Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by $p$-Laplacian

We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show $C^{1}$-regularity of the solution at the boundary of a facet where the gradient of the solution vanishes. For this purpose we blow-up the solution and prove that its limit is a constant function by establishing a Liouville-type result, which is proved by showing a strong maximum principle. Our argument is rather elementary since we assume that the solution is convex. A few generalization is also discussed.


Introduction
We consider a one-Laplace equation perturbed by -Laplacian of the form long-standing open problem whether its weak solution is 1 up to a facet, the place where the gradient ∇ vanishes, even if is smooth. This is a non-trivial question since a weak solution to the unperturbed one-Laplace equation, i.e., −Δ 1 = may not be 1 . This is because the ellipticity degenerates in the direction of ∇ for Δ 1 . Our goal in this paper is to solve this open problem under the assumption that a solution is convex.
Since the exponent 1 − is negative, the ellipticity ratio of ∇ 2 ( 0 ) blows up as 0 → 0. By this property, we can observe that the equation (1.2) becomes non-uniformly elliptic near the facet. It should be noted that our problem is substantially different from the ( , )-growth problem, since for ( , )-growth equations, non-uniform ellipticity appears as a norm of a gradient blows up [26,Section 6.2]. Although regularity of minimizers of double phase functionals, including were discussed in scalar and even in vectorial cases by Colombo and Mingione [6,7], their results do not recover our 1 -regularity results. This is basically derived from the fact that, unlike ∇ 2 with 1 < < ∞, the Hessian matrix ∇ 2 1 ( 0 ) ( 0 ≠ 0) always takes 0 as its eigenvalue. In other words, ellipticity of the operator Δ 1 degenerates in the direction of ∇ , which seems to be difficult to handle analytically.
On the other hand, the ellipticity ratio of ∇ 2 ( 0 ) is uniformly bounded over | 0 | > for each fixed > 0. In this sense we may regard the equation (1.3) as locally uniformly elliptic outside the facet. To show Lipschitz bound, we do not need to study over the facet. In fact, local Lipschitz continuity of solutions to (1.1) are already established in [32]; see also [33] for a weaker result. To study continuity of derivatives, we have to study regularity up to the facet. Thus, it seems to be impossible to apply standard arguments based on De Giorgi-Nash-Moser theory. In this paper, we would like to show continuity of derivatives of convex solutions by elementary arguments based on convex analysis.
Let us give a basic strategy to prove Theorem 1. Since the problem is local, we may assume that Ω is convex, or even a ball. By 1 -regularity criterion for a convex function, to show is 1 at ∈ Ω it suffices to prove that the subdifferential ( ) at ∈ Ω is a singleton; (1.4) see [1,Appendix D], [30, §25] and Remark 1 for more detail. Here the subdifferential of at 0 ∈ Ω is defined by Here · | · stands for the standard inner product in R . For a convex function : Ω → R, we can simply express the facet of as ≔ { ∈ Ω | ( ) ∋ 0} = { ∈ Ω | ( ) ≤ ( ) for all ∈ Ω}. By definition it is clear that the facet is non-empty if and only if a minimum of in Ω exists. By convexity of , we can easily check that ⊂ Ω is a relatively closed convex set in Ω. We also define an open set ≔ Ω \ = { ∈ Ω | ( ) < ( ) for some ∈ Ω}.
Our strategy to show (1.4) depends on whether is inside or not.
Throughout this paper, we use these well-known results without proofs.
We first discuss the case ∈ . Our goal is to show directly that is 1, near a neighborhood of and therefore ( ) = {∇ ( )} ≠ {0} for all ∈ . This strategy roughly consists of three steps. Among them the first step, a kind of separation of ∈ from the facet , plays an important role. Precisely speaking, we first find a neighborhood ( ) ⊂ , an open ball centered at with its radius > 0, such that for some direction and some constant > 0. In order to justify (1.5), we fully make use of convexity of (Lemma 8 in Section A), not elliptic regularity theory. Then with the aid of local Lipschitz continuity of , the inclusion ( ) ⊂ {0 < ≤ ≤ |∇ | ≤ } holds for some finite positive constant . Secondly, this inclusion allows us to check that admits local 2, 2 -regularity in ( ) by the standard difference quotient method. Therefore we are able to obtain the equation (1.3) in the distributional sense. Finally, we appeal to the classical De Giorgi-Nash-Moser theory to obtain local 1, -regularity at ∈ , since the equation (1.3) is uniformly elliptic in ( ). Here the constant ∈ (0, 1) we have obtained may depend on the location of ∈ through ellipticity, so may tend to zero as tends to the facet.
It takes much efforts to prove that ( ) = {0} for all ∈ . Our strategy for justifying this roughly consists of three parts; a blow-argument for solutions, a strong maximum principle, and a Liouville-type theorem. Here we describe each individual step.
We first make a blow-argument. Precisely speaking, for a given convex solution : Ω → R and a point 0 ∈ Ω, we set a sequence of rescaled functions { } >0 defined by We show that locally uniformly converges to some convex function 0 : R → R, which satisfies ( 0 ) ⊂ 0 ( 0 ) by construction. Moreover, we prove that 0 satisfies , 0 = 0 in R in the distributional sense. There we will face to justify a.e. convergence of gradients, and this is elementarily shown by regarding gradients in the classical sense as subgradients (Lemma 9 in the appendices). Next we prove that if 0 ∈ , then the convex weak solution 0 constructed as above satisfies 0 ( 0 ) = {0}. Moreover, we are going to prove that 0 is constant (a Liouville-type theorem). For this purpose we establish the maximum principle. It should be noted that this result is a kind of strong maximum principle in the sense that where ( ) ≔ ( 0 ) + ∇ ( 0 ) | − 0 and 0 is a connected component of . The affine function clearly satisfies , = 0 in the classical sense. In order to justify (1.6), we will face three problems. The first is a justification of the comparison principle, the second is regularity of , and the third is a construction of suitable barrier subsolutions, all of which are essentially needed in the classical proof of E. Hopf's strong maximum principle [20]. In order to overcome these obstacles, we appeal to both classical and distributional approaches, and restrict our analysis only over regular points. For details, see Section 1.2.
Even though our strong maximum principle is somewhat weakened in the sense that this holds only on each connected component of ⊂ Ω, we are able to show the following Liouville-type theorem.
Theorem 3 (Liouville-type theorem). Let be a convex weak solution to , = 0 in R . Then ⊂ R , the facet of , satisfies either = ∅ or = R . In particular, satisfies either of the followings.
1. If attains its minimum in R , then is constant.

If does not attains its minimum in R , then is a non-constant affine function in R .
In the proof of the Liouville-type theorem, our strong maximum principle plays an important role. Precisely speaking, if a convex solution in the total space does not satisfy ∅ R , then Theorem 2 and the supporting hyperplane theorem from convex analysis help us to determine the shape of convex solutions. In particular, the convex solution can be classified into three types of piecewise-linear functions of one-variable. These non-smooth piecewise-linear functions are, however, no longer weak solutions, which we will prove by some explicit calculations.
By applying the Liouville-type theorem and our blow-argument, we are able to show that subgradients at points of the facet are always 0, i.e., ( ) = {0} for all ∈ , and we complete the proof of the 1 -regularity theorem. Note that the statements in Theorem 2 and 3 should not hold for unperturbed one-Laplace equation −Δ 1 = , since any absolutely continuous non-decreasing function of one variable = ( 1 ) satisfies −Δ 1 = 0.
Finally we mention that we are able to refine our strategy, and obtain 1 -regularity of convex solutions to more general equations. We replace the one-Laplacian Δ 1 by another operator which is derived from a general convex functional of degree 1. This generalization requires us to modify some of our arguments, including a blow-up argument and the Liouville-type theorem. For further details, see Section 1.4 and Section 6.2.

Literature overview on maximum principles
We briefly introduce maximum principles related to the paper. We also describe our strategy to establish the strong maximum principle.
Maximum principles, including comparison principles and strong maximum principles, have been discussed by many mathematicians in various settings. In the classical settings, E. Hopf proved a variety of maximum principles on elliptic partial differential equations of second order, by elementary arguments based on constructions of auxiliary functions. E. Hopf's strong maximum principle is one of the well-known results on maximum principles. In Hopf's proof of the strong maximum principle [20], he defined an auxiliary function which becomes a classical subsolution in a fixed open annulus = ( * ) ≔ ( * ) \ /2 ( * ) for sufficiently large > 0. An alternative function is given in [ (1.6). A justification of comparison principles is easily obtained in the distributional schemes (see [29, Chapter 3] as a related material). However, the remaining two obstacles, the differentiability of and the construction of subsolutions, cannot be resolved affirmatively by just imitating arguments given in [29,. In the first place, it should be mentioned that convex weak solutions we treat in this paper are assumed to have only local Lipschitz regularity, whereas supersolutions treated in [29,Chapter 5] are required to be in 1 . We recall that 1 -regularity of convex weak solutions can be guaranteed in ⊂ Ω (the outside of the facet) by the classical De Giorgi-Nash-Moser theory, and this result enables us to overcome the problem whether is differentiable at certain points. This is the reason why Theorem 2 need to restrict on . Although the construction of distributional subsolutions is generally discussed in [29, Chapter 4], we do not appeal to this. Instead, we directly construct a function = ℎ + in R \ { * }, where > 0 is a constant and ℎ is defined as in (1.7) or (1.8). We will determine the constants , > 0 so precisely that satisfies , ≤ 0 in the classical sense over a fixed open annulus = ( * ). We also make |∇ | very close to |∇ | ≡ |∇ ( 0 )| > 0 over , so that ∇ no longer degenerates there. By direct calculation of , , we explicitly construct classical subsolutions to , = 0 in . Finally we are able to deduce (1.6) by an indirect proof. Another type of definitions of subsolutions and supersolutions to (1.1) in the distributional schemes can be found in F. Krügel's thesis in 2013 [25]. The significant difference is that Krügel did not regard the term ∇ /|∇ | as a subgradient vector field. Since monotonicity of | · | is not used at all, it seems that Krügel's proof of comparison principle [25,Theorem 4.8] needs further explanation. For details, see Remark 3.

Mathematical models and previous researches
Our problem is derived from a minimizing problem of a certain energy functional, which involves the total variation energy. The equation (1.1) is deduced from the following Euler-Lagrange equation; The energy functional often appears in fields of materials science and fluid mechanics. In [31], Spohn modeled the relaxation dynamics of a crystal surface below the roughening temperature. On ℎ describing the height of the crystal for a two-dimensional domain Ω is modeled as where is a chemical potential. In [31], its evolution is given as with > 0. This Φ is essentially the same as with = 3. Then, the resulting evolution equation for ℎ is of the form This equation can be defined as a limit of step motion, which is microscopic in the direction of height [23]; see also [27]. The initial value problem of this equation can be solved based on the theory of maximal monotone operators [12] under the periodic boundary condition. Subdifferentials describing the evolution are characterized by Kashima [21], [22]. Its evolution speed is calculated by [21] for one dimensional setting and by [22] for radial setting. It is known that the solution stops in finite time [13], [14]. In [27], numerical calculation based on step motion is calculated. If one considers a stationary solution, ℎ must satisfies If , 3 ℎ is a constant, our Theorem 1 implies that the height function ℎ is 1 provided that ℎ is convex. For a second order problem, i.e., ℎ = , ℎ, its analytic formulation goes back to [4], [8, Chapter VI] for = 2, and its numerical analysis is given in [19]. For the fourth order problem, its numerical study is more recent. The reader is referred to papers by [15], [16], [24]. Another important mathematical model for the equation (1.1) is found in fluid mechanics. Especially for = 2 and = 2, the energy functional appears when modeling stationary laminar incompressible flows of a material called Bingham fluid, which is a typical non Newtonian fluid. Bingham fluid reflects the effect of plasticity corresponding to Δ 1 as well as that of viscosity corresponding Δ 2 = Δ in (1.1). Let us consider a parallel stationary flow with velocity = (0, 0, ( 1 , 2 )) in a cylinder Ω × R. Of course, this is incompressible flow, i.e., div = 0. If this flow is the classical Newtonian fluid, then the Navier-Stokes equations become (1.1) in Ω with = 0 and = − 3 , where denotes the pressure. In the case that plasticity effects appears, one obtains (1.1), following [8, Chapter VI, Section 1]. There it is also mentioned that since the velocity is assumed to be uni-directional, the external force term in (1.1) is considered as constant in this laminar flow model. The significant difference is that motion of the Bingham fluid is blocked if the stress of the Bingham fluid exceeds a certain threshold. This physical phenomenon is essentially explained by the nonlinear term Δ 1 , which reflects rigidity of the Bingham fluid. For more details, see [8, Chapter VI] and the references therein.
On continuity of derivatives for solutions, less is known even for the second order elliptic case. Although Krügel gave an observation that solutions can be continuously differentiable [25, Theorem 1.2] on the boundary of a facet, mathematical justifications of 1 -regularity have not been well-understood. Our main result (Theorem 1) mathematically establishes continuity of gradient for convex solutions.

Organization of the paper
We outline the contents of the paper.
Section 2 establishes 1, -regularity at regular points of convex weak solutions (Lemma 1). In order to apply De Giorgi-Nash-Moser theory, we will need to justify local 2, 2 -regularity by the difference quotient method.
The key lemma, which is proved by convex analysis, is contained in the appendices (Lemma 8).
Section 3 provides a blow-up argument for convex weak solutions. The aim of Section 3 is to prove that 0 : R → R, a limit of rescaled solutions, satisfies , 0 = 0 in the weak sense over the whole space R (Proposition 1). To assure this, we will make use of an elementary result on a.e. convergence of gradients, which is given in the appendices (Lemma 9). Section 4 is devoted to justifications of maximum principles for the equation In Section 5, we will show the Liouville-type theorem (Theorem 3) by making use of Theorem 2, and complete the proof of our main theorem (Theorem 1).
Finally in Section 6, we discuss a few generalization of the operators Δ 1 and Δ . Since the general strategy for the proof is the same, we only indicate modification of our arguments. Among them, we especially treat with a Liouville-type theorem and a blow-up argument, since these proofs require basic facts of a general convex functional which is positively homogeneous of degree 1. These well-known facts are contained in the appendices for completeness.

Regularity outside the facet
In Section 2, we would like to show that is 1 at any ∈ , and therefore (1.4) holds for all ∈ . This result will be used in the proof of the strong maximum principle (Theorem 2).
We first give a precise definition of weak solutions to , = in a convex domain Ω ⊂ R , which is not necessarily bounded.
The aim of Section 2 is to show Lemma 1 below.

Lemma 1.
Let be a convex weak solution to (1.1) in a convex domain Ω ⊂ R , and ∈ loc (Ω) ( < ≤ ∞). If 0 ∈ , then we can take a small radius 0 > 0, a unit vector 0 ∈ R , and a small number 0 > 0 such that
In the proof of Lemma 1, we will use Lemma 7-8 without proofs. For precise proofs, see Section A.
Proof. For each fixed 0 ∈ , we may take and fix 1 . By ∈ (Ω), we may take a sufficiently small 0 > 0 such that For the first inequality in (2.5), we have used Lemma 8, which is basically derived from convexity of . Recall that ( ) = {∇ ( )} for a.e. ∈ Ω, and hence we are able to recover (2.3) from (2.5).

A blow-up argument
In order to show that (1.4) holds true even for ∈ , we first make a blow-argument and construct a convex weak solution in the whole space R , in the sense of Definition 1.

Proposition 1.
Let Ω ⊂ R be a convex domain, and ∈ loc (Ω) ( < ≤ ∞). Assume that is a convex weak solution to (1.1), and 0 ∈ Ω. Then there exists a convex function 0 : R → R such that In particular, if 0 ∈ , then the facet of 0 is non-empty.
It is clear that ∞ ( , R ) ≤ 1 for all 0 < < / . Hence by [5,Corollary 3.30], up to a subsequence, we may assume that * for some 0, ∈ ∞ ( , R ). By lower-semicontinuity of the norm with respect to the weak * topology and

Maximum principles
In Section 4, we justify maximum principles for the equation , = 0.
We first define subsolutions and supersolutions in the weak sense.

Remark 2.
We describe some remarks on our definitions of weak solutions, subsolutions and supersolutions.

Comparison principle
We justify the comparison principle, i.e., for any subsolution − and supersolution + , under the condition that + and − admits continuity properties in Ω.

Construction of classical subsolutions
In Section 4.2, we construct a classical subsolution to , = 0 in an open annulus. Here is an open annulus, and in (4.14) denotes the exterior unit vector normal to ( * ).
Before proving Lemma 2, we fix some notations on matrices. For a given × matrix , we write tr( ) as the trace of . We denote 1 by the × unit matrix. For column vectors = ( ) , = ( ) ∈ R , we define a tensor ⊗ , which is regarded as a real-valued × matrix Assume that ℎ satisfies (4.15). Then the triangle inequality implies that The estimate (4.17) allows us to calculate , in the classical sense over ( * ). By direct calculations we have We note that ∇ 2 = ∇ 2 ℎ by definition. Here we recall a well-known result on Pucci's extremal operators. For given constants 0 < ≤ Λ < ∞ and a fixed × symmetric matrix , we define where ∈ R are the eigenvalues of . The following formula is a well-known result [1, Remark 5.36] ; where A , Λ denotes the set of all symmetric matrices whose eigenvalues all belong to the closed interval [ , Λ]. By (4.17) , is an uniformly elliptic operator in ( * ). This enables us to find constants 0 < ≤ Λ < ∞, depending on 0 < < ∞, 1 < < ∞, | | > 0, such that ∇ 2 (∇ ) ∈ [ , Λ] in ( * ). Combining these results, it suffices to show that where ( ) ∈ R denotes the eigenvalues of ∇ 2 ℎ( ). Now we construct classical subsolutions. Our first construction is a modification of that by E. Hopf [20].
Proof. Without loss of generality we may assume * = 0. We define From (4.20), we can easily check (4.13). By direct calculation we get From this result, (4.14) is clear. Also, we have then we can check that ℎ satisfies (4.15). Now we prove (4.16) to complete the proof. For ≠ 0, the eigenvalues of ∇ 2 ℎ( ) are given by and the geometric multiplicities are 1, − 1.
Let , > 0 satisfy then we can check that ℎ satisfies (4.15). Now we prove (4.16) to complete the proof. For ≠ 0, the eigenvalues of ∇ 2 ℎ( ) are given by and the geometric multiplicities are 1, − 1.

Strong maximum principle
We prove the strong maximum principle (Theorem 2).

Proofs of main theorems
In Section 5, we give proofs of the Liouville-type theorem (Theorem 3) and the 1 -regularity theorem (Thorem 1).

Liouville-type theorem
For a preparation, we prove Lemma 3 below.

Lemma 3.
Let be a real-valued convex function in R . Assume that satisfies the following, 2. attains its minimum 0.

is affine in each connected component of ≔ R \ .
Then up to a rotation and a shift translation, can be expressed as either of the following three types of piecewise-linear functions.

Now we give the proof of Theorem 3.
Proof. Assume by contradiction that , the facet of , would satisfy ∅ R . Without loss of generality, we may assume that attains its minimum 0. By the strong maximum principle (Theorem 2), the convex weak solution is affine in each connected component of ≔ R \ . Therefore we are able to apply Lemma 3. By rotation and translation, can be expressed as (5.1)-(5.3). Now we prove that is no longer a weak solution to , = 0 in R . We set open cubes ′ ≔ (−1, 1) −1 ⊂ R −1 and ≔ (− , ) × ′ ⊂ R , where > 0 is to be chosen later. We claim that does not satisfy , = 0 in −1, ′ ( ). Assume by contradiction that there exists a vector field ∈ ∞ ( , R ) such that the pair ( , ) ∈ 1, ( ) × ∞ ( , R ) satisfies , = 0 in −1, ′ ( ). We define an admissible test function Test ∈ 1 ( ) into , = 0 in −1, ′ ( ), and divide the integration over into that over and . Then (5.4) implies that Here we have applied the Gauss-Green theorem to the integration over , and the Cauchy-Schwarz inequality to the integration over . For the integrations 1 and 2 , we make use of Fubini's theorem and (5.5). Then we have Finally we obtain From (5.6), we can easily deduce a contradiction by choosing sufficiently small = ( , , 1 , 2 ) > 0. Similarly we can prove that defined as in (5.3) does not satisfy , = 0 in −1, ′ ( ), since it suffices to restrict < 0 . We consider the remaining case (5.2). We have for a.e. ∈ , − 1 for a.e. ∈ . by definition of . We test the same function ∈ 1 ( ) in , = 0, then it follows that which is a contradiction. This completes the proof.

Remark 4.
The estimate (5.6) breaks for = 1, since the equation |0| −2 0 = 0 is no longer valid for = 1. This means that we have implicitly used differentiability of the function | | / at 0 ∈ R . Also it should be noted that for the one-variable case, functions as in (5.1), which are in general not in 1 , are one-harmonic in R.

1 -regularity theorem
We give the proof of Theorem 1.
Proof. We may assume that Ω is convex. By [30, Theorem 25.1 and 25.5] and Lemma 1, it suffices to show that ( 0 ) = {0} for all 0 ∈ . Let 0 ∈ . We get a convex function 0 : R → R as a blow-up limit as in Proposition 1. We note that the facet of 0 is non-empty by Proposition 1. Hence by the Liouville-type theorem (Theorem 3),

Generalization
In Section 6, we would like to discuss 1 -regularity of convex weak solutions to which covers (1.1). Precisely speaking, throughout Section 6, we make these following assumptions for Ψ and on regularity and ellipticity. For regularity, we only require For , we assume that for each fixed 0 < ≤ < ∞, there exist constants 0 < < Γ < ∞ such that satisfies for all 0 , , ∈ R with ≤ | 0 | ≤ . Also, there is no loss of generality in assuming that Finally, we assume that Ψ is positively homogeneous of degree 1. In other words, Ψ satisfies holds for all 0 ∈ R and > 0. This clearly yields Ψ(0) = 0. By modifying some of our arguments, we are able to show that Theorem 4 ( 1 -regularity theorem for general equations). Let Ω ⊂ R be a domain. Assume that ∈ loc (Ω) ( < ≤ ∞) and the functionals Ψ and satisfy (6.2)- (6.5). If is a convex weak solution to (6.1), then is in 1 (Ω).

Preliminaries
In Section 6.1, we mention some basic properties of Ψ and , which are derived from the assumptions (6.2)-(6.5). For , by (6.2)-(6.3) and (6.5) it is easy to check that the continuous mapping : R ∋ ↦ → ∇ ( ) ∈ R satisfies strict monotonicity (4.8). In particular, by (6.5) we have For the proof, see Lemma 10 in the appendices. For Ψ, we first note that Ψ satisfies the triangle inequality We define a functionΨ : Ψ is the support function for the closed convex set Ψ ≔ { ∈ R | Ψ( ) ≤ 1}. By definition it is easy to check thatΨ is convex and lower semicontinuous. Also, if ∈ R satisfiesΨ( ) < ∞, then the following Cauchy-Schwarz-type inequality holds; If a convex function Ψ is positively homogeneous of degree 1, then the subdifferential operator Ψ is explicitly given by for all ∈ R . In particular, we have the following formula which is often called Euler's identity. Also, assumptions (6.2) and (6.6) imply that for all > 0 and 0 ∈ R \ {0}. Proofs of (6.8)-(6.10) are given in Lemma 11 of the appendices for the reader's convenience.

For
for all , ∈ R . Here the finite constant is explicitly given by Lemma 4 states lower semicontinuity of a functional in the weak * topology of an ∞ -space. This result is used in the justification of a blow-up argument for the equation (6.1).

Lemma 4.
Let Ω ⊂ R be a Lebesgue measurable set, and let Ψ : R → [0, ∞) be a convex function which satisfies (6.6). Assume that a vector field ∈ ∞ (Ω, R ) and a sequence { } ⊂ ∞ (Ω, R ) satisfy * We give an elementary proof of Lemma 4, which is based on a definition ofΨ.
Since > 0 is arbitrary, this completes the proof of (6.15).

Sketches of the proofs
We first give definitions of weak solutions to (6.1). We also define weak subsolutions, and supersolutions to an equation = 0 in a bounded domain.

Definition 3.
Let Ω ⊂ R be a domain.

Remark 6.
We describe some remarks on Definition 3.
1. In this paper we treat a convex solution, which clearly satisfies local Lipschitz regularity. Hence it is not restrictive to assume local or global 1, ∞ -regularity for solutions in Definition 3. Also it should be noted that if a vector field satisfies (6.19), then is in ∞ by Remark 5. Hence our regularity assumptions of the pair ( , ) involve no loss of generality.
To prove Theorem 4, we may assume that Ω is a bounded convex domain, since our argument is local. As described in Section 1.1, we would like to prove that a convex solution to (6.1) satisfies (1.4) for all ∈ Ω.
For the case ∈ , we can show (1.4) by De Giorgi-Nash-Moser theory. This is basically due to the fact that the functional ( ) ≔ Ψ( ) + ( ) for ∈ R satisfy the following property. For each fixed constants 0 < ≤ < ∞, there exists constants 0 < ≤ Λ < ∞ such that the estimates (2.7)-(2.8) hold for all 0 , , ∈ R with ≤ | 0 | ≤ . In other words, the operator is locally uniformly elliptic outside a facet, in the sense that for a function the operator becomes uniformly elliptic in a place where 0 < ≤ |∇ | ≤ < ∞ holds. This ellipticity is an easy consequence of (6.3)-(6.4) and (6.13)-(6.14). Appealing to local uniform ellipticity of the operator outside the facet and De Giorgi-Nash-Moser theory, we are able to show that a convex solution to = is 1, near a neighborhood of each fixed point ∈ , similarly to the proof of Lemma 1.
For the case ∈ , we first make a blow-argument to construct a convex function 0 : R → R satisfying ( ) ⊂ 0 ( ), and 0 = 0 in R in the sense of Definition 3. Next we justify a maximum principle, which is described as in (1.6), holds on each connected component of . This result enables us to apply Lemma 3, and thus similarly in Section 5.1, we are able to prove a Liouville-type theorem. Hence it follows that a convex solution 0 , which is constructed by the previous blow-argument, should be constant. Finally the inclusions {0} ⊂ ( ) ⊂ 0 ( ) ⊂ {0} hold, and this completes the proof of (1.4), i.e., ( ) = {0}.
For maximum principles on the equation = 0, the proofs are almost similar to those in Section 4. Indeed, we first recall that the operator : R ∋ 0 ↦ → ∇ ( 0 ) ∈ R satisfies strict monotonicity (4.8). Combining with monotonicity of the subdifferential operator Ψ, we can easily prove a comparison principle as in Proposition 2. Also, similarly to Lemma 2, we can construct classical barrier subsolutions to = 0 in an open annulus, since the operator is locally uniformly elliptic outside a facet. These results enable us to prove a maximum principle outside a facet.
We are left to justify the remaining two problems, a blow-argument and the Liouville-type theorem. To show them, we have to make use of some basic facts on a convex functional which is homogeneous of degree 1. These fundamental results are contained in Section A.3.
For a blow-up argument as in Section 3, we similarly define rescaled solutions. Existence of a limit of these rescaled functions are guaranteed by the Arzelà-Ascoli theorem and a diagonal argument. By proving Lemma 5 below, we are able to demonstrate that 0 , a limit of rescaled solutions, is a weak solution to = 0 in R , and this finishes our blow-up argument.
We prove a Liouville-type theorem as in Theorem 3. In other words, for a convex solution to = 0 in R , we show that , the facet of , would satisfy either = ∅ or = R . Assume by contradiction that satisfies ∅ R . Then by Lemma 3, we may write a convex solution by either of (5.1)-(5.3). However, Lemma 6 below states that is no longer a weak solution, and this completes our proof.

Lemma 6. Let be a piecewise-linear function defined as in either of (5.1)-(5.3). Then is not a weak solution to
= 0 in R .

A.2 Convex analysis
Lemma 8 is used in the proof of Lemma 1 for a justification of local 2, 2 -regularity of a convex weak solution outside of the facet. Our proof of Lemma 9 is inspired by [11,Lemma A.3].

A.3 Convex functionals
We prove some basic property of convex functionals Ψ and in Section 6.