Asymptotic mean-value formulas for solutions of general second-order elliptic equations

1 Introduction

It is well-known that mean-value formulas characterize harmonic functions; in fact, a weaker statement, known as the asymptotic mean-value property, is enough to characterize harmonicity (see [7,18,31]). Furthermore, we have

from which we conclude that, when f is continuous, a function u satisfies the classical Poisson equation Δ u ( x ) = f ( x ) in Ω if and only if

for each x ∈ Ω . We use the standard notation o ( ε 2 ) to denote a quantity such that o ( ε 2 ) / ε 2 → 0 as ε → 0 . Analogous results hold for sub- and supersolutions replacing equalities by appropriate inequalities.

The classic mean-value property can be seen as a nonlocal formulation of the local Laplace and Poisson equations. It has deep connections with other fundamental properties such as the maximum principle, which allows proving existence results by Perron’s method or symmetry of solutions by the moving-planes method. In this sense, the mean-value property and the maximum principle are nothing more than a quantitative expression of the monotonicity inherent to the equation.

Over the past 10 years, there has been increasing interest in whether the mean value property holds in some form for nonlinear operators. This question was in part motivated by the surprising connection between harmonious extensions, the normalized infinity Laplacian, and dynamic programming principles for Random Tug-of-Wag games, discovered in [19,20,29]. Lately, a partial differential equation (PDE) revisitation of the dynamic programming principle for the p -Laplacian, introduced in [30], has been performed in [13], see also [25].

In our recent paper [4], we established asymptotic mean-value properties in the viscosity sense for solutions to the Monge-Ampère equation,

which is elliptic only in the class of convex functions and, consequently, requires f ≥ 0 . Our results in [4] are based on the formula

which holds whenever D 2 u ( x ) ≥ 0 . Mean-value properties in the viscosity sense require more geometrical arguments adapted to the operator. In particular, our methods do not require explicit representation formulas, making them flexible enough to be applied to various nonlinear problems.

In this article, we establish asymptotic mean-value formulas for a wide array of fully nonlinear equations that include degenerate operators such as the k -Hessians, [11], truncated Laplacians [3], and prescribed eigenvalues of the Hessian [5,28]. The operators we discuss can be written as an infimum of linear operators with coefficients chosen from a given set, see below for examples. Of course, there are corresponding statements for equations involving a supremum or combinations of infimum and supremum.

Mean-value formulas hold under more lenient regularity conditions than the corresponding PDEs and can provide a simple and unified approach to nonlinear equations in nonEuclidean contexts such as Carnot groups. As a proof of concept, we prove mean-value formulas for Monge-Ampère operators in the Heisenberg group in Section 6. Other directions for applications of the asymptotic mean-value formulas below concern game-theoretic interpretations of the corresponding PDEs and their numerical analysis. In this regard, there are convergent difference schemes for the normalized infinity Laplacian and p -Laplacian using mean-value formulas, see [27]. Hence, the mean-value formulas that we develop here could provide new numerical methods for the corresponding nonlinear equations.

Let us now describe our results. First, we consider differential operators of the form F ( D 2 u ) , with F : S n ( R ) → R ∪ { − ∞ } given by

Here, A is a subset of S + n ( R ) , the set of symmetric positive semi-definite matrices. We observe in Lemma 4.1 that the set A is bounded if and only if the operator F ( M ) is well-defined (finite) for all M ∈ S n ( R ) . Therefore, A determines the set of admissible solutions, functions for which F ( D 2 u ) is finite. We say that

where we define the cone

(see [11] for a related notion of admissibility). This condition plays an analogous role to the convexity for the Monge-Ampère equation; in fact,

for the unbounded set A = { A ∈ S + n ( R ) : det ( A ) = 1 } , see [4].

Observe that F is an infimum of linear functions, which are continuous; therefore, it is upper semi-continuous. Even more, F is concave and continuous in Γ A ∘ . We assume that

so that we have that

Condition (1.6) is not satisfied in Example 4.6 and the mean-value formula fails.

To deal with operators of the form (1.3), where the class of matrices A is unbounded, we consider matrices A such that A ∈ A and A ≤ ϕ ( ε ) I with ϕ ( ε ) a positive function such that

(an example of such a function is ϕ ( ε ) = ε − α for 0 < α < 1 ). Note that the condition A ≤ ϕ ( ε ) I becomes less restrictive as ε → 0 but is still enough to make the mean-value formula local, see Section 4. We obtain the following result.

As a consequence, we can characterize viscosity solutions to the equation F ( D 2 u ( x ) ) = f ( x ) in terms of an asymptotic mean-value formula in the viscosity sense. For the precise definition of the notion of viscosity solutions and viscosity mean-value formulas, see Section 2, [12], and [25]. Informally, an equation or a mean-value property holds in the viscosity sense when they hold with an appropriate inequality (instead of an equality) for smooth test functions that touch u from above or below at x .

The concept of a mean-value formula in the viscosity sense is weaker than a mean-value formula that holds pointwise. For the infinity Laplacian and the Monge-Ampère equation, there are instances of asymptotic mean-value formulas that hold in the viscosity sense but do not hold pointwise, see [25] and [4], respectively. An interesting question is, under what circumstances do the viscosity mean-value formulas outlined above hold pointwise. This question has an obvious affirmative answer for all equations for which viscosity solutions are known to be classical since, in that case, we can apply the pointwise formula for regular functions in (1.9). Nevertheless, there are examples of nonclassical viscosity solutions for which mean-value formulas hold pointwise; see [2,4,23].

A fundamental example of application of Theorems 1.1 and 1.2 is the k -Hessian operators, given by the elementary symmetric polynomials

evaluated in the eigenvalues of the Hessian, { λ i ( D 2 u ) } 1 ≤ i ≤ n . Here, k = 1 , 2 , … , n and the cases k = 1 and k = n correspond to the Laplacian and Monge-Ampère, respectively. For these operators to fit our framework we need to write them in the form

according to the following characterization, which we prove in Section 5.

An important consequence of Lemma 1.4 is that for the k -Hessian operators, A -admissibility is equivalent to the notion of k -convexity in [33,34,35], see Corollary 5.3. This means that

and

A condition such as (1.11) could be used in place of (1.6) to prove the mean-value property. In fact, (1.11) may even appear more natural at first; for instance, it was used in [15] in relation to the existence and uniqueness of solutions to general fully nonlinear second-order PDEs. However, condition (1.11) implies (1.6), but not vice versa (see Remark 4.4 and Example 4.5), making condition (1.6) more general.

In this case Theorem 1.1 reads as follows, see Section 5 for details.

It is also possible to consider operators where the class of matrices A depends on the point x . This happens naturally when we consider mean-value formulas for Monge-Ampère operators in the Heisenberg group in Section 6. In that case, the sets A depend on the point x and are unbounded.

We also obtain a rich family of examples when considering operators for which A x ⊂ S + n ( R ) is bounded for each x ∈ Ω . In fact, we consider sup-inf operators where for every x , the supremum is taken over a subset A x of P ( S + n ( R ) ) , the power set of S + n ( R ) . This is not done only for the sake of generality. It is motivated by examples that cannot be covered otherwise, such as prescribed eigenvalues of the Hessian and Isaacs operators. These operators can be degenerate elliptic since we do not impose any lower bounds on the eigenvalues of the matrices A ∈ A x . We provide a list of examples below, which shows the flexibility of the approach.

Therefore, let us now consider differential operators of the form F ( x , D 2 u ( x ) ) with F : Ω × S n ( R ) → R given by

Here, A x ⊂ P ( S + n ( R ) ) (the power set of S + n ( R ) ) is a nonempty subset for each x ∈ R n and we assume that

Observe that if A x contains only one element A x for each x , then (1.13) is equivalent to

with A x ⊂ S + n ( R ) bounded for each x ∈ R n . On the other hand, if every A x is a set of singletons, (1.13) is equivalent to

where A x = { A ∈ S + n ( R ) : A ∈ A for some A ∈ A x } . One can also consider inf-sup operators of the form

with straightforward adaptations in the statements and proofs.

By Lemma 4.1, (1.14) is equivalent to the operator M ↦ F ( x , M ) being well-defined and finite for every M ∈ S n ( R ) ; in particular, every u ∈ C 2 ( Ω ) is admissible. Moreover, we prove in Lemma 3.2 that M ↦ F ( x , M ) is Lipschitz continuous for every x ∈ Ω .

We obtain the following counterparts of Theorems 1.1 and 1.2. Observe that when ∪ A x is bounded, the condition A ≤ ϕ ( ε ) I in (1.9) becomes unnecessary because it is satisfied at every point for ε small enough.

We have a corresponding result in the line of Theorem 1.2, characterizing viscosity solutions to the equation F ( x , D 2 u ( x ) ) = f ( x ) by an asymptotic mean-value formula in the viscosity sense.

Let us now provide some examples of operators of the form (1.13), (1.15), and (1.16) and the corresponding sets A x and A x . In the sequel, we consider the eigenvalues of a matrix M ∈ S n ( R ) arranged in increasing order, that is, λ 1 ( M ) ≤ λ 2 ( M ) ≤ ⋯ ≤ λ n ( M ) .

1. Operators of the form (1.13), (1.17) include the usual Isaacs operators (see [8]) defined as

   F ( x , M ) = sup α ∈ A inf β ∈ ℬ trace ( A α β t M A α β ) and F ( x , M ) = inf α ∈ A sup β ∈ ℬ trace ( A α β t M A α β )

   for a given family of matrices { A α β } α ∈ A , β ∈ ℬ (and similarly in the inf-sup case). Here we take A = { { A α β } β ∈ ℬ } α ∈ A in (1.13), with A independent of the point x . In fact, operators of the form (1.13) could be seen as Issacs operators, but the usual Isaacs condition

   inf α ∈ A sup β ∈ ℬ trace ( A α β t M A α β ) = sup β ∈ ℬ inf α ∈ A trace ( A α β t M A α β )

   does not seem to have a clear counterpart. A typical requirement in the literature is that the family { A α β } α ∈ A , β ∈ ℬ is uniformly elliptic, i.e., it is uniformly bounded between two constants 0 < θ < Θ . In this case, the resulting Issacs operator is uniformly elliptic. We want to emphasize that here we are only requiring ∪ A x to be bounded and, therefore, the operators (1.13) may be degenerate elliptic.

2. We consider the truncated Laplacians [3], defined as

   P k − ( D 2 u ) = ∑ i = 1 k λ i ( D 2 u ) and P k + ( D 2 u ) = ∑ i = 1 k λ n + 1 − i ( D 2 u ) ,

   for k = 1 , 2 , … , n − 1 (for k = n these operators coincide with the Laplacian). These degenerate operators appear naturally in geometric problems, when considering manifolds of partially positive curvature [32,37], or mean curvature flow in arbitrary codimension [1], see also [3,10,15,16] and references therein. The operators P k − and P k + are of the form (1.15) and (1.16), respectively, for the set

   A = { A ∈ S + n ( R ) : λ 1 = ⋯ = λ n − k = 0 and λ n − k + 1 = ⋯ = λ n = 1 } .

3. Operators of the form (1.13) allow us to consider degenerate operators such as the kth smallest eigenvalue of the Hessian, given by the Courant-Fischer min–max principle

   (1.19) λ k ( D 2 u ( x ) ) = max V { min v ∈ V , ∣ v ∣ = 1 ⟨ D 2 u ( x ) v , v ⟩ : V ⊂ R n subspace , dim ( V ) = n − k + 1 } .

   We can write the operator λ k ( D 2 u ) in the form (1.13) for the set

   A = { { A ∈ S + n ( R ) : λ i ( A ) = 0 for i ≠ n , λ n ( A ) = 1 , v n ∈ V } : V ⊂ R n subspace of dimension n − k + 1 } ,

   where v n is the eigenvector corresponding to λ n ( A ) . The cases k = 1 and k = n were studied in [5,15,28] in connection with the convex and concave envelope of a function; i.e., the unique viscosity solutions of λ 1 ( D 2 u ( x ) ) = 0 and λ n ( D 2 u ( x ) ) = 0 are, respectively, the convex and concave envelopes of u ∣ ∂ Ω in Ω . These operators are of the form (1.15), (1.16) with the set of matrices

   A = { A ∈ S + n ( R ) : λ 1 ( A ) = ⋯ = λ n − 1 ( A ) = 0 and λ n ( A ) = 1 } .

4. When the mean-value formula involves averages over balls that are not centered at 0 but at ε 2 v with ∣ v ∣ = 1 , we obtain operators with first-order terms. For example, we have

   inf A ∈ A x ⨍ B ε ( ε 2 v ) u ( x + A y ) d y − u ( x ) = ε 2 inf A ∈ A x 1 2 ( n + 2 ) trace ( A t D 2 u ( x ) A ) + ⟨ D u ( x ) , A v ⟩ + o ( ε 2 ) ,

   as ε → 0 . We can also look for zero-order terms and consider mean-value properties like

   inf A ∈ A x ( 1 − α ε 2 ) ⨍ B ε ( 0 ) u ( x + A y ) d y − u ( x ) = ε 2 1 2 ( n + 2 ) inf A ∈ A x trace ( A t D 2 u ( x ) A ) − α u ( x ) + o ( ε 2 ) ,

   as ε → 0 .

5. Finally, we consider extremal Pucci operators for given ellipticity constants 0 < θ < Θ , defined as

   ℳ θ , Θ − ( D 2 u ) = θ ∑ λ i ( D 2 u ) > 0 λ i ( D 2 u ) + Θ ∑ λ i ( D 2 u ) < 0 λ i ( D 2 u )

   and

   ℳ θ , Θ + ( D 2 u ) = Θ ∑ λ i ( D 2 u ) > 0 λ i ( D 2 u ) + θ ∑ λ i ( D 2 u ) < 0 λ i ( D 2 u ) .

   These operators are of the form (1.15), (1.16) with

   A θ Θ = { A ∈ S + n ( R ) : θ ≤ λ i ( A ) ≤ Θ } ,

   which is bounded uniformly in x . From the definition of A θ Θ , it is clear that ℳ θ , Θ ± are uniformly elliptic operators. We apply similar ideas to general fully nonlinear uniformly elliptic operators F ( x , D 2 u ( x ) ) (with ellipticity constants 0 < θ ≤ Θ ) by means of the characterization

   F ( x , M ) = sup N ∈ S n ( R ) inf A ∈ A θ Θ { trace ( A t M A ) + F ( x , N ) − trace ( A t N A ) } ,

   see Lemma 3.4.

The paper is organized as follows. In Section 2, we gather some definitions and preliminary results, and in Section 3 we deal with concave and Isaacs operators, when the sets of coefficients are bounded. These include general uniformly elliptic operators and operators including lower-order terms. We consider bounded coefficients first for the sake of clarity, to establish the techniques and ideas before we dive into the unbounded case in Section 4. In Section 5, we study the k -Hessians. Finally, Section 6 is devoted to examples in the Heisenberg group, where the class A x is unbounded and naturally depends on x .

2 Preliminaries

We begin by stating the definition of a viscosity solution to a fully nonlinear, second-order, elliptic PDE. We refer to [12] for general results on viscosity solutions. Given a continuous function

we consider the PDE

Viscosity solutions use the monotonicity of ℱ in D 2 u (ellipticity) in order to “pass derivatives to smooth test functions.”

We will also need the definition of an asymptotic mean-value formula in the viscosity sense. In the next definition, M ( u , ε ) ( x ) stands for a mean-value operator (that depends on the parameter ε ) applied to u at the point x . For example, we can take

as in Theorem 1.2 and the next section.

Next, we include a lemma that is related to the simplest mean-value property for the usual Laplacian. A proof can be found in [4].

3 Bounded operators

In this section, we prove Theorems 1.6 and 1.7. We consider differential operators given by

where A x ⊂ P ( S + n ( R ) ) (the power set of S + n ( R ) ) is a nonempty subset for each x ∈ R n and

Due to (3.2), the operator F ( x , M ) is finite for every M ∈ S n ( R ) .

Moreover, since the set of matrices ∪ A x is bounded, the mean-value formula (1.18) is local. In fact, for every x ∈ Ω and y ∈ B ε ( 0 ) there exists C x > 0 such that A ≤ C x I for every A ∈ ∪ A x . We obtain

as ε → 0 . In particular, observe that for ε small enough x + A y ∈ Ω for every y ∈ B ε ( 0 ) and hence the integrals

are well-defined for integrable functions u : Ω → R .

In the next lemma, we prove an explicit continuity estimate that we use in the proof of Theorem 1.6.

We are now ready to prove Theorem 1.6.