Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators
-
Fausto Ferrari
Abstract
We consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator, in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such an operator generally enjoys the qualitative properties of the Laplace operator, as for instance maximum and comparison principles, ABP and Harnack inequalities, Liouville theorems for subsolutions or supersolutions. Existence and uniqueness for the Dirichlet problem are also proved as well as local and global Hölder estimates for viscosity solutions. All results are discussed for a more general class of weighted partial trace operators.
1 Introduction and main results
In this paper we investigate the properties of weighted partial trace operators
where
and
The class
considered by Harvey and Lawson [49, 50] and Caffarelli, Li and Nirenberg [23, 22].
Here we introduce the subclass
As we will see later,
Of course, the case
It also worth noticing that the operators
After introducing our main results we shall further come back to the original motivation for studying the operators of
A good number of results will depend on the dimension n and on the following two quantities, namely the minimum between the coefficients of the smallest and the greatest eigenvalue, and the arithmetic mean of the coefficients, namely
in the sense that the involved constants are uniformly bounded when a positive upper bound of the first one and a finite upper bound of the latter one are avalaible.
The constants which depend only on n,
The following result is a revisitation of the bilateral Alexandroff–Bakelman–Pucci (ABP) estimate for the class
Theorem 1.1.
Let
where
We emphasize the following difference between estimate (1.3) and the standard ABP estimates, see for instance [47, Theorem 9.1]: the denominator of the right-hand side is
The above result is obtained as a consequence of two unilateral ABP estimates for subsolutions (4.6) and supersolutions (4.7).
The ABP estimate stated before also underlies a corresponding Harnack inequality for the equation
Theorem 1.2 (Harnack inequality).
Let
where C is a positive constant depending only on n,
We prove Theorem 1.2 via two inequalities for subsolutions and non-negative supersolutions, known in literature respectively as the local maximum principle (Theorem 5.1) and the weak Harnack inequality (Theorem 5.2), suitably adapted to this framework, by comparison with Pucci extremal operators.
From the Harnack inequality, the interior
Here, in Lemma 5.4, we get boundary Hölder estimates assuming for Ω a uniform exterior sphere property, with radius
for all
We obtain the following estimates for the Hölder seminorm
Theorem 1.3 (Global Hölder estimates).
Let
Suppose that Ω satisfies a uniform exterior sphere condition (s) with radius
(1.4)where C is a positive constants depending only on n,
Suppose in addition that Ω has a uniform Lipschitz boundary with Lipschitz constant L. If
(1.5)
A global estimate for the Hölder norm
In some cases, we can obtain an explicit interior Hölder exponent. For instance, in the case of asymmetric distributions of weights, concentrated on the smallest or the largest eigenvalue, as for the upper and lower partial trace operators
Theorem 1.4.
Let
In particular, we deduce the following
Suppose that Ω satisfies a uniform exterior sphere condition (s) with radius
(1.6)where C is a positive constant depending only on n,
Suppose in addition that Ω has a uniform Lipschitz boundary with Lipschitz constant L. If
(1.7)where C is a positive constant also depending on L.
The optimal regularity of solutions in the case of degenerate, non-uniform ellipticity, is an open problem.
Partial answers are contained for instance in [65] for the special case
Concerning higher regularity, one could borrow the techniques of [11, 54, 75, 24, 19, 73, 27, 28, 56, 53, 57], which however do not seem at the moment directly applicable in the more general non-uniformly elliptic setting.
It is remarkable the particular case of the interior
Further aspects of the qualitative theory, like the strong maximum principle and Liouville theorems, will be discussed in the last sections of the paper. New results for operators
Turning to the motivations about the importance of this research, we recall that the partial trace operators
where
Upper and lower partial trace operators arise in geometric problems of mean partial curvature considered by Wu [80] and Sha [71, 72]. Following the interest generated by the previous works, a number of papers has been devoted to the properties of these operators, we recall for instance [3, 26, 46, 79].
On the other hand, it is also worth noticing that Bellman equations arise in stochastic control problems, see Krylov [58], Fleming and Rishel [42], Fleming and Soner [43], Fleming and Souganidis [44] and the references therein.
As well as the partial trace operators
where
The alternative representation
suggests the relationship between
Following the main stream of the mean value properties of solution to linear equations, as well as in the case of the
As a consequence, if we consider a continuous function u, the operator given by the limit
whenever it exists, may be considered as the weak version of our operator
The paper is organized as follows. In Section 2 we introduce the main definitions about elliptic operators and viscosity solutions. In Section 3 we discuss in detail the properties of the weighted partial trace operators, in particular
2 General Preliminaries
This section is organized in some subsections, mainly for introducing common notation about viscosity theory of elliptic nonlinear PDEs, see Sections 2.1 and 2.5. In Sections 2.2 and 2.3 we introduce our class of operators
2.1 Ellipticity and Viscosity Solutions
We start recalling some ellipticity notions. Let
A fully nonlinear operator, that is a mapping
and uniformly elliptic if
for positive constants λ and Λ, called ellipticity constants. Note indeed that, by the left-hand side inequality in (2.2), a uniformly elliptic operator
Suppose now
The class of uniformly elliptic operators with given ellipticity constants λ and Λ is bounded by two estremal operators, the maximal and minimal Pucci operator, which are in turn uniformly elliptic with the same ellipticity constants, respectively:
where
In view of this definition, the uniformly ellipticity (2.2) of
From this it also follows that, if
which shows the extremality of Pucci operators.
Throughout this paper we will assume in fact
Of course, the results can be applied, in the case
Let Ω be an open set of
Let f be a function defined in Ω. A solution
Let
We briefly recall what it means to solve the equation
Let
and
We denote by
If
A viscosity solution of the equation
It is worth noticing that classical solutions are viscosity solutions. Viceversa, viscosity solutions of class
2.2 The Operator Class 𝒜
Let
Let
where
We observe that
Very recently, recalling the pioneeristic paper [66], Blanc and Rossi [15] have shown that it is possible to define a game satisfying a dynamic programming principle (DPP) which leads to the Dirichlet problem
Moreover, an associated evolution problem is considered in [14].
We point out that
Actually,
which can be seen as
Setting in addition
we notice that
We remark for instance that, while the min-max operator
On the other hand, every
so that every
we conclude that
We also observe that the operators
Next, we investigate more closely the peculiar properties of the min-max operator
2.3 The Min-Max Operator ℳ
In the previous subsection, we claimed that
Nonetheless, we present a few counterexamples that support the above claim.
Remark 2.1.
Let us consider the matrices
Then
The operator
and therefore
so that
The operator
against the strictly increasing property on ordered chains observed in Section 2.1 for the uniformly elliptic case.
The operator
and therefore
so that
while it is plain that for every
Thus
Since
On the other hand,
and therefore (2.7) continues to hold for
The next remark contains a few comments on the representation (1.9).
Remark 2.2.
The operator
In order to prove this, we start observing that plainly
To have also the reverse inequality, and so (2.8), we observe that the representative matrix
and thus
To compute the inf in the latter equation, we may assume that X is diagonal, by rotational invariance, with the eigenvalues
Using the Lagrange multipliers λ and μ, the inf is obtained in correspondence of a critical point of the function
which solve the system
or equivalently
We can show that
Since
If instead of “sup inf” as in (2.8) we consider “inf sup”, we re-obtain
with or without the constraint
2.4 Comparison with the Partial Trace Operators Operator 𝒫 k ±
Let us give a comparative look to the partial trace operators (1.2):
Remark 2.3.
If in (2.9) we consider “sup sup” or “inf inf” instead of “sup inf” or “inf sup”, it is not difficult to recognize, from (1.8), that we obtain the above partial trace operators with
Next, we list some properties of operators
Moreover,
and from the right-hand
In particular, since
and
We recall that the inequality stated above for the partial trace operators
where I is the
2.5 Duality
Let
If we deal with an arbitrary fully nonlinear operator and u is a subsolution to
which is in general different from
Computing the dual of the operators introduced above, we note that by homogeneity for the min-max operator
3 Auxiliary Results
In this section we apply the Perron method, well known in the literature, see for instance [33] and [48], in order to show: weak maximum and comparison principles, existence and uniqueness of solutions, see respectively Sections 3.1 and 3.2. The proofs are based on the properties of our operators, suitably exploited, and an appropriate adaptation of arguments used for the uniformly elliptic case. In Section 3.3 we obtain the radial representation of the operators
3.1 Weak Maximum and Comparison Principles
The following comparison principle holds between viscosity subsolutions and supersolutions of the equation
Theorem 3.1 (Comparison Principle).
Let
Letting
Corollary 3.2 (Weak Maximum Principle).
Let
On the other hand, if
Proof of Theorem 3.1.
The case of
For sake of completeness, we give an analytic proof based on the device contained in the proof of [33, Theorem 3.3] by Crandall, Ishii and Lions. See also [8].
We have to show that, under the given assumptions, the maximum of
(i) Firstly, setting
where
and
Moreover,
Noting that (3.1) implies
Taking the limit as
(ii) From (i) it follows, for all
where
From Corollary 3.2 we deduce the following uniform estimates for viscosity solutions of the equation
Proposition 3.3 (Uniform Estimate).
Let
where C is a positive constant, which can be chosen equal to
On the other hand, if
Proof.
Let us prove the first one. Setting
which yields the first inequality of the estatement. ∎
3.2 Existence and Uniqueness
As a consequence of the above comparison principle, we can also prove an existence and uniqueness result for the Dirichlet problem in bounded domains Ω via the Perron method, assuming that Ω has a uniform exterior cone condition, see [60] and [21]: there exist
where
Theorem 3.4.
Let Ω be a bounded domain of
has a unique viscosity solution
Proof.
According to the Perron method [33, Theorem 4.1], we need a comparison principle, and the existence of a subsolution and a supersolution of the equation
To do this, we will use the following inequalities, see (2.4) and (2.5):
and
If
so that
and, if
so that
Therefore, if
by the extremality properties (2.3) of Pucci operators, from (3.5) and (3.6) we have
Next, setting
and
Since obviously
An existence and uniqueness result is provided for all the class
This condition does not require smooth boundary, but it is nevertheless stronger than the exterior cone property.
Theorem 3.5.
Let Ω be a bounded domain of
has a unique viscosity solution
Proof.
Following the same lines of the proof of Theorem 3.4, we only need to look for a viscosity subsolution
and
Next, setting
and
As in the proof of Theorem 3.4,
3.3 Radial Solutions
We compute
where
As a consequence,
From this we deduce useful properties which are collected in the following remark.
Remark 3.6.
The operator
Any function of the form
with a and b constant, is a solution of
Recall that the k-th Hessian operator,
In case
are just the radial solutions of
Recalling that
where
4 The ABP Estimate
The celebrated ABP estimate provides a uniform estimate for the solution of an elliptic equation
In [18] it was also proved for the first time an ABP estimate for solutions in
It is worth noticing that an ABP estimate for degenerate elliptic equations of p-Laplacian type has been proved by Imbert [52].
An extension of this inequality to unbounded domains Ω of cylindrical type for bounded solutions in
The proof of the ABP estimates of Theorem 1.1 is based on the geometrical argument used in [47, proof of Theorem 9.1] for classical solutions.
We denote by
Lemma 4.1.
Let Ω be a bounded domain with diameter d, and
where
On the other hand, let us assume
Proof.
Let us prove the first estimate (4.1). We argue following [47, proof of Lemma 9.2] and [25, proof of Lemma 3.], denoting by
We remark that on the upper contact set
If
Let κ be the function whose graph is the cone K with vertex
Since on the contact set we have
From (4.4) and (4.5) we obtain the estimate from above (4.1).
For the estimate from below, we can apply (4.1) with
Then
Theorem 1.1 is obtained combining the two unilateral ABP estimates, which hold separately for subsolutions and supersolutions, contained in the following result.
Theorem 4.2.
Let Ω be a bounded domain of diameter d. Let f be continuous and bounded in Ω. There exist an universal constant
for viscosity subsolutions
(4.6)for viscosity supersolutions
(4.7)
For classical solutions the proof follows directly from Lemma 4.1.
Proof of Theorem 4.2: Classical Solutions.
For subsolutions, supposing
From Lemma 4.1, passing to
From Lemma 4.1, passing to
To consider viscosity subsolutions, we extend
In what follows we will refer to
The key tool is the following lemma, which allows to apply the classical ABP estimates obtained before to viscosity subsolutions and supersolutions and is the counterpart of [25, Lemma 3.3].
Lemma 4.3.
Let
in the viscosity sense, and w be a convex function such that
For sufficiently small
where
Here
Proof.
The first one inequality in (4.8) depends on the fact that
Concerning the second one, we may proceed assuming
(i) Subtracting
with
and
which implies
(ii) Let
where
and the paraboloid
Evaluating
(iii) Since
where
(iv) We claim that
In fact, arguing by contradiction, suppose that
By (4.10) and (4.12), the comparison principle would imply
Setting
Proof of Theorem 4.2: Viscosity Solutions.
We follow the lines of the proof of [25, Theorem 3.6], considering subsolutions
Let
in the viscosity sense. Let w be a concave function such that
For sufficiently small
where
Here
Using [25, Lemma 3.5], we deduce from the above that
If
Reasoning as in the proof of [25, Theorem 3.6], that is observing that the upper contact points are in Ω and
Estimating (4.15) with (4.16), we get the ABP estimate (4.6) for
5 Harnack Inequality and C α
The Harnack inequality, classically related to the mean properties of the Laplace operator, is a powerful nonlinear technique for regularity in the framework of fully nonlinear equations. We refer to [47] for solutions of linear uniformly elliptic equations in Sobolev spaces, to [74] for quasi-linear uniformly elliptic equations and to [24, 25] for viscosity solutions of fully nonlinear equations. See also [4, 52, 61] for further contributions.
In order to prove the Harnack inequality for non-negative solutions and the related local estimates for subsolutions and non-negative supersolutions, respectively known in literature (see for instance [47]) as the local maximum principle and the weak Harnack inequality, we could employ the same strategy of [25, Chapter 4].
A quicker way, sufficient for the applications, is based on inequalities (3.7) obtained in Section 3. The results are given in cubes, and here
but they could be equivalently stated in balls.
Theorem 5.1 (Local Maximum Principle).
Let
in
where
Proof.
In view of inequalities (3.7), we have
Theorem 5.2 (Weak Harnack Inequality).
Let
in
where
Proof.
In view of inequalities (3.7), we have
The proof of Theorem 1.2 (Harnack inequality) follows at once.
Proof of Theorem 1.2.
Let
which yields the result. ∎
From the Harnack inequality, in a standard way, using the technique for the proof of [25, Proposition 4.10] and [47, Lemma 8.23], the following Hölder regularity results and
Theorem 5.3 (Interior Hölder Continuity).
Let
in
where C is a positive constant depending only on n,
Global Hölder estimates can be proved for domains with the uniform exterior sphere condition (s), see Section 1, via the boundary Hölder estimates of the lemma below. We adopt the following notations, for the Hölder seminorm (
Lemma 5.4.
Let
If
with
Assume in addition that Ω has a uniform Lipschitz boundary with Lipschitz constant L. If
(5.4)with
If u is a viscosity supersolution, then (i) and (ii) hold with
Proof.
We treat in detail the case of subsolutions. The result for subsolutions will follow by duality. Therefore, suppose that
Case (i). By assumption on g and (5.5), we have
To simplify, we may suppose
Next, we define
where ε is any positive number and
Thus from (5.6)
Moreover, φ is a supersolution in Ω:
By the comparison principle
Then for an arbitrary
from which (5.4) follows.
Case (ii). By assumption on g and on
where
Therefore
where
where ε is any positive number and
Therefore by (5.7):
Moreover, φ is a supersolution in Ω:
By the comparison principle, we get
Then for arbitrary
for all
We are ready to show the global Hölder estimates of Theorem 1.3.
Proof of Theorem 1.3.
Let
where
(i) Suppose
Recall that
Since also
(ii) Suppose now
From (5.10) and (5.11), letting
In some cases, when the weights
Lemma 5.5.
Let
resp.
where
Proof.
We only treat the case of subsolution, when
Next, we define
where
On the other hand,
By (5.14) and (5.13), using the comparison principle, we get
Interchanging the role of
Combining Lemma 5.4 with Lemma 5.5, we obtain the global estimates of Theorem 1.4.
Proof of Theorem 1.4.
To obtain (1.4) and (1.5), it is sufficient to follow the proof of Theorem 1.3. The above estimates (1.6) and (1.7) are in particular obtained from this proof taking
Remark 5.6.
Note that Theorem 1.4 provides Lipschitz estimates only in the case
Remark 5.7.
Asking for higher regularity of viscosity solutions, we cannot expect viscosity solutions more regular than
so that
6 The Strong Maximum Principle
The strong maximum principle for an elliptic operator F, such that
One of the most elegant proof of the strong maximum principle, also known for this reason as the celebrated Hopf maximum principle [51], is based on boundary point lemma, which we establishes here below for the class of weigthed partial trace operators
Lemma 6.1 (Hopf Boundary Point Lemma).
Let
On the other hand, let
Proof.
We just prove the theorem for subsolutions. We may suppose that B is centered at the origin, i.e.
for
Taking
which proves (6.1). ∎
Following [47], we remark that, whether or not the normal derivative exists, we have instead of (6.1) and (6.2), respectively, the inequalities
and
where Σ is any circular cone of vertex
The Hopf boundary point lemma can be used to prove the strong maximum principle for classical subsolutions or viscosity subsolutions which are differentiable. A strong maximum principle, valid also for nonsmooth viscosity solutions, can be obtained through the weak Harnack inequality of Lemma 5.2.
For a detailed discussion on the strong maximum principle, we refer to the paper [70] and the papers quoted therein. In the case of fully nonlinear elliptic operators, see for instance [6, 7]. For further references see [76, 36, 69, 62].
Theorem 6.2 (Strong Maximum Principle).
Let u be a non-negative continuous viscosity supersolution of the equation
Proof.
For the proof in the case of differentiable solutions u, based on the Hopf lemma, we refer to the proof of [47, Theorem 3.5].
Concerning viscosity supersolutions (strong minimum principle), let
We are left with proving that A is open. Let
The function
In the case of viscosity subsolutions (strong maximum principle), we argue in a similar manner, considering the set
It follows that for elliptic operators
It is plain that the strong maximum principle implies the weak maximum principle (see Section 2) in bounded domains. This is no more true in unbounded domains, where the strong maximum principle may hold while the weak maximum principle fails to hold. An elementary example of this fact is given by the function
Turning to bounded domains, as observed in Section 2, it is sufficient that
Actually, the strong maximum and minimum principle may fail when
From the proof of Theorem 6.2, the weak Harnack inequality, which would imply the strong minimum principle, fails to hold in general for the partial trace operator
7 Liouville Theorems
A direct application of the Harnack inequality yields in a standard fashion the following Liouville result for entire solutions, defined in the whole
Theorem 7.1 (Liouville Theorem).
Let
It is well known that the above Liouville theorem holds in a stronger unilateral version for the Laplace operator in dimension
On the other hand, this is no longer true in higher dimension. For instance, the function
is a non-constant subharmonic, bounded function in
As well, the unilateral Liouville theorem does not hold for general elliptic operators even in dimension
Therefore, the uniform ellipticity is not sufficient by itself to guarantee such an unilateral Liouville property, even in dimension 2.
However, for particular uniformly elliptic operators as the minimal Pucci operators
We notice here that the same is true for the min-max operator
Theorem 7.2 (Hadamard Three Circles Theorem).
Let
with
Proof.
Actually, by Remark 3.6 the function
satisfies the equation
Note that
From Theorem 7.2 it follows that such operators satisfy the same Liouville unilateral property which holds for the Laplace operator in dimension
Theorem 7.3 (Unilateral Liouville Property).
Let
On the other hand, suppose
Proof.
Let
concluding that
Supposing
with
choosing
Concerning supersolutions v, bounded below, of the same equation in
Funding source: Istituto Nazionale di Alta Matematica
Award Identifier / Grant number: project 2018
Award Identifier / Grant number: project 2019
Funding statement: Fausto Ferrari is partially funded by INDAM-GNAMPA project 2018: Costanti critiche e problemi asintotici per equazioni completamente non lineari and INDAM-GNAMPA project 2019: Proprietà di regolarità delle soluzioni viscose con applicazioni a problemi di frontiera libera.
References
[1] A. D. Aleksandrov, Certain estimates for the Dirichlet problem (in Russian), Dokl. Akad. Nauk. SSSR 134 (1960), 1001–1004; translation in Soviet Math. Dokl. 1 (1960), 1151–1154. Search in Google Scholar
[2] A. D. Aleksandrov, Uniqueness conditions and bounds for the solution of the Dirichlet problem (in Russian), Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom. 18 (1963), no. 3, 5–29; translation in Amer. Math. Soc. Transl. (2) 68 (1968), 89–119. Search in Google Scholar
[3] M. E. Amendola, G. Galise and A. Vitolo, Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators, Differential Integral Equations 26 (2013), no. 7–8, 845–866. 10.57262/die/1369057820Search in Google Scholar
[4] M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal. 2008 (2008), Article ID 178534. 10.1155/2008/178534Search in Google Scholar
[5] I. J. Bakel’man, On the theory of quasilinear elliptic equations (in Russian), Sibirsk. Mat. Z. 2 (1961), 179–186. Search in Google Scholar
[6] M. Bardi and F. Da Lio, Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. I. Convex operators, Nonlinear Anal. 44 (2001), no. 8, 991–1006. 10.1016/S0362-546X(99)00317-XSearch in Google Scholar
[7] M. Bardi and F. Da Lio, Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. II. Concave operators, Indiana Univ. Math. J. 52 (2003), no. 3, 607–627. 10.1512/iumj.2003.52.2147Search in Google Scholar
[8] M. Bardi and P. Mannucci, On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal. 5 (2006), no. 4, 709–731. 10.3934/cpaa.2006.5.709Search in Google Scholar
[9] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92. 10.1002/cpa.3160470105Search in Google Scholar
[10] I. Birindelli, I. Capuzzo Dolcetta and A. Vitolo, ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains, Commun. Contemp. Math. 18 (2016), no. 4, Article ID 1550075. 10.1142/S0219199715500753Search in Google Scholar
[11] I. Birindelli and F. Demengel,
[12] I. Birindelli, G. Galise and H. Ishii, A family of degenerate elliptic operators: maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 417–441. 10.1016/j.anihpc.2017.05.003Search in Google Scholar
[13] I. Birindelli, G. Galise and F. Leoni, Liouville theorems for a family of very degenerate elliptic nonlinear operators, Nonlinear Anal. 161 (2017), 198–211. 10.1016/j.na.2017.06.002Search in Google Scholar
[14] P. Blanc, C. Esteve and J. D. Rossi, The evolution problem associated with eigenvalues of the Hessian, preprint (2019), https://arxiv.org/abs/1901.01052. 10.1112/jlms.12363Search in Google Scholar
[15] P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, J. Math. Pures Appl. (9) 127 (2019), 192–215. 10.1016/j.matpur.2018.08.007Search in Google Scholar
[16] R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dyn. Games Appl. 1 (2011), no. 1, 74–114. 10.1007/s13235-010-0005-0Search in Google Scholar
[17] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations, SIAM J. Control Optim. 47 (2008), no. 1, 444–475. 10.1137/060671954Search in Google Scholar
[18] X. Cabré, On the Alexandroff–Bakel’man–Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48 (1995), no. 5, 539–570. 10.1002/cpa.3160480504Search in Google Scholar
[19] X. Cabré and L. A. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations
[20] V. Cafagna and A. Vitolo, On the maximum principle for second-order elliptic operators in unbounded domains, C. R. Math. Acad. Sci. Paris 334 (2002), no. 5, 359–363. 10.1016/S1631-073X(02)02267-7Search in Google Scholar
[21] L. Caffarelli, M. G. Crandall, M. Kocan and A. Świȩch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), no. 4, 365–397. 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-ASearch in Google Scholar
[22] L. Caffarelli, Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators, Comm. Pure Appl. Math. 66 (2013), no. 1, 109–143. 10.1002/cpa.21412Search in Google Scholar
[23] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl. 5 (2009), no. 2, 353–395. 10.1007/s11784-009-0107-8Search in Google Scholar
[24] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. 10.2307/1971480Search in Google Scholar
[25] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, American Mathematical Society, Providence, 1995. 10.1090/coll/043Search in Google Scholar
[26] I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, On the inequality
[27] I. Capuzzo Dolcetta and A. Vitolo,
[28] I. Capuzzo Dolcetta and A. Vitolo, Glaeser’s type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 539–557. 10.3934/dcds.2010.28.539Search in Google Scholar
[29] I. Capuzzo Dolcetta and A. Vitolo, The weak maximum principle for degenerate elliptic operators in unbounded domains, Int. Math. Res. Not. IMRN 2018 (2018), no. 2, 412–431. 10.1093/imrn/rnw249Search in Google Scholar
[30] I. Capuzzo Dolcetta and A. Vitolo, Directional ellipticity on special domains: Weak maximum and Phragmén–Lindelöf principles, Nonlinear Anal. 184 (2019), 69–82. 10.1016/j.na.2019.02.004Search in Google Scholar
[31] S. Cho and M. Safonov, Hölder regularity of solutions to second-order elliptic equations in nonsmooth domains, Bound. Value Probl. 2007 (2007), Article ID 57928. 10.1155/2007/57928Search in Google Scholar
[32] M. G. Crandall, Viscosity solutions: A primer, Viscosity Solutions and Applications, Lecture Notes in Math. 1660, Springer, Berlin (1997), 1–47. 10.1007/BFb0094294Search in Google Scholar
[33] M. G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N. S.) 27 (1992), no. 1, 1–67. 10.1090/S0273-0979-1992-00266-5Search in Google Scholar
[34] M. G. Crandall and A. Świȩch, A note on generalized maximum principles for elliptic and parabolic PDE, Evolution Equations, Lecture Notes Pure Appl. Math. 234, Dekker, New York (2003), 121–127. 10.1201/9780429187599-8Search in Google Scholar
[35] A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 2, 219–245. 10.1016/s0294-1449(00)00109-8Search in Google Scholar
[36] F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 317–330. 10.4171/JEMS/81Search in Google Scholar
[37] F. Ferrari, An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations, Proc. Roy. Soc. Edinburgh Sect. A (2019), 10.1017/prm.2018.147. 10.1017/prm.2018.147Search in Google Scholar
[38] F. Ferrari, Q. Liu and J. Manfredi, On the characterization of p-harmonic functions on the Heisenberg group by mean value properties, Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2779–2793. 10.3934/dcds.2014.34.2779Search in Google Scholar
[39] F. Ferrari and A. Pinamonti, Characterization by asymptotic mean formulas of q-harmonic functions in Carnot groups, Potential Anal. 42 (2015), no. 1, 203–227. 10.1007/s11118-014-9430-9Search in Google Scholar
[40] F. Ferrari and E. Vecchi, Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group, Topol. Methods Nonlinear Anal., to appear. 10.12775/TMNA.2019.073Search in Google Scholar
[41] W. H. Fleming and M. Nisio, Differential games for stochastic partial differential equations, Nagoya Math. J. 131 (1993), 75–107. 10.1017/S0027763000004554Search in Google Scholar
[42] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Appl. Math. 1, Springer, Berlin, 1975. 10.1007/978-1-4612-6380-7Search in Google Scholar
[43] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Appl. Math. (New York) 25, Springer, New York, 1993. Search in Google Scholar
[44] W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38 (1989), no. 2, 293–314. 10.1512/iumj.1989.38.38015Search in Google Scholar
[45] K. Fok, A nonlinear Fabes–Stroock result, Comm. Partial Differential Equations 23 (1998), no. 5–6, 967–983. 10.1080/03605309808821375Search in Google Scholar
[46] G. Galise and A. Vitolo, Removable singularities for degenerate elliptic Pucci operators, Adv. Differential Equations 22 (2017), no. 1–2, 77–100. 10.57262/ade/1484881286Search in Google Scholar
[47] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar
[48] F. R. Harvey and H. B. Lawson, Jr., Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), no. 3, 396–443. 10.1002/cpa.20265Search in Google Scholar
[49] F. R. Harvey and H. B. Lawson, Jr., Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry, Surveys in Differential Geometry. Geometry and Topology, Surv. Differ. Geom. 18, International Press, Somerville (2013), 103–156. 10.4310/SDG.2013.v18.n1.a3Search in Google Scholar
[50] F. R. Harvey and H. B. Lawson, Jr., Removable singularities for nonlinear subequations, Indiana Univ. Math. J. 63 (2014), no. 5, 1525–1552. 10.1512/iumj.2014.63.5398Search in Google Scholar
[51] E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungsberichte Akad. Berlin 1927 (1927), 147–152. 10.1515/crll.1927.158.80Search in Google Scholar
[52] C. Imbert, Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations, J. Differential Equations 250 (2011), no. 3, 1553–1574. 10.1016/j.jde.2010.07.005Search in Google Scholar
[53] C. Imbert and L. Silvestre,
[54] H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), no. 1, 26–78. 10.1016/0022-0396(90)90068-ZSearch in Google Scholar
[55] S. Koike and A. Świȩch, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann. 339 (2007), no. 2, 461–484. 10.1007/s00208-007-0125-zSearch in Google Scholar
[56] J. Kovats, The minmax principle and
[57] J. Kovats, On the second order derivatives of solutions of a special Isaacs equation, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1523–1533. 10.1090/proc/12956Search in Google Scholar
[58] N. V. Krylov, Controlled Diffusion Processes, Springer, New York, 1977. Search in Google Scholar
[59] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc. 138 (2010), no. 3, 881–889. 10.1090/S0002-9939-09-10183-1Search in Google Scholar
[60] K. Miller, Extremal barriers on cones with Phragmén–Lindelöf theorems and other applications, Ann. Mat. Pura Appl. (4) 90 (1971), 297–329. 10.1007/BF02415053Search in Google Scholar
[61] A. Mohammed, G. Porru and A. Vitolo, Harnack inequality for nonlinear elliptic equations with strong absorption, J. Differential Equations 263 (2017), no. 10, 6821–6843. 10.1016/j.jde.2017.07.035Search in Google Scholar
[62] A. Mohammed and A. Vitolo, On the strong maximum principle, Complex Var. Elliptic Equ. (2019), 10.1080/17476933.2019.1594207. 10.1080/17476933.2019.1594207Search in Google Scholar
[63] I. Netuka and J. Veselý, Mean value property and harmonic functions, Classical and Modern Potential Theory and Applications (Chateau de Bonas 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 430, Kluwer Academic, Dordrecht (1994), 359–398. 10.1007/978-94-011-1138-6_29Search in Google Scholar
[64] M. Nisio, Stochastic differential games and viscosity solutions of Isaacs equations, Nagoya Math. J. 110 (1988), 163–184. 10.1017/S0027763000002932Search in Google Scholar
[65] A. M. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5871–5886. 10.1090/S0002-9947-2011-05240-2Search in Google Scholar
[66] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. 10.1007/978-1-4419-9675-6_18Search in Google Scholar
[67] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 1984. 10.1007/978-1-4612-5282-5Search in Google Scholar
[68] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15–30. 10.1007/BF02416445Search in Google Scholar
[69] P. Pucci and V. D. Rădulescu, The maximum principle with lack of monotonicity, Electron. J. Qual. Theory Differ. Equ. 2018 (2018), Paper No. 58. 10.14232/ejqtde.2018.1.58Search in Google Scholar
[70] P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), no. 1, 1–66. 10.1016/j.jde.2003.05.001Search in Google Scholar
[71] J.-P. Sha, p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437–447. 10.1007/BF01394417Search in Google Scholar
[72] J.-P. Sha, Handlebodies and p-convexity, J. Differential Geom. 25 (1987), no. 3, 353–361. 10.4310/jdg/1214440980Search in Google Scholar
[73] A. Świech,
[74] N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61 (1980), no. 1, 67–79. 10.1007/BF01389895Search in Google Scholar
[75] N. S. Trudinger, Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 1–2, 57–65. 10.1017/S0308210500026512Search in Google Scholar
[76] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. 10.1007/BF01449041Search in Google Scholar
[77] A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J. Differential Equations 194 (2003), no. 1, 166–184. 10.1016/S0022-0396(03)00193-1Search in Google Scholar
[78] A. Vitolo, A note on the maximum principle for second-order elliptic equations in general domains, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 1955–1966. 10.1007/s10114-007-0976-ySearch in Google Scholar
[79] A. Vitolo, Removable singularities for degenerate elliptic equations without conditions on the growth of the solution, Trans. Amer. Math. Soc. 370 (2018), no. 4, 2679–2705. 10.1090/tran/7095Search in Google Scholar
[80] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525–548. 10.1512/iumj.1987.36.36029Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- A Morse Lemma for Degenerate Critical Points of Solutions of Nonlinear Equations in ℝ2
- Non-local Degenerate Diffusion Coefficients Break Down the Components of Positive Solutions
- Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions
- Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System
- Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior
- Sharp Caffarelli–Kohn–Nirenberg-Type Inequalities on Carnot Groups
- A new Singular Trudinger–Moser Type Inequality with Logarithmic Weights and Applications
- Coron Problem for Nonlocal Equations Involving Choquard Nonlinearity
- Fractional Choquard Equations with Confining Potential With or Without Subcritical Perturbations
- Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities
- Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators
Articles in the same Issue
- Frontmatter
- A Morse Lemma for Degenerate Critical Points of Solutions of Nonlinear Equations in ℝ2
- Non-local Degenerate Diffusion Coefficients Break Down the Components of Positive Solutions
- Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions
- Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System
- Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior
- Sharp Caffarelli–Kohn–Nirenberg-Type Inequalities on Carnot Groups
- A new Singular Trudinger–Moser Type Inequality with Logarithmic Weights and Applications
- Coron Problem for Nonlocal Equations Involving Choquard Nonlinearity
- Fractional Choquard Equations with Confining Potential With or Without Subcritical Perturbations
- Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities
- Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators
