A mean value formula for the variational $p$-Laplacian

We prove a new asymptotic mean value formula for the $p$-Laplace operator, $$ \Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u), $$ valid in the viscosity sense. In the plane, and for a certain range of $p$, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.

In this paper, we study a new asymptotic mean value property for p-harmonic functions, i.e., solutions of the equation ∆ p u = 0.
Our result implies in particular that a function is p-harmonic at a point x if and only if it is satisfies Br |u(x + y) − u(x)| p−2 (u(x + y) − u(x))dy = o(r p ), as r → 0.
The major strength and novelty of our mean value formula is that it recovers the variational p-Laplace operator (1.1) in contrast to the other known mean value formulas that recover the normalized p-Laplacian, In particular, it allows us to deal with non-homogeneous problems of the form −∆ p u = f with f = 0, which was not possible with previous approaches. The drawback is that it cannot be written in the form for some monotone operator A r . However, the mean value formula is still monotonically increasing in u and monotonically decreasing in u(x), which is decisive in the context of viscosity solutions.
Our first result, that will be proved in Section 6, provides the mean value formula for C 2 functions. It reads: Theorem 2.1. Let p ∈ (1, ∞), x ∈ R d and φ ∈ C 2 (B R (x)) for some R > 0. If p ∈ (1, 2) assume also that |∇φ(x)| = 0. Then We refer to Section 7 for the proof of the above result, and to Section 5 for the definition of viscosity solutions.
Our third result states that in the plane, and for a certain range of p, functions that satisfy the (homogeneous) mean value property in a pointwise sense are the same as the p-harmonic functions. Let p 0 be the root of 2 −p + (36(p − 1) + (p − 2) 2 − p + 16(p − 1) + (p − 2) 2 2 = 1 p − 1 that lies in the interval (1,2). We have p 0 ≈ 1.117. in Ω.
We refer to Section 8 for the proof of this theorem and to page 4 for a heuristic explanation on the technical limitation p > p 0 .
Remark 2.4. Theorem 2.2 and Theorem 2.3 remain true if one replaces I p r by M p r . The fourth and the last of our main results concerns the associated dynamic programming principle. Consider the following boundary value problem where G is a continuous extension of g from ∂Ω to ∂Ω r .
Theorem 2.5. Let Ω ⊂ R d be a bounded, open and C 2 domain, p ∈ (1, ∞), f be a continuous function in Ω and g a continuous function on ∂Ω. Then i) there is a unique classical solution U r of (2.1), ii) U r → u as r → 0 uniformly in Ω, where u is the viscosity solution of Remark 2.6. We have stated all our results in the context of viscosity solutions. Since weak and viscosity solutions are equivalent (cf. [12] and [11]), the same results hold true for weak solutions.

Related results
Recently, there has been a surge of interest around mean value properties of equations involving the p-Laplacian. In [19], it is proved that a function is p-harmonic if and only if This was first proved to be valid in the viscosity sense. In [18], this was proved to hold in the pointwise sense, in the plane and for 1 < p <p ≈ 9.52. Shortly after, this was extended to all p ∈ (1, ∞), in [3]. Linked to a mean value formula, there is a corresponding dynamic programming principle (DPP), which is the solution U r of the problem U r = A r [U r ] subject to the corresponding boundary conditions. The typical result is to show that U r → u where u is a viscosity solution of the boundary value problem associated to the p-Laplacian.
The above mentioned results are based on the following identity for the so-called normalized p-Laplacian and the now well-known mean value formulas for the Laplacian and ∞-Laplacian. More precisely, for a smooth function φ, for some constant C d,p > 0. In the last years, several other mean value formulas for the normalized p-Laplacian have been found, and the corresponding program (equivalence of solutions in the viscosity and classical sense and study of the associated dynamic programming principle) has been developed. See for instance [2], [6], [8], [13], [15], [16], [17], and [21].
We also want to mention [7] and [9], where two other nonlinear mean value formulas are studied, with some similarities with ours.
It is noteworthy to mention that our results are also related to asymptotic mean value formulas for nonlocal operators involving for instance fractional or non-local versions of the p-Laplacian. See [1] and [5].

Comments on our results
Comments on Theorem 2.3. The curious reader might wonder why we are not able to prove the pointwise validity for the mean value formula for the full range p ∈ (1, ∞), as in [3]. To make a long story short this has to do with the fact that the mean value formula considered in [3] has quadratic scaling. It is therefore enough with an error term of order strictly larger than 2. The mean value formula in the present paper however, has scaling p/(p − 1), which makes it necessary with an error term of order strictly larger than p/(p − 1). When p < 2, this certainly comes with some difficulties that for the moment forces us to assume the larger lower bound p > p 0 . However, we still believe that such a result holds in the full range p ∈ (1, ∞).
Comments on the limit p → 1. Formally, when p = 1 the mean value formula becomes Br sign(u(x + y) − u(x))dy = o(r), as r → 0, which could give a characterization of 1-harmonic functions. We plan to study this possibility in the future.
More general datum. It would also be interesting to study problems where f = f (x, u, ∇u(x)) has the right monotonicity assumptions as described in [20]. Theorem 2.2 follows in a straightforward way. However, the convergence of dynamic programming principles like in Theorem 2.5 would require a more delicate study, both in terms of existence and properties of the r-scheme, and the study of convergence based on the Barles-Souganidis approach.
Plan of the paper. The plan of the paper is as follows. In Section 5, we introduce some notation and the notions of viscosity solutions. This is followed by Section 6, where we prove the mean value formula for C 2 functions. This result is then used in Section 7, where we prove the mean value formula for viscosity solutions. In Section 8, we prove that in dimension d = 2, and for a certain range of p, functions that satisfy the (homogeneous) mean value property in a pointwise sense are the same as the p-harmonic functions. In Section 9, we study existence, uniqueness and convergence for the dynamical programming principle. Finally, in the Appendix, we prove and state some auxiliary inequalities.

Notation and prerequisites
Throughout this paper, d will denote the dimension and we will for p ∈ (1, ∞) use the notation We now define viscosity solutions of the related equations and mean value properties: Definition 5.1 (Viscosity solutions of the equation). Suppose that f is continuous function in Ω. We say that a lower (resp. upper) semicontinuous function u in Ω is a viscosity supersolution (resp. subsolution) of the equation in Ω if the following holds: whenever x 0 ∈ Ω and ϕ ∈ C 2 (B R (x 0 )) for some R > 0 are such that |∇ϕ(x 0 )| = 0, A viscosity solution is a continuous function being both a viscosity supersolution and a viscosity subsolution.
It is well known that regarding viscosity solutions of equations involving the p-Laplace operator, there is no need to test at critical point of the test function, see for instance page 703 in [12].
Definition 5.2 (The mean value property in the viscosity sense). Suppose that f is continuous function in Ω. We say that a lower (resp. upper) semicontinuous function u in Ω is a viscosity supersolution (resp. subsolution) of the equation in Ω if the following holds: whenever x 0 ∈ Ω and ϕ ∈ C 2 (B R (x 0 )) for some R > 0 are such that |∇ϕ(x 0 )| = 0, . A viscosity solution is a continuous function being both a viscosity supersolution and a viscosity subsolution.
Remark 5.1. The above definition can also be considered with M p r instead of I p r . Finally, we define the concept of viscosity solution for the the boundary value problem (2.2).

Definition 5.3 (Viscosity solutions of the boundary value problem)
. Suppose that f is continuous function in Ω, and that g is a continuous function in ∂Ω. We say that a lower (resp. upper) semicontinuous function u in Ω is a viscosity supersolution (resp. subsolution) of (2.2) if (a) u is a viscosity supersolution (resp. subsolution) of −∆ p u = f in Ω (as in Definition 5.1); A viscosity solution of (2.2) is a continuous function in Ω being both a viscosity supersolution and a viscosity subsolution.

The mean value formula for C 2 -functions
In this section we prove the mean value formulas for C 2 -functions as presented in Theorem 2.1. The proof is split into two different cases: p > 2 and p < 2. The case p = 2 is well known so we leave that out. We restate the results for convenience.
Since the first term is odd and we are integrating over a sphere in (6.1), we get Without loss of generality, assume that, ∇φ(x) = ce 1 for some c ≥ 0. Note that this assumption implies that |∇φ(x)| = c and ∆ ∞ φ(x) = c 2 D 11 φ(x). The symmetry of the integral and the term y T D 2 φ(x)y imply that Note that if d ≥ 2, for all i = 1 we have that Thus, Now, from identity (3.1) we get which concludes the proof.
We now proceed to the case p < 2, which is slightly more involved.
Proof. We keep the notation A r of (6.1). Without loss of generality, we assume that ∇φ(x) = ce 1 for some c > 0. We split the proof into several parts. Part 1: First we prove an estimate that will be used several times along the proof. Let α ∈ (0, 1) and ρ ≥ 0 small enough. Then To prove (6.2), we first note that its left hand side is equal to Part 2: In this part, we prove By Taylor expansion, whereŷ := y/|y|. Thus, where the last identity follows from applying (6.2) with ρ = r (choosing r small enough). Part 3: This part amounts to proving that where C γ → 0 as γ → 0. First we note that with our notation we have By antisymmetry ∂Br ∩{|ŷ·e1|≤γ} This, (6.2) with α = (p − 2)(1 + δ) > −1 and ρ = 0, and Hölder's inequality imply

Part 4:
We will now prove that for fixed γ > 0, Here it is crucial that the integrals are restricted to the set {|ŷ · e 1 | > γ}.
We observe that outside ξ = 0 the function ξ → J p (ξ) is smooth. In particular, for a = 0 and b such that |b| < |a|/2, we have the following estimate For any y such that |ŷ · e 1 | > γ, the above estimate with a =ŷ · e 1 and b = 1 2 c −1 |y|ŷ T D 2 φ(x)ŷ (since a = 0 and b < γ/2 < |a|/2 by choosing r = |y| small enough), together with (6.3) imply where R(y) is bounded by For some constants C 3 , C 4 ≥ 0 and r small enough (depending on γ). Moreover, by antisymmetry, We apply (6.2) with α = −p + 2 + δ ∈ (0, 1) two times, first with ρ = r and later with ρ = 0 to get where the bound is uniform for fixed γ. This implies (6.4). Part 5: From parts 2 and 3 we have Since the last term is independent of r and converges to as γ → 0, where the last equality follows from the proof of Theorem 6.1, the result follows.

Viscosity solutions
Now we prove that satisfying the asymptotic mean value property in the viscosity sense is equivalent to being a viscosity solution of the corresponding PDE.
Proof of Theorem 2.2. We only prove that the notion of supersolutions are equivalent. The case of a subsolution can be treated similarly. Suppose first that u is a viscosity supersolution of −∆ p u = f in Ω. Take x 0 ∈ Ω and ϕ ∈ C 2 (B R (x 0 )) for some R > 0 such that |∇ϕ(x 0 )| = 0, By the definition of a supersolution, and the proof is complete.

The pointwise property in the plane
Now we are ready to prove that the mean value property is satisfied in a pointwise sense in the aforementioned range of p. Assume now instead that u is a viscosity solution of −∆ p u = 0 and let x 0 ∈ Ω. If |∇u(x 0 )| = 0, then u is real analytic near x 0 and the mean value formula holds trivially at x 0 by Theorem 2.1. If |∇u(x 0 )| = 0 we need different arguments depending on p.
Case p ≥ 2: The case p = 2 is well-known and we do not comment on it. If p > 2, Theorem 1 in [10] implies that u ∈ C 1,α for some 1 > α > 1/(p − 1). Then (1), which ends the proof in this case.
Case p 0 < p < 2: First we use that on page 146 in [18] it is proved that for some integer n ≥ 1 we have that In particular, when n ≥ 3 we have that 1/η n < p − 1 which implies |D 2 u| = o r 1 p−1 −1 in B r (x 0 ). By Taylor expansion we thus get, (8.2). We still need to check the cases n = 1 and n = 2. We do it in several steps.
Step 3. Now we go back to u. Using (8.3), we have together with Lemma A.2 with γ given in (8.4). By Step 2, A satisfies the mean value property at x 0 and thus |I p r [u](x 0 )| ≤ Cr −p+(p−1)γ . The proof will be finished if we verify that γ > p/(p − 1), that is, First we verify (8.5) when n = 1. In this case This inequality is exactly true when p ∈ (p 0 , 2). If n = 2 then This inequality turns out to be true for p > 1.06 and therefore it is true for p > p 0 .

Study of the dynamic programming principle
Recall the notation Given an open domain Ω and r > 0, we will in this section denote by and Ω r = Ω ∪ ∂Ω r . We want to study solutions of the (extended) boundary value problem where f ∈ C(Ω) and G ∈ C(∂Ω r ) (a continuous extension of g ∈ C(∂Ω)). These will be our running assumptions in this section.
9.1. Existence and uniqueness: The proof of Theorem 2.5 i). For convenience, we will write M p instead of M p r when the subindex r plays no role. We first prove a comparison principle which immediately implies uniqueness and then we prove the existence.
Proof. Assume by contradiction that U (x) > V (x) for some x ∈ Ω. It has to be in the interior of Ω since by definition U ≤ G ≤ V in ∂Ω r . Let M > 0 and x 0 ∈ Ω be such that By the monotonicity of J p From the equations satisfied by U and V we have Hence, the average of the non-negative integrand is non-positive. This means that By the strict monotonicity of J p this implies that is, V (x 0 + y) =Ũ (x 0 + y) for all y ∈ B r . This means thatŨ (x) = V (x) for all x ∈ B r (x 0 ). Repeating this process in the contact points ofŨ and V and iterating, we will eventually arrive at the conclusion thatŨ (x) = V (x) for some x ∈ ∂Ω r . This contradicts the factŨ < V in ∂Ω r .
In order to prove the existence and to study the limit as r → 0, we will first derive uniform bounds (in r) for the solution of (9.1).
Proposition 9.2 (L ∞ -bound). Let p ∈ (1, ∞), let R > 0 and U r be the solution of (9.1) corresponding to some r ≤ R. Then U r ∞ ≤ A with A > 0 depending on p, Ω, f, g and R (but not on r).
Proof. Consider the function h(x) = |x| Let C, D ∈ R and z ∈ R d to be chosen later and define Then ψ ∈ C ∞ (Ω R ). By Finally, we choose C such that ψ(x) ≥ G ∞ for all x ∈ ∂Ω R . Thus for all r ≤ R. Then, by comparison (Proposition 9.1) Note that this bound depends on R but not on r. A similar argument with −ψ as barrier shows that U (x) ≥ − ψ ∞ and thus, which concludes the proof.
The aim is now to prove the existence of a solution of (9.1). Before doing that, we need some auxiliary results. Define (b) Let ψ 1 and ψ 2 be such that Proof. We start by proving the comparison principle. This will imply uniqueness. Assume that ψ 1 (x) > ψ 2 (x) for some x ∈ Ω. Then which is a contradiction. To prove existence we start by defining

Since sup
By defining ψ I (x) = inf we may prove that −L[ψ I , φ](x) ≤ f (x) in a similar manner. By continuity we can conclude that for every x ∈ Ω, there exists a value We may then define ψ(x) := a x for all x ∈ Ω. Clearly, for all x ∈ Ω, so the existence is proved.
We are now ready to prove the existence.
Proof. Consider h to be the barrier function constructed in Proposition 9.2 (denoted there by ψ), i.e., h is such that Note that if x ∈ ∂Ω r then .
We define the sequence U k as the sequence of solutions of As long as U k−1 is bounded, U k exists by Lemma 9.3(a). We now prove that U k+1 (x) ≥ U k (x) in Ω r by induction. We start by proving that Assume towards a contradiction that Thus, −f (x) > f ∞ , which is clearly a contradiction. We conclude that By comparison (Lemma 9.3(b)), U k+1 ≥ U k . Thus the induction is complete and the claim is proved.
We will now verify that U k is uniformly bounded from above by G ∞ . We argue that U 0 (x) ≤ h(x) as follows. If x ∈ ∂Ω r , then If instead x ∈ Ω, then On the other hand, if x ∈ Ω, then (Lemma 9.3(b)) implies that U k+1 ≤ h and thus proves the claim.
We conclude that for every x ∈ Ω r , the sequence U k (x) is non-decreasing and bounded from above. We can then define the limit By the monotone convergence theorem so that U is a solution of (9.1). 9.2. Convergence: The proof of Theorem 2.5 ii). The proof of the convergence is based on the numerical analysis technique introduced by Barles and Souganidis in [4]. We follow the outline of [6], where this technique was adapted to problems involving the p-Laplacian. 9.2.1. The strong uniqueness property for the boundary value problem. Our approximate problem (9.1) will produce a sequence of solutions that converges to a so-called generalized viscosity solution (see below). To complete our program we need to ensure that this solution is unique and coincides with the usual viscosity solution.
Definition 9.1 (Generalized viscosity solutions of the boundary value problem). Let f be a continuous function in Ω and g a continuous function in ∂Ω. We say that a lower (resp. upper) semicontinuous function u in Ω is a generalized viscosity supersolution (resp. subsolution) of (2.2) in Ω if whenever x 0 ∈ Ω and ϕ ∈ C 2 (B R (x 0 )) for some R > 0 are such that |∇ϕ(x 0 )| = 0, We need the following uniqueness results for the generalized concept of viscosity solutions.
Theorem 9.5 (Strong uniqueness property). Let Ω be a C 2 domain. If u and u are generalized viscosity subsolutions and supersolutions of (2.2) respectively, then u ≤ u.
The above result for standard viscosity solutions is well known (see Theorem 2.7 in [12]). The proof of Theorem 9.5 follows from this fact together with the following equivalence result between the two notions of viscosity solutions. Proposition 9.6. Let Ω be a C 2 domain. Then u is a viscosity subsolution (resp. supersolution) of (2.2) if and only if u is a generalized viscosity subsolution (resp. supersolution) of (2.2).
Proof. We prove the statement for subsolutions. Clearly if u is a viscosity subsolution, then it is also a generalized viscosity subsolution since The proof of the other implication is essentially contained in [6]. We spell out the details below.
Assume u is a generalized viscosity subsolution. Fix a point x 0 ∈ ∂Ω and define, for ε > 0 small enough, the following function As it is shown in the proof of Theorem 3.4 in [6], this is a suitable test function at some point y ε ∈ Ω as in Definition 9.1. Moreover, u(x 0 ) ≤ u(y ε ) for all ε > 0 small enough and y ε → x 0 as ε → 0. By direct computations, it is also shown in step four of the proof of Theorem 3.4 in [6] that for constants C 1 , C 2 , C 3 > 0. From here, it is standard to get that there exists a constant C > 0 and ε 0 > 0 such that for all ε < ε 0 This implies that y ε ∈ ∂Ω. Indeed, if y ε ∈ Ω then by definition of generalized viscosity subsolution we have −∆ p ϕ ε (y ε ) − f (y ε ) ≤ 0. Since y ε ∈ ∂Ω, then we have by definition that min{−∆ p ϕ(y ε ) − f (y ε ), u(y ε ) − g(y ε )} ≤ 0 which implies that u(y ε ) − g(y ε ) ≤ 0. Finally, using the fact that u(x 0 ) ≤ u(y ε ) and taking the limit as ε → 0, we obtain u(x 0 ) − g(x 0 ) ≤ 0, since g is continuous. This shows u is a viscosity subsolution.
Note that the restriction of having a C 2 domain in the proposition above comes from the fact that we need the distance function to be C 2 close to the boundary. 9.2.2. Monotonicity and consistency of the approximation. For convenience we define Note that (9.1) can then be equivalently formulated as S(r, x, U r (x), U r ) = 0 x ∈ Ω r .

9.2.3.
Proof of the convergence. The only thing left to show is the convergence stated in Theorem 2.5. Once we have proved monotonicity and consistency as stated in Lemma 9.7, the proof follows as explained in Section 4.3 of [6].
Proof of Theorem 2.5 ii). Define By definition u ≤ u in Ω. If we show that u (resp. u) is a generalized viscosity subsolution (resp. supersolution) of (2.2), the strong uniqueness property of Theorem 9.5 ensures that u ≤ u.
Thus, u := u = u is a generalized viscosity solution of (2.2) and U r → u uniformly in Ω (see [4]). Proposition 9.6 ensures that u is a viscosity solution (2.2). We need to show that u is a generalized viscosity subsolution. First note that u is an upper semicontinuous function by definition, and it is also bounded since U r is uniformly bounded by Proposition 9.2. Take x 0 ∈ Ω and ϕ ∈ C 2 (B R (x 0 )) such that u( We claim that we can find a sequence (r n , y n ) → (0, x 0 ) as n → ∞ such that To show this, we consider a sequence (r j , x j ) → (0, x 0 ) as j → ∞ such that U rj (x j ) → u(x 0 ), which exists by definition of u. For each j, there exists y j such that where we in the third inequality have used (9.4). This together with (9.2) implies thatŷ = x 0 and thus finishes proof of the claim. Choose now ξ n := U rn (y n ) − ϕ(y n ). We have from (9.3) that, U rn (x) ≤ ϕ(x) + ξ n + e −1/rn for all x ∈ Ω ∩ B R (x 0 ).

Appendix A. Auxiliary inequalities
We need some technical results.
Proof. It follows from the Taylor expansion of the function J p (t) = |t| p−2 t.
The following inequality is Lemma 3.4 in [14].
Here C only depends on p.
We also need the following lemma. where C only depends on s and d.
Hence the integral over that interval is bounded by some constant.