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<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document}. We prove that a weak solution is continuously differentiable (C1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C^1)$$\end{document} if it is convex. Note that a similar result fails to hold for the unperturbed one-Laplace equation. The main difficulty is to show C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1}$$\end{document}-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 generalizations are also discussed.


Introduction
We consider a one-Laplace equation perturbed by p-Laplacian of the form where 1 u := div (∇u/|∇u|) , p u = div |∇u| p−2 ∇u in a domain in R n , ∇u = (∂ x 1 u, . . . , ∂ x n u) with ∂ x j u = ∂u/∂ x j for a function u = u(x 1 , . . . , x n ), and div X = n i=1 ∂ x i X i for a vector field X = (X 1 , . . . , X n ). The constants b > 0 and p ∈ (1, ∞) are given and fixed. It has been a long-standing open problem whether its weak solution is C 1 up to a facet, the place where the gradient ∇u vanishes, even if f is smooth. This is a non-trivial question since a weak solution to the unperturbed one-Laplace equation, that is, − 1 u = f may not be C 1 . This is because the ellipticity degenerates in the direction of ∇u for 1 u. Our goal in this paper is to solve this open problem under the assumption that a solution is convex.

Main theorems and our strategy
Throughout the paper, we assume that f ∈ L q loc ( ) (n < q ∞), that is, | f | q is locally integrable in . Our main result is Theorem 1. (C 1 -regularity theorem) Let u be a convex weak solution to (1.1) with f ∈ L q loc ( ) (n < q ≤ ∞). Then u is in C 1 ( ). Difficulty on proving regularity on gradients of solutions to (1.1) can be explained from a viewpoint of ellipticity ratio. We set a convex function E : R n → [0, ∞) by where E s (1 ≤ s < ∞) is defined by were discussed in scalar and even in vectorial cases by Colombo and Mingione [6,7], their results do not recover our C 1 -regularity results. This is basically derived from the fact that, unlike ∇ 2 z E p with 1 < p < ∞, the Hessian matrix ∇ 2 z E 1 (z 0 ) (z 0 = 0) always takes 0 as its eigenvalue. In other words, ellipticity of the operator 1 u degenerates in the direction of ∇u, which seems to be difficult to handle analytically.
On the other hand, the ellipticity ratio of ∇ 2 z E(z 0 ) is uniformly bounded over |z 0 | > δ for each fixed δ > 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 is convex, or even a ball. By C 1 -regularity criterion for a convex function, to show u is C 1 at x ∈ it suffices to prove that the subdifferential ∂u(x) at x ∈ is a singleton; (1.4) see [1,Appendix D], [30, §25] and Remark 1 for more detail. Here the subdifferential of u at x 0 ∈ is defined by Here · | · stands for the standard inner product in R n . For a convex function u : → R, we can simply express the facet of u as F := {x ∈ | ∂u(x) 0} = {x ∈ | u(x) ≤ u(y) for all y ∈ }.
By definition it is clear that the facet F is non-empty if and only if a minimum of u in exists. By convexity of u, we can easily check that F ⊂ is a relatively closed convex set in . We also define an open set D := \ F = {x ∈ | u(y) < u(x) for some y ∈ }.
Our strategy to show (1.4) depends on whether x is inside F or not.

Remark 1. (Some properties on differentiability of convex functions)
Let v a realvalued convex function in a convex domain ⊂ R n , then following properties hold. 1. v is locally Lipschitz continuous in , and therefore v is almost everywhere differentiable in by Rademacher's theorem ( [1,Theorem 1.19], see also [9, [30,Theorem 25.1]).
Throughout this paper, we use these well-known results without proofs.
We first discuss the case x ∈ D. Our goal is to show directly that u is C 1, α near a neighborhood of x and therefore ∂u(x) = {∇u(x)} = {0} for all x ∈ D. This strategy roughly consists of three steps. Among them the first step, a kind of separation of x ∈ D from the facet F, plays an important role. Precisely speaking, we first find a neighborhood B r (x) ⊂ D, an open ball centered at x with its radius r > 0, such that ∂ ν u ≥ μ > 0 almost everywhere in B r (x) (1.5) for some direction ν and some constant μ > 0. In order to justify (1.5), we fully make use of convexity of u (Lemma 8 in Section A), not elliptic regularity theory. Then with the aid of local Lipschitz continuity of u, the inclusion B r (x) ⊂ {0 < μ ≤ ∂ ν u ≤ |∇u| ≤ M} holds for some finite positive constant M. Secondly, this inclusion allows us to check that u admits local W 2, 2 -regularity in B r (x) 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 C 1, α -regularity at x ∈ D, since the Equation (1.3) is uniformly elliptic in B r (x). Here the constant α ∈ (0, 1) we have obtained may depend on the location of x ∈ D through ellipticity, so α may tend to zero as x tends to the facet. It takes much efforts to prove that ∂u(x) = {0} for all x ∈ F. 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 u : → R and a point x 0 ∈ , we set a sequence of rescaled functions {u a } a>0 defined by We show that u a locally uniformly converges to some convex function u 0 : R n → R, which satisfies ∂u(x 0 ) ⊂ ∂u 0 (x 0 ) by construction. Moreover, we prove that u 0 satisfies L b, p u 0 = 0 in R n in the distributional sense. There we will face to justify almost everywhere 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 x 0 ∈ F, then the convex weak solution u 0 constructed as above satisfies ∂u 0 (x 0 ) = {0}. Moreover, we are going to prove that u 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 The affine function a clearly satisfies L b, p a = 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 u, 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 D ⊂ , we are able to show the following Liouville-type theorem.
Theorem 3. (Liouville-type theorem) Let u be a convex weak solution to L b, p u = 0 in R n . Then F ⊂ R n , the facet of u, satisfies either F = ∅ or F = R n . In particular, u satisfies either of the following: 1. If u attains its minimum in R n , then u is constant. 2. If u does not attains its minimum in R n , then u is a non-constant affine function in R n .
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 ∅ F R n , 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, that is, ∂u(x) = {0} for all x ∈ F, and we complete the proof of the C 1 -regularity theorem. Note that the statements in Theorem 2 and 3 should not hold for unperturbed one-Laplace equation − 1 u = f , since any absolutely continuous non-decreasing function of Finally we mention that we are able to refine our strategy, and obtain C 1regularity of convex solutions to more general equations. We replace the one-Laplacian 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 Sections 1.4 and 6.2.

Literature overview on maximum principles
We briefly introduce maximum principles related to the paper. We also describe our strategy for establishing 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 i.e., − div(A(x, ∇u(x))) = 0, which covers the p-Laplace equation with 1 < p < ∞. Even in the distributional schemes, the proof of the maximum principle [29,Theorem 5.4.1] is partially similar to E. Hopf's classical one, in the sense that it is completed by calculating directional derivatives of auxiliary functions. The significant difference is, however, the construction of spherically symmetric subsolutions of C 1 class, which is given in [29,Chapter 4], is based on Leray-Schauder's fixed point theorem [17,Theorem 11.6]. Also it should be noted that the proofs of comparison principles [29, Theorem 2.4.1 and 3.4.1] are just based on strict monotonicity of the mapping A(x, · ) : R n → R n , whereas Hopf's proof appeals to direct constructions of auxiliary functions.
With our literature overview in mind, we describe our strategy for showing (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 u 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 C 1 . We recall that C 1 -regularity of convex weak solutions can be guaranteed in D ⊂ (the outside of the facet) by the classical De Giorgi-Nash-Moser theory, and this result enables us to overcome the problem whether u is differentiable at certain points. This is the reason why Theorem 2 need to restrict on D. 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 v = βh + a in R n \ {x * }, where β > 0 is a constant and h is defined as in (1.7) or (1.8). We will determine the constants α, β > 0 so precisely that v satisfies L b, p v ≤ 0 in the classical sense over a fixed open annulus E R = E R (x * ). We also make |∇v| very close to |∇a| ≡ |∇u(x 0 )| > 0 over E R , so that ∇v no longer degenerates there. By direct calculation of L b, p v, we explicitly construct classical subsolutions to L b, p u = 0 in E R . 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 thesis in 2013 [25]. The significant difference is that Krügel did not regard the term ∇u/|∇u| as a subgradient vector field. Since monotonicity of ∂| · | is not used at all, it seems that Krügel's proof of the 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 G 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 twodimensional domain is modeled as h t + div j = 0 with j = −∇μ, where μ is a chemical potential. In [31], its evolution is given as with κ > 0. This is essentially the same as G with p = 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 satisfies 3 h is a constant, our Theorem 1 implies that the height function h is C 1 provided that h is convex. For a second order problem, i.e., bh t = L b, p h, its analytic formulation goes back to [4], [8,Chapter VI] for p = 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 p = 2 and n = 2, the energy functional G 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 1 u as well as that of viscosity corresponding 2 u = u in (1.1). Let us consider a parallel stationary flow with velocity U = (0, 0, u(x 1 , x 2 )) in a cylinder × R. Of course, this is incompressible flow, that is, div U = 0. If this flow is the classical Newtonian fluid, then the Navier-Stokes equations become (1.1) in with b = 0 and f = −∂ x 3 π , where π 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 b 1 u, 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 C 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 C 1, α -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 W 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 u 0 : R n → R, a limit of rescaled solutions, satisfies L b, p u 0 = 0 in the weak sense over the whole space R n (Proposition 1). To assure this, we will make use of an elementary result on almost everywhere convergence of gradients, which is given in the appendices (Lemma 9). Section 4 is devoted to justifications of maximum principles for the equation L b, p u = 0. We first give definitions of sub-and supersolution in the weak sense. Section 4.1 provides a justification of the comparison principle (Proposition 2). Section 4.2 establishes an existence result of classical barrier subsolutions in an open annulus (Lemma 2). Applying these results in Section 4.1-4.2, we prove the strong maximum principle outside the facet (Theorem 2).
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 1 and p . 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 u is C 1 at any x ∈ D, and therefore (1.4) holds for all x ∈ D. 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 L b, p u = f in a convex domain ⊂ R n , which is not necessarily bounded.

Definition 1. Let
⊂ R n be a domain, which is not necessarily bounded, and f ∈ L q loc ( ) (n < q ≤ ∞). We say that a function u ∈ W 1, p loc ( ) is a weak solution to (1.1), when for any bounded Lipschitz domain ω , there exists a vector field Z ∈ L ∞ (ω, R n ) such that the pair (u, Here p ∈ (1, ∞) denotes the Hölder conjugate exponent of p ∈ (1, ∞).
The aim of Section 2 is to show Lemma 1, below.

Lemma 1. Let u be a convex weak solution to
. If x 0 ∈ D, then we can take a small radius r 0 > 0, a unit vector ν 0 ∈ R n , and a small number μ 0 > 0 such that (2.3) and there exists a small number Before proving Lemma 1, we introduce difference quotients. For given g : where e j ∈ R n denotes the unit vector in the direction of the x j -axis.
In the proof of Lemma 1, we will use Lemmas 7-8 without proofs. For precise proofs, see Section A.
Proof. For each fixed x 0 ∈ D, we may take and fix From (2.4), the inclusion B r 0 (x 0 ) ⊂ D clearly holds. (2.4) also allows us to check that for all y 0 ∈ B r 0 (x 0 ), z 0 ∈ ∂u(y 0 ), For the first inequality in (2.5), we have used Lemma 8, which is basically derived from convexity of u. Recall that ∂u(x) = {∇u(x)} for almost everywhere x ∈ , and hence we are able to recover (2.3) from (2.5).
In order to obtain C 1 -regularity in D, we will appeal to the classical De Giorgi-Nash-Moser theory. For preliminaries, we check that the operator L b, p u assures uniform ellipticity in B r 0 (x 0 ). Local Lipschitz continuity of u implies that there exists a sufficiently large number M 0 ∈ (0, ∞) such that ess sup For notational simplicity, we write subdomains by Now we check that u ∈ W 2, 2 (U 4 ) by the difference quotient method. We refer the reader to [ Here we note that ∇u no longer degenerates in U 1 by (2.3). We fix a cutoff function is an admissible test function. By testing φ, we have ) denotes a matrix-valued function in U 2 given by We note that with the aid of (2.3)-(2.6), we obtain for all ζ, ω ∈ R n and for almost everywhere x ∈ U 2 . We set an integral By (2.13), it is clear that I 1 ≥ λJ . By Young's inequality and applying a Poincarétype inequality (Lemma 7) to η 2 j, h u ∈ W 1, 2 0 (U 2 ), we obtain for any ε > 0, Here we have invoked the property 0 ≤ η ≤ 1 in U 2 . We fix ε := λ/6 > 0. By (2.14) and Young's inequality, we have The estimate (2.9) follows from this, and therefore u ∈ W 2, 2 (U 4 ).
Integrating by parts, we obtain

A Blow-Up Argument
In order to show that (1.4) holds true even for x ∈ F, we first make a blowargument and construct a convex weak solution in the whole space R n , in the sense of Definition 1.
Assume that u is a convex weak solution to (1.1), and x 0 ∈ . Then there exists a convex function u 0 : R n → R such that In particular, if x 0 ∈ F, then the facet of u 0 is non-empty.
Proof. Without loss of generality, we may assume that x 0 = 0 and u(x 0 ) = 0. First we fix a closed ball B R (0) = B R ⊂ . We note that u ∈ Lip(B R ) since u is convex. Hence there exists a sufficiently large number M ∈ (0, ∞) such that ess sup We take and fix a vector field For each a > 0, we define a rescaled convex function u a : B R/a → R and a dilated vector field We also set f a ∈ L q (B R/a ) by Then it is easy to check that the pair by the definition of u a . Hence by the Arzelà-Ascoli theorem and a diagonal argument, we can take a decreasing sequence {a N } ∞ N =1 ⊂ (0, ∞), such that a N → 0 as N → ∞, and u a N → u 0 locally uniformly in R n . (3.2) for some function u 0 : R n → R. Clearly u 0 is convex in R n , and the inclusion ∂u(x 0 ) ⊂ ∂u 0 (x 0 ) holds true by the construction of rescaled functions u a . If and therefore x 0 lies in the facet of u 0 . We are left to show that u 0 is a weak solution to L b, p u 0 = 0 in R n . Before proving this, we note that from (3.1)-(3.2) and Lemma 9, it follows that It is clear that Z a L ∞ (B r , R n ) ≤ 1 for all 0 < a < R/r . Hence by [5,Corollary 3.30], up to a subsequence, we may assume that By lower-semicontinuity of the norm with respect to the weak * topology and (3.3)-(3.4), we get (3.9) Letting a = a N in (3.5) and Since B r ⊂ R n is arbitrary, (3.9)-(3.10) means that u 0 is a weak solution to L b, p u 0 = 0 in R n , in the sense of Definition 1.

Maximum Principles
In Section 4, we justify maximum principles for the equation L b, p u = 0. We first define subsolutions and supersolutions in the weak sense.
For u ∈ W 1, p ( ), we simply say that u is respectively a subsolution and a supersolution to L b, p u = 0 in the weak sense if there is Z ∈ L ∞ ( , R n ) such that the pair (u, Z ) is a weak subsolution and a weak supersolution to L b, p u = 0 in . Remark 2. We describe some remarks on our definitions of weak solutions, subsolutions and supersolutions.

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

Proposition 2.
Let ⊂ R n be a bounded domain. Assume that u + , u − ∈ C( ) ∩ W 1, p ( ) is a subsolution and a supersolution to L b, p u = 0 in the weak sense respectively. If u + , u − satisfies then u − ≤ u + in .
By testing (u − − u + ) + ∈ D + ( ) into (4.9)(4.10) and substracting the two inequalities, Krügel claims that ∇u − = ∇u + over := {x ∈ | u − (x) ≥ u + (x)} and hence u − = u + almost everywhere in . Despite Krügel's comment that integrals over F − and F + cancel out, however, it seems unclear whether is valid. This problem is essentially due to the fact that Krügel did not appeal to monotonicity of the subdifferential operator ∂| · | and did not regard the term ∇u/|∇u| as an L ∞ -vector field satisfying the property (4.2). In our proof of the comparison principle (Proposition 2), we make use of monotonicity of the operator ∂| · |. Compared to our argument based on monotonicity, the equality (4.12) itself seems to be too strong to hold true.

Construction of classical subsolutions
In Section 4.2, we construct a classical subsolution to L b, p u = 0 in an open annulus.  Here is an open annulus, and ν in (4.14) denotes the exterior unit vector normal to B R (x * ).
Before proving Lemma 2, we fix some notations on matrices. For a given n × n matrix A, we write tr(A) as the trace of A. We denote 1 n by the n × n unit matrix. For column vectors x = (x i ) i , y = (y i ) i ∈ R n , we define a tensor x ⊗ y, which is regarded as a real-valued n × n matrix x n y 1 · · · x n y n ⎞ ⎟ ⎠ .
Assume that h satisfies (4.15). Then the triangle inequality implies that The estimate (4.17) allows us to calculate L b, p v in the classical sense over E R (x * ). By direct calculations we have We note that ∇ 2 v = ∇ 2 h by definition. Here we recall a well-known result on Pucci's extremal operators. For given constants 0 < λ ≤ < ∞ and a fixed n × n symmetric matrix M, we define where A λ, denotes the set of all symmetric matrices whose eigenvalues all belong to the closed interval [λ, ]. By (4.17) L b, p v is an uniformly elliptic operator in E R (x * ). This enables us to find constants 0 < λ ≤ < ∞, depending on 0 < b < ∞, 1 < p < ∞, |c| > 0, such that ∇ 2 z E(∇v) ∈ [λ, ] in E R (x * ). Combining these results, it suffices to show that where λ i (x) ∈ R denotes the eigenvalues of ∇ 2 h(x). 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 x * = 0. We define Here α = α(b, n, p, |c|, R) > 0 is a sufficiently large constant to be chosen later. It is clear that 0 ≤ h(x) ≤ e −α R 2 /4 − e −α R 2 in E R (0). We first let α > 0 be so large that From (4.20), we can easily check (4.13). By direct calculation we get From this result, (4.14) is clear. Also, we have Let α > 0 be so large that Assume that α satisfies so that λ > 0 > λ ⊥ in E R (0). Therefore we get We can take sufficiently large α = α(|c|, m, n, R, λ, ) > 0 so that α satisfies It is possible to construct an alternative function h ∈ C ∞ (R n \ {x 0 }) which satisfies (4.13)-(4.16). We give another proof of Lemma 2, which is derived from [29, Chapter 2.8].

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

Proof. Let
and we will prove that = D 0 . It is also clear that 0 ∈ and hence = ∅. Suppose for contradiction that D 0 . Then it follows that ∂ ∩ D 0 = ∅, since D 0 is connected. We may take and fix a point x * ∈ D 0 \ such that dist(x * , ) < dist(x * , ∂ D 0 ). By extending a closed ball centered at x * until it hits , we can take a point y * ∈ D 0 and a closed ball . This inequality becomes equality at y * ∈ ∂ E R (x * ) by (4.13) and (4.26). Therefore the function u( takes its minimum at y * ∈ ∂ B R (x * ). Also by y * ∈ and the subgradient inequality it is clear that the function w(x) := u(x) − c | x (x ∈ D 0 ) takes its minimum 0 at y * ∈ D 0 . We note that w, w − h ∈ C 1 (D 0 ) by Lemma 1. By calculating classical partial derivatives at y * in the direction ν 0 := (y * − x * )/R, we obtain This is a contradiction, and therefore = D 0 .

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

Liouville-type theorem
For a preparation, we prove Lemma 3 below.
Before starting the proof of Lemma 3, we introduce notations on affine hyperplanes. For c ∈ R n \ {0} and x 0 ∈ R n , we define In order to prove the Liouville-type theorem, we will make use of the supporting hyperplane theorem, which states that for any non-empty closed convex set C ⊂ R n and x 0 ∈ ∂C, there exists c ∈ R n \ {0} such that Proof. Since R n is connected and F ⊂ R n is a closed convex set, it follows that ∂ F = ∅. Without loss of generality we may assume that 0 ∈ ∂ F and u(0) = 0. By the supporting hyperplane theorem, we can take and fix a supporting hyperplane for F at the boundary point 0, which we write H c, 0 ⊂ R n . By rotation, we may assume that c = e 1 . Let D 1 be the connected component of D which contains H + e 1 , 0 ⊂ R n \ F = D. By the assumption 3 and u(0) = 0, it follows that there exists c ∈ R n \ {0} such that u(x) = c | x for all x ∈ D 1 . We should note that H c, 0 = H e 1 , 0 and hence c = t 1 e 1 for some t 1 ∈ (0, ∞), since otherwise it follows Again, by the condition 3 and similar arguments as those above, we can determine u| D 2 as u(x) = −t 2 e 1 | x for all x ∈ D 2 . Here t 2 ∈ (0, ∞) is a constant. Hence we obtain (5.2). For the case H e 1 , 0 ∂ F, we take and fix z 0 ∈ ∂ F \ H e 1 , 0 and a supporting hyperplane for F at z 0 , which we write by H c , z 0 . Let D 2 be the connected component of D which contains H + c , z 0 ⊂ D. By the assumption 3 and u(z 0 ) = 0, it follows that there exists c ∈ R n \ {0} such that u(x) = c x − z 0 for all x ∈ D 2 . Completely similarly to the arguments above for showing that H c, 0 = H e 1 , 0 , we can easily notice that H c , z 0 = H c , z 0 and hence c = t 1 c for some constant t 1 ∈ (0, ∞). Moreover, we also realize that c = t * e 1 for some t * ∈ R \ {0}. Otherwise it follows that the two hyperplanes H e 1 , 0 and H c , z 0 cross, and hence we get D 1 = D 2 and H + e 1 , 0 ∩ H − c , z 0 = ∅, which implies that there exists a point x 0 ∈ D such that u(x 0 ) < 0. This is clearly a contradiction. This result and convexity of u imply that D consists of two connected components D 1 = H + e 1 , 0 and D 2 = H + −e 1 , z 0 , and that F = {x ∈ R n | −l 0 ≤ x 1 ≤ 0}. Here l 0 := dist(H e 1 , 0 , H −e 1 , z 0 ) > 0. Finally we obtain the last possible expression (5.3). u can be expressed by either of (5.1)-(5.3).

Now we give the proof of Theorem 3.
Proof. Assume by contradiction that F, the facet of u, would satisfy ∅ F R n . Without loss of generality, we may assume that u attains its minimum 0. By the strong maximum principle (Theorem 2), the convex weak solution u is affine in each connected component of D := R n \ F. Therefore we are able to apply Lemma 3. By rotation and translation, u can be expressed as (5.1)-(5.3). Now we prove that u is no longer a weak solution to L b, p u = 0 in R n . We set open cubes Q := (−1, 1) n−1 ⊂ R n−1 and Q := (−d, d) × Q ⊂ R n , where d > 0 is to be chosen later. We claim that u does not satisfy L b, p u = 0 in W −1, p (Q). Assume by contradiction that there exists a vector field Z ∈ L ∞ (Q, R n ) such that the pair For the first case (5.1), we have |Z (x)| ≤ 1 for almost everywhere x ∈ Q, and by definition of Z . We also set another open cube Q l := (−d, 0) × Q ⊂ R n . We take and fix non-negative functions , and divide the integration over Q into that over Q l and Q r . Then (5.4) implies that Here we have applied the Gauss-Green theorem to the integration over Q r , and the Cauchy-Schwarz inequality to the integration over Q l . For the integrations I 1 and I 2 , we make use of Fubini's theorem and (5.5). Then we have Finally we obtain (5.6) From (5.6), we can easily deduce a contradiction by choosing sufficiently small d = d(b, p, t 1 , φ 2 ) > 0. Similarly we can prove that u defined as in (5.3) does not satisfy L b, p u = 0 in W −1, p (Q), since it suffices to restrict d < l 0 . We consider the remaining case (5.2). We have by definition of Z . We test the same function φ ∈ C 1 c (Q) in L b, p u = 0, then it follows that which is a contradiction. This completes the proof.
Remark 4. The estimate (5.6) breaks for p = 1, since the equation |0| p−2 0 = 0 is no longer valid for p = 1. This means that we have implicitly used differentiability of the function |z| p / p at 0 ∈ R n . Also it should be noted that for the one-variable case, functions as in (5.1), which are in general not in C 1 , are one-harmonic in R.

C 1 -Regularity theorem
We give the proof of Theorem 1.
Proof. We may assume that is convex. By [30, Theorem 25.1 and 25.5] and Lemma 1, it suffices to show that ∂u(x 0 ) = {0} for all x 0 ∈ F. Let x 0 ∈ F. We get a convex function u 0 : R n → R as a blow-up limit as in Proposition 1. We note that the facet of u 0 is non-empty by Proposition 1. Hence by the Liouville-type theorem (Theorem 3), u 0 is constant and we obtain ∂u 0 (x 0 ) = {0}. Combining these results, we have {0} ⊂ ∂u(x 0 ) ⊂ ∂u 0 (x 0 ) = {0} and therefore ∂u(x 0 ) = {0}. This completes the proof.

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

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

2.
For z 0 ∈ R n \ {0}, the Hessian matrix ∇ 2 z (z 0 ) satisfies 0 ≤ ∇ 2 z (z 0 )ζ ζ , (6.13) for all ζ, ω ∈ R n . Here the finite constant C is explicitly given by Lemma 4 states lower semicontinuity of a functional in the weak * topology of an L ∞ -space. This result is used in the justification of a blow-up argument for the Equation (6.1).

Lemma 4.
Let ⊂ R m be a Lebesgue measurable set, and let : R n → [0, ∞) be a convex function which satisfies (6.6). Assume that a vector field Z ∈ L ∞ ( , R n ) and a sequence (6.15) where˜ denotes the support function of the closed convex set We give an elementary proof of Lemma 4, which is based on a definition of˜ .
Proof. We consider the case C ∞ := lim inf Take arbitrary 0 ≤ φ ∈ L 1 ( ) and w ∈ C . Then with the aid of (6.9), we have for all j ∈ N and for almost everywhere x ∈ , which yields Since 0 ≤ φ ∈ L 1 ( ) is arbitrary, for each w ∈ C , there exists an L n -measurable set U w ⊂ , such that L n (U w ) = 0 and Here we denote L n by the n-dimensional Lebesgue measure. Since C ⊂ R n is separable, we may take a countable and dense set D ⊂ C ψ . We set an L nmeasurable set which clearly satisfies L n (U ) = 0. Then we conclude that Hence by definition of˜ , it is clear that Since ε > 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 Lu = 0 in a bounded domain.

Definition 3.
Let ⊂ R n be a domain.
1. Let f ∈ L q loc ( ) (n < q ≤ ∞). We say that a function u ∈ W 1, ∞ loc ( ) is a weak solution to (6.1), when for any bounded Lipschitz domain ω , there exists a vector field Z ∈ L ∞ (ω, R n ) such that the pair (u, for all φ ∈ W 1, 1 0 (ω), and Z (x) ∈ ∂ (∇u(x)) (6.19) for almost everywhere x ∈ ω. Here A denotes the continuous mapping A : R n x → ∇ z W (x) ∈ R n . For such pair (u, Z ), we say that (u, for all 0 ≤ φ ∈ C ∞ c ( ), and Z (x) ∈ ∂ (∇u(x)) for almost everywhere x ∈ . (6.21) Similarly we call a pair (u, Z ) ∈ W 1, p ( )× L ∞ ( , R n ) a weak supersolution L b, p u = 0 in , if it satisfies (6.21) and For u ∈ W 1, p ( ), we simply say that u is respectively a subsolution and a supersolution to Lu = 0 in the weak sense if there is Z ∈ L ∞ ( , R n ) such that the pair (u, Z ) is a weak subsolution and a weak supersolution to Lu = 0 in . 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 W 1, ∞ -regularity for solutions in Definition 3. Also it should be noted that if a vector field Z satisfies (6.19), then Z is in L ∞ by Remark 5. Hence our regularity assumptions of the pair (u, Z ) involve no loss of generality. 2. Integrals in (6.18) make sense by Z , ∇u ∈ L ∞ (ω, R n ), A ∈ C(R n , R n ), and the continuous embedding Then the pair (u, ∇ z (∇u)) ∈ W 1, p ( )× L ∞ ( , R n ) satisfies (6.20)-(6.21). For such u, we simply say that u satisfies Lu ≤ 0 in in the classical sense.
To prove Theorem 4, we may assume that 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 u to (6.1) satisfies (1.4) for all x ∈ .
For the case x ∈ D, we can show (1.4) by De Giorgi-Nash-Moser theory. This is basically due to the fact that the functional E(z) := (z) + W (z) for z ∈ R n satisfy the following property. For each fixed constants 0 < μ ≤ M < ∞, there exists constants 0 < λ ≤ < ∞ such that the estimates (2.7)-(2.8) hold for all z 0 , ζ, ω ∈ R n with μ ≤ |z 0 | ≤ M. In other words, the operator L is locally uniformly elliptic outside a facet, in the sense that for a function v the operator Lv becomes uniformly elliptic in a place where 0 < μ ≤ |∇v| ≤ M < ∞ 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 L outside the facet and De Giorgi-Nash-Moser theory, we are able to show that a convex solution to Lu = f is C 1, α near a neighborhood of each fixed point x ∈ D, similarly to the proof of Lemma 1.
For the case x ∈ F, we first make a blow-argument to construct a convex function u 0 : R n → R satisfying ∂u(x) ⊂ ∂u 0 (x), and Lu 0 = 0 in R n 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 D. 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 u 0 , which is constructed by the previous blow-argument, should be constant. Finally the inclusions {0} ⊂ ∂u(x) ⊂ ∂u 0 (x) ⊂ {0} hold, and this completes the proof of (1.4), that is, ∂u(x) = {0}.
For maximum principles on the equation Lu = 0, the proofs are almost similar to those in Section 4. Indeed, we first recall that the operator A : R n z 0 → ∇ z W (z 0 ) ∈ R n satisfies strict monotonicity (4.8). Combining with monotonicity of the subdifferential operator ∂ , we can easily prove a comparison principle as in Proposition 2. Also, similarly to Lemma 2, we can construct classical barrier subsolutions to Lu = 0 in an open annulus, since the operator L 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 u 0 , a limit of rescaled solutions, is a weak solution to Lu = 0 in R n , and this finishes our blow-up argument. Proof. For each N ∈ N, there exists a vector field Z N ∈ L ∞ (U, R n ) such that

Lemma 5. Let U ⊂ R n be a bounded domain. Assume that sequences of functions
Combining the assumption f N → f in L q (U ) with the continuous embedding By A ∈ C(R n , R n ) and (6.22), the vector fields where C is independent of N ∈ N. From these and Lebesgue's dominated convergence theorem, it follows that for some Z ∈ L ∞ (U, R n ). By (6.25)-(6.28) we obtain Now we are left to prove that By (6.10), it suffices to show that Z satisfies for almost everywhere x ∈ U . Similarly, it follows that for each N ∈ N, the vector field Z N satisfies Hence (6.29) is an easy consequence of Lemma 4. We recall (6.2), and thus ∂ (z 0 ) = {∇ z (z 0 )} holds for all z 0 ∈ R n \ {0}. Combining (6.23), we can check that Z N (x) → Z (x) for almost everywhere x ∈ D := {x ∈ U | ∇u(x) = 0}. Hence (6.30) holds for almost everywhere x ∈ D. Note that (6.30) is clear for x ∈ U \ D, and this completes the proof.
We prove a Liouville-type theorem as in Theorem 3. In other words, for a convex solution to Lu = 0 in R n , we show that F, the facet of u, would satisfy either F = ∅ or F = R n . Assume by contradiction that F satisfies ∅ F R n . Then by Lemma 3, we may write a convex solution u by either of (5.1)-(5.3). However, Lemma 6 below states that u is no longer a weak solution, and this completes our proof. Lemma 6. Let u be a piecewise-linear function defined as in either of (5.1)-(5.3). Then u is not a weak solution to Lu = 0 in R n .
Proof. As in the proof of Theorem 3, we introduce a constant d > 0, and set open cubes Q ⊂ R n−1 and Q, Q l , Q r ⊂ R n . By choosing sufficiently small d > 0, we show that u does not satisfy Lu = 0 in W −1, ∞ (Q). Assume by contradiction that there exists a vector field Z ∈ L ∞ (Q, R n ) such that the pair (u, Z ) satisfies We first show that a function u defined as in (5.1) is not a weak solution. For this case, (6.12) implies that Z satisfies Z (x) = ∇ z (e 1 ) for almost everywhere x ∈ Q r . We take and fix non-negative functions φ 1 ∈ C 1 c ((−d, d)), φ 2 ∈ C 1 c Q such that (5.5) holds, and define φ ∈ C 1 Here we have used the Cauchy-Schwarz-type inequality (6.5) for the integral over Q l , and applied the Gauss-Green theorem to the integration over Q r . For I 1 , we make use of (6.9)-(6.8), Fubini's theorem and (5.5). Then we have where ∇ x φ 2 := (∂ x 2 φ 2 , . . . , ∂ x n φ 2 ). For I 2 , recalling Euler's identity (6.11), we get I 2 = −φ 1 (0) (e 1 ) φ 2 L 1 (Q ) . We set a constant μ := A(t 1 e 1 ) | e 1 , which is positive by (6.7). Then we obtain Choosing d = d(μ, , φ 2 ) > 0 sufficiently small, we have 0 ≤ I 1 + I 2 + I 3 < 0, which is a contradiction. Similarly we can deduce that u defined as in (5.3) does not satisfy Lu = 0 in W −1, ∞ (Q), since it suffices to restrict d < l 0 . For the remaining case (5.2), we have already known that Z (x) = ∇ z (e 1 ) for almost everywhere x ∈ Q r , ∇ z (−e 1 ) for almost everywhere x ∈ Q l by definition of Z and (6.12). We set two constants μ 1 := A(t 1 e 1 ) | e 1 , μ 2 := A(−t 2 e 1 ) | −e 1 , both of which are positive by (6.7). Testing the same function which is a contradiction. Here we have used the Gauss-Green theorem and Euler's identity (6.11). This completes the proof.

A. Proofs for a Few Basic Facts
In this section, we give proofs for a few basic facts used in this paper for completeness.

A.1. A Poincaré-Type inequality
We give a precise proof of Lemma 7, a Poincaré-type inequality for difference quotients of functions in W Before the proof of Lemma 7, we note that j, h u(x) makes sense for almost everywhere x ∈ by the zero extension of u ∈ W 1, p 0 (U ). That is, for a given u ∈ W 1, p 0 (U ), we set u ∈ W 1, p (R n ) by Proof. We fix j ∈ { 1, . . . , n }, h ∈ R\{0}. We first note that the operator j, h : W Hence we obtain (A.1) for all u ∈ C ∞ c (U ), and this completes the proof.

A.2. Convex analysis
Lemma 8 is used in the proof of Lemma 1 for a justification of local W 2, 2 -regularity of a convex weak solution outside of the facet.
Remark 7. Instead of subgradient inequalities, we are able to show (A.3) by monotonicity of ∂u. For each fixed x 1 , x 2 ∈ with x 1 = x 2 , we may take and fix x 3 := x 1 + t (x 2 − x 1 ) for some 0 < t < 1 and z 3 ∈ ∂u(x 3 ) such that with the aid of the mean value theorem for non-smooth convex functions [1,Theorem D.6].
The following lemma is used in the proof of Proposition 1. Our proof of Lemma 9 is inspired by [11,Lemma A.3].
Proof. We define L n -measurable sets P N := {x ∈ U | u N is not differentiable at x} for N ∈ N ∪ {∞}.
Clearly P N (N ∈ N ∪ {∞}) satisfies L n (P N ) = 0 by Lipschitz continuity of u N , and therefore the L n -measurable set P := N ∈N∪{∞} P N ⊂ U also satisfies L n (P) = 0. We claim that ∇u N (x 0 ) → ∇u ∞ (x 0 ) for all x 0 ∈ U \ P.

(A.7)
We take and fix arbitrary x 0 ∈ U \ P. We note that ∇u N (x 0 ) exists for each N ∈ N since x 0 ∈ P N , and we obtain sup N ∈N |∇u N (x 0 )| ≤ L with the aid of (A.5). Hence it suffices to check that, if a subsequence {u N k } k ⊂ {u N } N satisfies then v = ∇u ∞ (x 0 ). Since x 0 ∈ P N k and therefore ∂u N k (x 0 ) = {∇u N k (x 0 )} for each k ∈ N, we easily get Letting k → ∞, we have by (A.6) and (A.8). This means that v ∈ ∂u ∞ (x 0 ). Note again that x 0 ∈ P ∞ and therefore ∂u ∞ (x 0 ) = {∇u ∞ (x 0 )}, which yields v = ∇u ∞ (x 0 ). This completes the proof of (A.7).

A.3. Convex functionals
We prove some basic property of convex functionals and W in Section 6. Lemma 10. Let W be a convex function which satisfies (6.2)-(6.3) and (6.5). Then the mapping A : R n z → ∇W (z) ∈ R n satisfies strict monotonicity (4.8).