). In Section 3.2, we would like to give the proof of Theorem 1.3. There we use key Propositions 3.2-3.3, the proofs of which are given after deducing a weak formulation in Section 3.3. Throughout Sections 3.4-3.6, we would like to justify that the vector field G 2 , (∇ ) satisfies a Campanato-type growth estimate under suitable conditions where a freezing coefficient method works (Proposition 3.2). In Section 3.7, we show a De Giorgi-type lemma on the scalar-valued function |G , (∇ )| (Proposition 3.3). Some lemmata in Section 3.7 are easy to deduce, appealing to standard arguments given in [3,Chapter 3], [23,Chapter 5] and [17,Chapter 10,[4][5]. For the reader's convenience, we provide detailed computations in the appendix (Section 4).
Finally, we would like to mention that our result is based on adaptations of [42], the author's recent work that focuses on system problems. This paper also aims to provide full computations and proofs of some estimates that are omitted in [42]. Compared with [42], some computations herein might become somewhat complicated, since the density function = 1 + is not assumed to be symmetric in scalar problems.

Approximation schemes and the proof of Theorem 1.3
The aim of Section 2 is to provide suitable approximation schemes for the equation (1.2). This approximation is necessary since the ellipticity ratio of (1.2) blows up near the facets, which makes it difficult to show local Sobolev regularity on second order derivatives. Therefore, we have to introduce relaxed equations whose ellipticity ratio is bounded even across the facets. When considering the operators L = L 1 + L , we have to give a suitable relaxation of the anisotropic operator L 1 . This is possible by convolving the density function 1 by the Friedrichs mollifier. The novelty concerning our relaxation is that this works as long as 1 is positively one-homogeneous.

Preliminaries
First in Section 2.1, we prove quantitative estimates related to gradients or Hessian matrices for preliminaries. The results herein will play an important role in Sections 2-3.
First, we would like to prove Lemma 2.1 below.
Secondly, we would like to show that some growth estimates given in Lemma 2.1 will be kept under the convolution by mollifiers (Lemma 2.2). Also, we will check that ellipticity or monotonicity estimates can also be obtained (Lemma 2.3).
Finally, we conclude Section 2.1 by deducing continuity estimates of truncating mappings. For Similarly, for 0 < < < 1, we set Lemma 2.4 belows states that the mapping G 2 , has Lipschitz continuity, uniformly for sufficiently small > 0. In the limiting case = 0, Lipschitz continuity of G is found in [10, Lemma 2.3]. Following the arguments therein, we would like to prove Lemma 2.4 by elementary computations. Before the proof, we recall an inequality which is easy to deduce by the triangle inequality.
In the remaining case, without loss of generality we may assume that which completes the proof.

Basic facts of positively one-homogeneous convex functions
In Section 2.2, we mention some basic properties of the positively one-homogeneous convex function 1 without proofs. All of the proofs are already given in [4, Section 1.3], [36, §13], [24,Sections 6.1 & A.3]. Also, we briefly give some estimates that immediately follows from (1.9) and Lemma 2.1.
We define a closed convex set 1 ⊂ R by often called the support function of the convex set 1 . The inequality˜ 1 ≥ 0 is clear by the inclusion 0 ∈ 1 . By (1.9) and the definition of˜ 1 , it is easy to check the Cauchy-Schwarz-type inequality: Also, under the condition (1.9), it is well-known that 1 is given by (2.28) In particular, is often called Euler's identity. We briefly give some estimates related to derivatives of 1 outside the origin. First, when 1 ∈ 1 (R \ {0}), it is easy to check by (1.9). In particular, we have ∇ 1 ∈ ∞ (R ) and Moreover, from Euler's identity (2.29), it follows that Finally, it is noted that when 1 ∈ 2, 0 loc (R \ {0}) with 0 ∈ (0, 1], we are able to define a constant Then, by (2.33), it is easy to check that for all ∈ (0, ∞) and 1 , 2 ∈ R \ {0} enjoying (1.13).

Approximation based on mollifiers
Section 2.3 is focused on the approximation of = 1 + , which is based on the convolution by the Friedrichs mollifier. We would like to check that the relaxed density = * satisfies some quantitative estimates.
It will be worth mentioning that under these settings, the inequality (2.57) in Lemma 2.6 can be deduced rather easily with 0 = 1. This is possible by applying the following Lemma 2.7 with = , ( ∈ { 1, }). There it should be noted that these relaxed densities satisfy the assumption (2.64) by (2.60)-(2.61).
Finally, when considering the special case where the operators are of the form where satisfies (2.58), we may introduce relaxed operators alternatively by We note that all of the proofs after Section 2.4 work for this different approximation scheme. Also, it should be worth mentioning that, under this setting, everywhere 1 -regularity of weak solutions for elliptic system problems have been established in another recent paper [42] by the author. There our method works under the assumption ∈ 2, 0 loc ((0, ∞)) with 0 ∈ (0, 1].

A fundamental result on convergences of vector fields
In this section, we prove Lemma 2.8 below, which plays important roles in a mathematical justification of convergence for approximated solutions. (1). For each fixed ∈ ( ; R ), we have
We set for each ∈ 1, (Ω). We also set for each ∈ 1, (Ω), ∈ (0, 1). As mentioned in Section 1, the equation (1.2) is connected with a minimizing problem of the functional F 0 given by (2.79). However, it should be noted that this functional is, in general, neither Gâteaux differentiable nor Fréchet differentiable. This is substantially due to nonsmoothness of the density function 1 at the origin. Therefore, it is natural to regard the term ∇ 1 (∇ ) as a vector field that satisfies (  (1.11). Assume that K ⊂ 1, (Ω) is a non-empty closed convex set, and there exists a positive constant K ∈ (0, ∞) such that for all 1 , 2 ∈ K. (2.81) Then, for a function ∈ K, these following are equivalent: (1). The function satisfies = arg min{F 0 ( ) | ∈ K}.

(2.82)
In other words, is the unique minimizer of the functional F 0 : K → R.
Proof. We first mention that the right hand sides of (2.82) and (2.84) are well-defined. To prove this, we will check coerciveness of F . For each fixed ∈ (0, 1), , satisfies In fact, by (2.42)-(2.43) we can compute It is easy to get the last inequality in the case 2 ≤ < ∞. When 1 < < 2, we should note that for all ∈ [0, 1], which yields the desired inequality. Hence, for each ∈ 1, (Ω), F satisfies for all ∈ K. By the estimate (2.85), which implies that F is a coercive functional in K, it is easy to construct a minimizer of F in K by direct methods. Here we note that the convex functional F is continuous with respect the strong topology of 1, (Ω), and hence sequentially lower-semicontinuous with respect to the weak topology of 1, (Ω). We note that the density is strictly convex by (2.54). From this fact and (2.81), we can easily conclude uniqueness of a minimizer of the functional F . Therefore, the right hand side of (2.84) is well-defined. This result similarly holds for F 0 . In fact, (2.85) is valid even when = 0, and strict convexity of also follows from (1.11). For detailed arguments and computations omitted in the proof, see e.g., [21,Chapter 8], [25,Chapter 4], [41, §3].
Let ( , ) ∈ K × ∞ (Ω; R ) satisfy (1.17) and (2.83). Then, for each ∈ K, it is easy to get which is so called a subgradient inequality. Similarly, for each ∈ K, we have another subgradient inequality From these, we can easily check 0 ≤ F( ) − F( ) for all ∈ K, which implies that satisfies (2.82).
We have already proved the implication (2) ⇒ (1), and unique existence of the minimizer (2.82). Hence, in order to complete the proof, it suffices to show the following two claims. The first is that the function ∈ K (0 < < 1) defined by (2.84) converges to a certain function ∈ K strongly in 1, (Ω) up to a sequence. The second is that this limit function satisfies (2). We would like to prove these assertions by applying Lemma 2.8. For each ∈ (0, 1), we set a function ∈ K by (2.84). Then, for each ∈ K, we have 0 ≤ F ( + ( − )) − F ( ) for all ∈ (0, 1).

Proof of main theorem
Corollary 2.12 in the previous Section 2.6 implies that a weak solution to (1.2) can be approximated by a weak solution to − div(∇ (∇ )) = with = * ( 1 + ), (2.89) under a suitable Dirichlet boundary condition, where satisfies (2.78). In the proof of Theorem 1.3, we aim to prove G (∇ ) is Hölder continuous. There Theorem 2.14 below plays an important role.
Theorem 2.14. In addition to assumptions of Theorem 1.3, we let positive numbers , satisfy (2.11), and a function ∈ 1, (Ω) be a weak solution to (2.89) in Ω with
By the definition of G , it is clear that In particular, there exists a continuous vector field 0 ∈ 0 (Ω; R ) such that G (∇ ) → 0 uniformly in Ω. On the other hand, there clearly holds G (∇ ) → ∇ a.e. in Ω as → 0+. Thus, ∇ = 0 ∈ 0 (Ω; R ) is realized, and this completes the proof of Theorem 1.3.

A priori Hölder estimates of the mapping G 2 , (∇ )
In Section 3, we consider weak solutions to the regularized equation (2.89), and show a priori Hölder bounds of truncated gradients (Theorem 2.14).
We set an exponent ∈ (0, 1) by whereˆ 0 is an arbitrary number satisfying 0 <ˆ 0 < 1. The number often appears when one considers regularity of weak solutions to the Poisson equation −Δ = ∈ . It is well-known that this weak solution is locally -Hölder continuous, which can be proved by the standard freezing coefficient method (see e.g., [28,Theorem 3.13]; see also [3,Chapter 3] and [23,Chapter 5] as related items).
To show local a priori Hölder estimates (Theorem 2.14), we apply Propositions 3.2-3.3 below. then the limit
Our analysis broadly depends on whether a scalar function degenerates or not, which can be judged by measure assumptions as in (3.5) or (3.9). We would like to describe each individual part.
The assumption (3.5) in Proposition 3.2 suggests that should be non-degenerate near the point 0 . This expectation enables us to apply freezing coefficient methods to show Campanato-type growth estimates as in (3.7). In other words, Proposition 3.2 is based on analysis over non-degenerate points of a scalar function or a gradient ∇ . To justify this, however, we will face to check that the average integral (∇ ) 0 , ∈ R never vanishes even when the radius tends to 0. To answer this affirmatively, we would like to find the suitable numbers and ★ as in Proposition 3.2. These numbers are the key criteria for judging whether it is possible to separate degenerate and non-degenerate points of a gradient ∇ . When choosing these important numbers, we will need a variety of energy estimates. There, the assumptions (3.3) and (3.5) are used to show these energy estimates.
With this in mind, we compute and by Young's inequality. Combining these results with | 7 | ≤ 6 , we easily conclude (3.26).

Perturbation results from a higher integrability lemma
For a given ball ( 0 ) ⋐ Ω, we consider an 2 -mean oscillation of the gradient ∇ , which is given by  and Here the constant ∈ (0, ∞) depends at most on , , , 0 , , Λ, , , and .
In the proof of Lemma 3.6, we use so called Gehring's lemma without proofs.
The proof of Gehring's lemma is found in [45,Theorem 3.3], which is based on ball decompositions [45, Lemma 3.1] and generally works for a metric space with a doubling measure. As related items, see also [25,Section 6.4], where the classical Calderon-Zygmund cube decomposition is fully applied to prove Gehring's lemma.
Now we would like to give perturbation estimates based on comparisons to harmonic functions (Lemma 3.8). This result can be obtained by the regularity assumption  and consider the Dirichlet boundary problem

Energy estimates
We go back to the weak formulation given in Lemma 3.5, and deduce some energy estimates under suitable assumptions as in Proposition 3.2. In Section 3.5, we introduce a vector field , ∈ ∞ (R ; R ) by for each ∈ (0, 1). By definition, it is clear that the mapping , : R → R is bĳective. By direct computations, it is easy to notice that the Jacobian matrix , satisfies for all ∈ (0, 1) and ∈ R . In particular, similarly to (2.42), we can find a constant = ( ) ∈ (0, ∞) such that there holds which clearly yields By this estimate and , (0) = 0, the inverse mapping for all ∈ (0, 1). Here , is the mapping defined by (3.37), and the constants ∈ (0, ∞) in (3.40)- (3.41) depend at most on , , , , Λ, , and .

Campanato-type decay estimates by shrinking arguments
The aim of Section 3.6 is to give the proof of Proposition 3.2 by standard shrinking methods. A significant point therein is to verify that the average integral (∇ ) 0 , never vanishes even when the radius tends to zero. We will justify this expectation by similar computations as in Lemma 3.12.

Appendix: Proofs for some basic estimates
In Section 4, we would like to provide the proofs Lemmata 3.16-3.17 in Section 3.7 for the reader's convenience. These results can be deduced by the fact that the function , satisfies Caccioppoli-type energy bounds since it is a weak subsolution to a uniformly elliptic problem (Lemma 3.15). We mention that the strategy herein are based on De Giorgi's levelset argument given in [17,Chapter 10,[4][5]. See also [10, §7] as a related item.