Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues

1 Introduction and main results

A growing attention has been received by the Hessian partial trace operators in the last few decades. Motivated by geometric problems of mean partial curvature [1,2,3], many works have been devoted to analytical and geometrical aspects of equations involving partial trace operators. For instance, just to mention a few ones, see the papers of Caffarelli et al. [4,5,6] and of Harvey and Lawson [7,8,9].

In this article, we are interested in the more general weighted Hessian partial trace operators:

where λ i is the eigenvalue of the Hessian matrix, in non-decreasing order, and a = ( a i ) i = 1 n is an n -tuple of numbers a i ≥ 0 such that a ¯ = max 1 ≤ i ≤ n a i > 0 .

For a list of symbols used here we refer to the end of this section.

Equations P ( D 2 u ) = 0 have been recently recognized to govern two-player zero-sum differential games. See the study of Blanc and Rossi [10], in which an existence and uniqueness theorem for solutions of boundary value problems for the equation λ i ( D 2 u ) = 0 is proved. See also [11] for a time-dependent version.

Such operators share a number of qualitative properties with uniformly elliptic operators. See for instance, recent papers [12,13,14,15,16, 17,18]. We expect that other properties, typical of uniformly elliptic operators, can be extended to the partial trace operators, like blow-up boundary results or Liouville properties for the Lane-Emden equation, see [19] as we plan to investigate further on.

Regularity properties seem to be a difficult task. Few Hölder estimates are known, see [14], depending on the coefficients a i . See also [20] for an effort to estimate Hölder exponents. Lipschitz regularity is only known for viscosity supersolutions of λ 1 ( D 2 u ) = f with f bounded below, and supersolutions of λ n ( D 2 u ) = f with f bounded above. See for instance [14,21]. It is worth noting that classical solutions would be semi-convex and semi-concave, respectively.

An interior C 1 , α regularity result has been shown by Oberman and Silvestre [32, Theorem 5.1] for solutions of the equation λ 1 ( D 2 u ) = 0 in a ball B with boundary values g ∈ C 1 , α . The result is qualified as “unusual” by the same authors because the C 1 , α -regularity is transmitted from the boundary to the interior but it cannot be extended up to the boundary, generally.

The main scope of this article is to show an interior Lipschitz regularity result for weighted partial trace operators with positive coefficients of the smallest and the biggest eigenvalue, in dimension n ≥ 3 .

Generally, interior regularity, in a compact set, is obtained supposing that the solution exists in a larger bounded open set containing it. In our case, we have to assume that solutions exist in a larger unbounded set, like a slab or a finite union of slabs, and a boundary condition far away, that the solution vanishes at infinity, is needed.

Here are the contributions of this article, where the measure-geometric condition G is given by Definition 2.6.

The set of functions u considered in the above theorem is not empty. We will establish an existence (and uniqueness) result in Theorem 1.3.

The previous result is applicable to weighted partial trace operators P with non-zero coefficients a 1 and a n of the extremal eigenvalues λ 1 and λ n , respectively. Among such ones, we recall:

introduced in a previous paper [14].

Note that ℳ is neither concave nor convex. It is degenerate, non-uniformly elliptic except for n = 2 , when ℳ ( D 2 u ) is the Laplace operator Δ u .

We shall also consider unbounded subsets of slabs S d ; i = { x ∈ R n : ∣ x i ∣ < d } , star-shaped domains S d = ⋃ i = 1 n S d ; i , and their translates S d ; i ∗ = x ∗ + S d ; i , S d ∗ = x ∗ + S d , in view of Theorem 1.1.

With respect to the bounded case, for unbounded domains the uniqueness will be proved under the additional assumption that the boundary values tend to zero at infinity.

At the same time, we point out that the following results show that solutions in unbounded sets, like slabs or a finite union of slabs, vanishing at infinity, do exist under assumptions that are not so restrictive.

The same assumptions allow us to obtain the so-called improved Alexandroff-Bakelman-Pucci (ABP) estimate.

See [23] for linear uniformly elliptic equations in non-divergence form, [24,25] for generalizations, and [26] for the case of fully nonlinear uniformly elliptic equations; see also [12] for further generalizations and [27] for other kinds of degenerate elliptic operators.

This article is organized as follows: in Section 2 we collect and rearrange a few results on elliptic operators, viscosity solutions, and geometric properties of a domain, which will be useful in the sequel, in particular we establish some connections between uniformly elliptic operators and partial trace operators; in Section 3 we prove existence and uniqueness results; in Section 4 we estimate the gradient of approximate solutions; and finally, in Section 5 we prove the convergence of approximate solutions.

List of symbols

• S n : set of the n × n real symmetric matrices

• X ≤ Y : Y − X positive semidefinite

• λ i ( X ) : eigenvalues of X ∈ S n , λ i ≤ λ i + 1

• a = ( a i ) i = 1 n , a ̲ ≔ min 1 ≤ i ≤ n a i , a ¯ ≔ max 1 ≤ i ≤ n a i

• P : X ∈ S n → ∑ i = 1 n a i λ i ( X ) ∈ R ; P ε ≔ P + ε Δ , ε > 0

• A ≔ { P : a ̲ ≥ 0 , a ¯ > 0 } , A u e ≔ { P : a ̲ > 0 }

• x = ( x 1 , … , x n ) , ∣ x ∣ = ∑ i = 1 n x i 2 1 / 2

• x ˆ i = ( x 1 , … , x i − 1 , x i + 1 , … , x n ) , ∣ x ˆ i ∣ = ∑ j ≠ i x j 2 1 / 2

• S d ; i ≔ { x ∈ ℛ n : ∣ x i ∣ < d } , d > 0 ; S d ≔ ⋃ i = 1 n S d ; i

• C ρ , d ; i : { x ∈ S d ; i : ∣ x ˆ i ∣ < ρ } , ρ > 0 ; S ρ , d ≔ ⋃ i = 1 n C ρ , d ; i

• Q d ∗ = ⋂ i = 1 n S d ; i ∗ ≡ ∏ i = 1 n ] x i ∗ − d , x i ∗ + d [ .

2 Review of known results and preparatory work

Let S n be the set of n × n real matrices, with the partial ordering X ≤ Y that means Y − X positive semidefinite. We call λ 1 ( X ) , … , λ n ( X ) the eigenvalues of X in the non-decreasing order.

We will consider in the sequel mappings F : S n → R . We say that ℱ is a degenerate elliptic operator if X ≤ Y implies ℱ ( X ) ≤ ℱ ( Y ) .

We assume f ∈ C 0 ( Ω ) . Let u be an upper semicontinuous function in Ω , for short u ∈ usc ( Ω ) . Then u is a viscosity subsolution of the equation ℱ ( D 2 u ) = f in Ω if we have

for all x ∈ Ω and all φ ∈ C 2 ( Ω ) touching u at x from above.

A lower semicontinuous function u in Ω , for short u ∈ lsc ( Ω ) , is a supersolution of the same equation if we have

for all x ∈ Ω and all φ ∈ C 2 ( Ω ) touching u at x from below.

A function u ∈ C 0 ( Ω ) is a solution of the equation ℱ ( D 2 u ) = f if it is a subsolution and a supersolution.

We also use the notation ℱ ( D 2 u ) ≥ f for subsolutions and ℱ ( D 2 u ) ≤ f for supersolutions. Such inequalities correspond to pointwise inequalities in the case u ∈ C 2 ( Ω ) . But in the case of lower regularity, when the Hessian matrices do not exist, we can only use the viscosity sense.

Let ℱ and G be degenerate elliptic operators such that ℱ ( X + Y ) ≤ ℱ ( X ) + G ( Y ) . We use in the sequel the fact that if ℱ ( D 2 u ) ≤ f and G ( D 2 v ) ≤ g in the viscosity sense, then we can still say that ℱ ( D 2 ( u + v ) ) ≤ f + g , as we would do for classical solutions, provided at least one between u and v is C 2 .

This article is focused on the weighted partial trace operators

with coefficients a i ≥ 0 for i = 1 , … , n , which are degenerate elliptic.

Depending only on the eigenvalues, they are invariant by reflection.

We refer to [28], and more diffusely [29], for major generality.

Here we consider a sample case in order to obtain to the point of interest. Suppose H = { x n = 0 } , so that

where I ˆ n is the ( n − 1 ) × ( n − 1 ) identity matrix.

Let u ( x ) = u ( x ˆ n , x n ) be a C 2 - function, and u ˜ ( x ) = u ( R x ) ≡ u ( x ˆ n , − x n ) , then

Let a ¯ = max i a i and a ̲ = min i a i . We denote by A the class of weighted partial trace operators P → R such that a ¯ > 0 , which are degenerate, non-uniformly elliptic if a ̲ = 0 , and by A u e the subset which consists of P ∈ A such that a ̲ > 0 , which will be seen uniformly elliptic.

Deferring later the proof of the uniform ellipticity in such cases, we recall that for any P ∈ A , the comparison principle holds in bounded domains. See [14, Theorem 3.1].

The most important elliptic operators, which depend only on the eigenvalues, are the well-known Pucci extremal operators, the maximal and the minimal one, depending on positive constants λ and Λ ≥ λ , respectively:

where λ i + = max ( λ i , 0 ) and λ i − = − min ( λ i , 0 ) .

For any degenerate elliptic operator ℱ we define the dual operator as

It is not hard to verify that the maximal and the minimal Pucci operators ℳ λ , Λ + and ℳ λ , Λ − , with the same constants λ and Λ , are the dual of each other.

The dual of the weighted partial trace operator P = a 1 λ 1 + ⋯ + a n λ n is again a weighted partial trace operator, P ∗ = a n λ 1 + ⋯ + a 1 λ n .

If P ∈ A , resp. P ∈ A e u , then P ∗ ∈ A , resp. P ∗ ∈ A e u . Self-dual operators, that is to say, P = P ∗ , are ℳ 1 , 1 + = ℳ 1 , 1 − = λ 1 + ⋯ + λ n , the Laplace operator, and the partial trace operator ℳ = λ 1 + λ n .

In this respect, note that, if ℱ ( D 2 u ) ≥ f , that is, u is a subsolution of the equation ℱ ( D 2 u ) = f , then v = − u is a supersolution of the equation ℱ ∗ ( D 2 v ) = f , namely, ℱ ∗ ( D 2 v ) ≤ f .

The above inequalities also hold in the case a ̲ = 0 , when the extremal Pucci operators are degenerate, non-uniformly elliptic.

But in this case (2.9) are almost useless. Conversely, if we consider weighted partial trace operators P with coefficients a 1 > 0 or a n > 0 , we can dispose of more useful inequalities; see also [14, Section 3.2].

In this case, we can control P with Pucci extremal operators ℳ λ , Λ ± having λ > 0 .

Let ℱ : S n → R . We say that ℱ is uniformly elliptic if

with positive constants λ and Λ ≥ λ , called ellipticity constants.

The extremal Pucci operators are uniformly elliptic, in particular ℳ 1 , 1 ± ( X ) = Tr ( X ) , which corresponds to the Laplace operator: Δ u = Tr ( D 2 u ) .

Differently, the partial trace operator ℳ = λ 1 + λ n is not uniformly elliptic when n > 2 . In fact, let X = e 1 ⊗ e 1 and Y = e 2 ⊗ e 2 . Then ( λ 1 + λ n ) ( X + Y ) = 1 = ( λ 1 + λ n ) ( X ) , while Tr ( Y ) = 1 , violating the left-hand-side inequality (2.16).

A fundamental tool for dealing with uniformly elliptic operators is the well-known ABP estimate. But in view of the inequalities of Proposition 2.4, it can be used for weighted partial trace operators P with a 1 > 0 and/or a n > 0 . See [14, Lemma 4.1].

Note that in the case f ≡ 0 , the above inequalities yield the maximum and the minimum principle.

We also recall that the ABP estimate has been improved by Cabré [30] and extended to domains, possibly unbounded, with the measure-geometric property G .

The result of Cabré [30, Theorem 1.1] provides the estimate (2.17) with R 0 instead of the diameter d , and a constant C which also depends on σ and τ . A refinement [26, Theorem 1] allow us to substitute ‖ f ‖ L n ( Ω ) with a local norm sup y ∗ ∈ Ω ‖ f ‖ L n ( B R 0 ( y ∗ ) ∩ Ω ) .

Here we extend the result to weighted partial trace operators P . Theorem 1.4 is a direct consequence of [26, Theorem 1] and Proposition 2.4.

Let d , ρ > 0 , i ∈ { 1 , … , n } . Examples of domains satisfying condition G are for instance the following domains, which will be used in the sequel:

• Finite cylinders C ρ , d ; i : { x = ( x ˆ i , x ) ∈ R n : ∣ x ˆ i ∣ < ρ , ∣ x i ∣ < d } ,

• Bounded star-shaped domains S ρ , d ≔ ⋃ i = 1 n C ρ , d ; i ,

• Slabs S d ; i ≔ { x ∈ R n : ∣ x i ∣ < d } ,

• Unbounded star-shaped domains S d ≔ ⋃ i = 1 n S d ; i .

But very different domains with respect to cylinders and slabs or finite unions are possible. For instance, in the plane, the complement of a regular lattice of balls with centers at integer coordinates, and the complement of the spiral ρ = e θ , θ ∈ R , in polar coordinates, are domains Ω endowed with a condition G . Note that in the latter case that R 2 \ Ω has empty interior.

An application of the ABP estimate, which will be used in the sequel, is the local Hölder continuity of solutions of uniformly elliptic equations. See [23]. Using Proposition 2.4, this can be extended to weighted partial trace operators. See [14, Theorem 5.3].

We recall that for Pucci extremal operators with ellipticity constants λ > 0 and Λ ≥ λ the Hölder inequality (2.20) holds with α and C depending on the same quantities but λ and Λ in the place of a 1 and a n .

The issue of the boundary regularity will be treated using the uniform exterior cone property and the Miller barrier functions.

A further application of the ABP estimate, which will be helpful in the uniqueness issue, is the boundary weak Harnack inequality, which follows from [14, Theorem 5.2] suitably extending non-negative supersolutions outside the domain Ω .

Finally, a useful tool in the approximation argument that will be used in the sequel is the uniform ellipticity of weighted partial trace operators with all positive weights.

3 Existence and uniqueness

In this section, we show the existence and the uniqueness of solution for the equation P ( D 2 u ) = 0 in Ω with the boundary condition u = g on ∂ Ω , where Ω is a domain with an exterior cone property, possibly unbounded.