Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues

: We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two - player zero - sum di ﬀ erential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non - uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non - standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at in ﬁ nity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satis ﬁ ed for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic opera -tors, to partial trace operators.


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: 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 D u 0 i 2 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 D u f 1 2 with f bounded below, and supersolutions of ( ) = λ D u f n 2 with f bounded above. See for instance [14,21]. It is worth noting that classical solutions would be semiconvex and semi-concave, respectively. An interior C α 1, regularity result has been shown by Oberman and Silvestre [32, Theorem 5.1] for solutions of the equation ( ) = λ D u 0 1 2 in a ball B with boundary values ∈ g C α 1, . The result is qualified as "unusual" by the same authors because the C α 1, -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.
x x x d : Then u is locally Lipschitz continuous in , and 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 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 is neither concave nor convex. It is degenerate, non-uniformly elliptic except for = n 2, when ( ) D u 2 is the Laplace operator u Δ .
Idea of the proof. First, we perturb to obtain an operator ε , with a small amount of uniform ellipticity, such that → ε as → + ε 0 . Next, we show a uniform estimate for the gradient of the solution u ε of the equation . Then we show that → u u ε as → + ε 0 .
Pursuing this program, we have the occasion to extend a few qualitative results, which have not yet been established elsewhere, from the uniformly elliptic case to weighted partial trace equations.
First, in [14, Section 3.2] the existence of solutions was proved for general weighted Hessian partial trace operators in a bounded domain with a suitable convexity assumption derived from [10].
Here we show existence and uniqueness for weighted partial trace operators with extremal eigenvalues in domains satisfying a uniform exterior cone property. See Definition 2.21.
Theorem 1.2. Suppose that Ω is a bounded domain of n . Let Ω be endowed with a uniform exterior cone property. Assume that = ∑ = a λ We shall also consider unbounded subsets of slabs Then the Dirichlet problem (1.8) has a viscosity solution ( ) ∈ u C Ω 0 , and (1.9) holds. Suppose in addition and the solution of the Dirichlet problem (1.8) is unique.
The same assumptions allow us to obtain the so-called improved Alexandroff-Bakelman-Pucci (ABP) estimate.
, a covering of Ω with balls of radius R 0 . Let ( ) ∈ u usc Ω be a subsolution of the equation where the positive constant C only depends on the dimension n, σ, τ, a 1 , and it is bounded when a 1 , σ are bounded away from zero, τ is bounded away from 1.
where the positive constant C only depends on the dimension n, σ. τ, a n , and it is bounded when a n , σ are bounded away from zero, τ is bounded away from 1.
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 • n : set of the × n n real symmetric matrices

Review of known results and preparatory work
Let n be the set of × n n real matrices, with the partial ordering ≤ X Y that means − Y X positive semidefinite. We call ( ) ( ) … λ X λ X , , n 1 the eigenvalues of X in the non-decreasing order. We will consider in the sequel mappings → F : n . We say that is a degenerate elliptic operator if Let u be an upper semicontinuous function in Ω, for short ( ) ∈ u usc Ω . Then u is a viscosity subsolution of the equation for all ∈ x Ω and all ( ) ∈ φ C Ω 2 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 if it is a subsolution and a supersolution. We also use the notation ( . But in the case of lower regularity, when the Hessian matrices do not exist, we can only use the viscosity sense.
Let and be degenerate elliptic operators such that ( in the viscosity sense, then we can still say that ( ( )) , 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 Depending only on the eigenvalues, they are invariant by reflection.
Lemma 2.1. The weighted partial trace operator are invariant by reflection with respect to any hyperplane H of n , that is: if R is the reflection matrix with respect to H , 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 (2.6) Let = a a max i i and = a a min i i . We denote by the class of weighted partial trace operators → such that > a 0, which are degenerate, non-uniformly elliptic if = a 0, and by ue the subset which consists of ∈ 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 ∈ , 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:

For any degenerate elliptic operator
we define the dual operator as It is not hard to verify that the maximal and the minimal Pucci operators + λ,Λ and − λ,Λ , with the same constants λ and Λ, are the dual of each other.
In this respect, note that, if ( Proposition 2.3. Let be a weighted partial trace operator with ≥ a 0 i . Then a a a a n , ,

9)
Proof. We show only the right-hand inequality, since the left-hand one can be obtained by duality. It is enough to note that ( ) ( ) On the other hand, Comparing (2.11) and (2.12) we obtain the right-hand inequality of (2.9) under consideration. □ The above inequalities also hold in the case = a 0, when the extremal Pucci operators are degenerate, non-uniformly elliptic. But X. a n a a n , 1 1 (2.14) Proof. For (2.13), it is sufficient to observe that X aλ X a a n λ X a n λ X X , where the last inequality is a consequence of Proposition 2.3. The proof of (2.14) is similar. One can also use duality, observing that, if > a 0 1 the inequality (2.13) holds for the dual operator * with a 1 instead of a n . □ Let → : n . We say that is uniformly elliptic if  Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues  1187 where the positive constant C depends only on the dimension n and a 1 , and it is bounded when a 1 is bounded away from zero. Let ( ) ∈ u lsc Ω be a supersolution of the equation where the positive constant C only depends on the dimension n and a n , and it is bounded when a n is bounded away from zero.
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. where Ω y τ , is the connected component of ( ) We point out that all the bounded sets satisfy condition G. Moreover, any subset of a domain with condition G satisfies in turn condition G. Let Examples of domains satisfying condition G are for instance the following domains, which will be used in the sequel: 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 , in polar coordinates, are domains Ω endowed with a condition G. Note that in the latter case that \Ω 2 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 The Hölder exponent α depends on n, a 1 , and a n . The constant C also depends on δ.
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.
Definition 2.8. (Exterior cone condition) Let Ω be a domain of n . Following [31, Section 3], we say that Ω is endowed with a uniform exterior cone property if there exists a circular cone is the angle with the negative half-line in the direction x n , such that for any point on ∈ ∂ z Ω we have for a suitable rotation matrix R.
Multiplying by a constant, we may suppose ≥

26)
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.
There exist positive constants p 0 and C such that The constants p 0 and C only depend on n, a 1 , a n , and τ.
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. . In particular, is uniformly elliptic with ellipticity constants = λ a and = a Λ , that is, for ≥ Y 0: Tr .
(2.29)  Analogously,  The proof for unbounded domains will proceed by approximation.
Proof. (Theorem 1.3) First, by means of the Tietze-Urysohn-Brouwer extension theorem we extend g to a continuous function on n , called in turn g , such that

Existence and interior continuity
Since Ω and Q ρ satisfy condition G and a uniform exterior cone condition, so does their intersection.
From the bounded case, then we can solve the Dirichlet problem in Ω ρ k . That is, we find such that, in the viscosity sense: The pointwise estimate (1.9) of the bounded case, combined with (3.3), yields which implies the full inequality (1.9) even in the unbounded case.
We know that the functions u k are locally Hölder. See Lemma 2.7. In fact, let K 1 be a compact subset of Ω.
Such estimate is uniform with respect to ∈ k , depending on the Hölder coefficient ∈ + C and the Hölder exponent ( ) ∈ α 0, 1 only on n, a 1 , a n , and δ 1 . Therefore, the functions u k are equi-bounded and equi-continuous on each compact subset of Ω. Let ℓ δ sequence of positive number such that ↓ ℓ δ 0 as ℓ → ∞. We invade Ω with the increasing sequence of bounded subsets From the Ascoli-Arzelà theorem it follows that the sequence u k uniformly converges over ℓ ℓ Ω δ , up to a subsequence for any ℓ ∈ . So by a Cantor diagonalization process, we can extract from { } ∈ u k k a subsequence (called again u k ) that converges, locally uniformly in Ω, to a continuous function u.
By construction, given ∈ ∂ x Ω, we have ( ) ( ) = u x g x k for sufficiently large ∈ k , so that the limit function exists for all points of Ω, and = u g on ∂Ω. Recalling (3.5), we also obtain the pointwise estimate (1.9). By stability theorems on viscosity solutions, for instance [23, Proposition 2.9], we can also conclude that the limit function u is a viscosity solution of the equation

Continuity up to the boundary
In order to show that we need to prove that for every ∈ ∂ y S 0 : ( ) ( ) ( ) = ≡ → ∈ u x u y g y lim .
x y x Ω 0 0 0 (3.9) For this purpose, up to translations, we may suppose = y 0 0 . Let us fix > ε 0. By the continuity of g we can choose We want to show (3.9) with = y 0 0 , that is, To this end, observe that Ω is endowed with a uniform exterior cone property. See Definition 2.8. We may suppose, up to rotations, that the axis of cone, with vertex at the boundary point = x 0, has direction x n . Thus, a n a a n , (3.14) Next, let > δ 0 0 be such that ( ) < δ δ r min , g 0 0 . We compare the function u with the function On the boundary, we have ≤ u w: in fact, 2 g on ∩ ∂B Ω δ0 , by (1.9) and (2.26). So, by (3.13) and (3.14), u and w are a subsolution and a supersolution, respectively, of the same uniformly elliptic equation in ∩ B Ω δ0 , and ≤ u w on ( ) ∂ ∩ B Ω δ0 . Therefore, by the comparison principle for uniformly elliptic equations (see [23] (3.17) In order to complete the proof of the continuity at the point = ∈ ∂ x 0 Ω, we also need to show that there exists > δ 0 2 such that for ∈ ∩ x B Ω δ2 : . and we take ( ) ∈ γ 0, 1 such that a n a a n , in the construction of the barrier function ( ) We compare now the function * u with thereby proving inequality (3.18), and so (3.11).
Since this can be done for an arbitrary point ∈ ∂ y Ω 0 , the above shows that the solution ( ) ∈ u C Ω 0 , and the proof of the continuity up to the boundary is complete.
Behavior of solutions at infinity Here we show that the solution u vanishes at infinity, under the assumption that g vanishes at infinity.

Uniqueness
Let u w , be two solutions of the Dirichlet problem under consideration. Then both u and w vanish at infinity. That is, for all > ε 0 there exists We compare u and + w ε 2 in ( ) * B x Rε as a subsolution and a supersolution of the equation u w ε 2 on the boundary, by the comparison principle (Lemma 2.2) we obtain: On the other hand, by (3.40): On the other hand, we can interchange the role of u and w, obtaining the same inequality for − w u. Therefore, | | − ≤ u w ε for all > ε 0. Letting → + ε 0 , we obtain = u w. That is, the solution is unique as claimed. □

Approximate solutions and Lipschitz constants
In this section, we consider perturbations of the weighted partial trace operators , which leads to uniformly elliptic operators, namely: , which are bounded in a cylinder of center * x and axis of fixed height with sufficiently large section, depending on ε, have a partial derivative bound in the axis direction, uniform with respect to ε, at the point * x . Note indeed that in the uniformly elliptic case the argument works without any constraint on the diameter of the section of the cylinder. In the degenerate case, instead, a larger and larger radius ρ ε is needed as → + ε 0 . See , ; ε ε , the finite cylinder, with center * x and axis direction x i : Let u ε be a bounded continuous viscosity solution, namely, Proof. We may suppose = * a ε a ε ε ε a ε a ε , ,
The operator ε is also invariant by reflection. As observed in Lemma 2.1 and in the subsequent discussion, leading to (2.6), we have therefore both where ( ε n n ε n n . Next, we introduce the function , ; ε ε (4.9) using the comparison principle (Lemma 2.2). By uniform ellipticity (4.6): This shows, together with (4.7) and (4.10), that By (4.7) and (4.13), it turns out that u ε and v ε are a subsolution and a supersolution of the uniformly elliptic equation , respectively. Next, we compare u ε and v ε on the boundary of + C ρ d n , ; n n and therefore by (4.3) Recalling that u ε is C α 1, and therefore the derivative exists, we take the limit as → x 0 n in the above, and we obtain Convergence of u ε (by subsequence) to a solution ( ) The solutions u ε are uniformly bounded by (1.9). Consequently, the Hölder estimates (3.7) are uniform with respect to ( ) ∈ ε 0, 1 , depending on the Hölder coefficient and the Hölder exponent only on n, a 1 , and a n . See Lemma 2.7.
Hence, the u ε s uniformly converge by subsequence in any compact subset of Ω.
Proceeding as in the proof of Theorem 1.3 (Existence and interior continuity), we can use the Ascoli-Arzelà theorem and a Cantor diagonalization process to obtain a sequence u εk , locally uniformly converging to a function By the locally uniform convergence, using stability results in the viscosity setting [23], the limit function v is a solution of the limit equation, that is, Continuity of the limit v up to the boundary Arguing as in the proof of Theorem 1.3, due to the uniform bound (1.9) and the uniform exterior cone property, the u εk s have the same modulus of continuity at any point ∈ ∂ y Ω 0 , that is: and we are pointing out that δ does not depend on ∈ k . Taking the limit as → ∞ k , we have therefore So v is continuous at an arbitrary point ∈ ∂ y Ω 0 . We conclude that End of the proof