Abstract
In this paper, we prove global and interior second derivative estimates of Pogorelov type for certain Monge–Ampère type equations, arising in optimal transportation and geometric optics, under sharp hypotheses. Specifically, for the case of generated prescribed Jacobian equations, as developed recently by the second author, we remove barrier or subsolution hypotheses assumed in previous work by Trudinger and Wang (Arch Ration Mech Anal 192:403–418, 2009), Liu and Trudinger (Discret Contin Dyn Syst Ser A 28:1121–1363, 2010), Jiang et al. (Calc Var Partial Differ Equ 49:1223–1236, 2014), resulting in new applications to optimal transportation and near field geometric optics.
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1 Introduction
This paper is concerned with global and interior second derivative estimates of solutions to partial differential equations of Monge–Ampère type, (MATEs) that arise in optimal transportation and geometric optics and are embraced by the notion of “generated prescribed Jacobian equation” (GPJE) introduced in [16]. Such equations can be written in the general form
where and are respectively symmetric matrix valued and scalar functions on a domain , and denote respectively the gradient vector and Hessian matrix of the scalar function , which is defined on a bounded domain for which the sets
are non-empty for all . A solution of (1.1) is called elliptic, (degenerate elliptic), whenever , which implies .
We shall establish the second derivative bounds for generated prescribed Jacobian equations under the hypotheses G1, G2, G1*, G3w and G4w introduced in [16], which extend the conditions A1, A2 and A3w for optimal transportation equations introduced in [20]. These are degenerate versions of the conditions A1, A2 and A3 for regularity originating in [11]. In fact we will relabel them here replacing G in [16] by A to make this extension clearer. First we need to suppose that there exists a smooth generating function defined on a domain for which the sets
are convex, (and hence open intervals). Setting
we then assume
-
A1: For each , there exists a unique point satisfying
-
A2: , , in , where is the matrix given by
Defining in A1, we thus obtain mappings with . The matrix function in the corresponding generated prescribed Jacobian equation is then given by
where and . Note that the Jacobian determinant of the mapping is , by A2, so that and are accordingly smooth. In particular, it suffices to have to define . The reader is referred to [16] for more details. We also mention though that the special case of optimal transportation is given by taking
where is a cost function defined on a domain in so that and conditions A1 and A2 are equivalent to those in [11, 19]. Note that here we follow the same sign convention as in [15, 20] so that our cost functions are the negatives of those usually considered. We remark also that the case when is independent of is equivalent to the optimal transportation case.
Crucial to us in this paper is the dual condition to A1, namely
-
A1*: The mapping is one-to-one in , for all .
We will treat the implications of condition A1* as needed in Sect. 2. Meanwhile we note that the Jacobian matrix of the mapping is , where is the transpose of , so its determinant is automatically non-zero when condition A2 holds. Note that in the optimal transportation case (1.3), we simply have but the duality is more complicated in the general situation and will be amplified further in Sect. 2.
The next conditions are expressed in terms of the matrix function and extend condition A3w in the optimal transportation case [20]. For these we assume that is twice differentiable in and once in .
-
A3w: The matrix function is regular in , that is is codimension one convex with respect to in the sense that,
in , for all such that .
-
A4w: The matrix is monotone increasing with respect to in , that is
in , for all .
An explicit formula for in terms of the generating function is given in [16]. In order to use conditions A3w and A4w for our barrier constructions in Sect. 2, we also assume that is sufficiently large in that the sets are convex in , for each , (for A3w), and , for each , where is an open interval in , (possibly empty), and , (for A4w). When we use both A3w and A4w, we would also assume to be convex. We note that in [4], so that these conditions are automatically satisfied and in the optimal transportation case . Similar conditions are also needed for the convexity theory in [16].
To formulate our second derivative estimates, we first denote the one-jet of a function by
and recall from [16] that a -affine function in is a function of the form
where and are fixed so that for all .
Theorem 1.1
Let be an elliptic solution of Eq. (1.1) in , where , and is given by (1.2) with generating function g satisfying conditions A1, A2, A1*, A3w and A4w. Suppose also in for some -affine function with the one jets . Then we have the estimate
where the constant depends on and .
Note that the inequality is trivially satisfied if there exists a -affine function on which is greater than and moreover can be dispensed with in the optimal transportation case, where Theorem 1.1 improves Theorems 3.1 and 3.2 in [20] and Theorem 1.1 in [4]. Also we may assume in place of A4w the reversed monotonicity condition:
-
A4*w: The matrix is monotone decreasing with respect to in , that is
in which case Theorem 1.1 continues to hold provided the inequality is replaced by .
Next we formulate the corresponding extension of the Pogorelov interior estimate [2] for generated prescribed Jacobian equations.
Theorem 1.2
Let be an elliptic solution of (1.1) satisfying on , for some -affine function , where , and is given by (1.2) with generating function g satisfying conditions A1, A2, A1*, A3w, A4w and . Then we have for any ,
where the constant depends on and .
In the optimal transportation case, Theorem 1.2 improves Theorem 1.1 in [9] and Theorem 2.1 in [4] by removing the barrier and subsolution hypotheses assumed there. As in [9], we may also replace by any strict supersolution. We also remark that the dependence on in Theorems 1.1 and 1.2 is more specifically determined by and dist.
This paper is organised as follows. In the next section we will construct a function in Lemma 2.1 for which the Eq. (1.1) is uniformly elliptic and use it to prove an appropriate version, Lemma 2.2, of the key Lemma 2.1 in [4]. From here we obtain Theorems 1.1 and 1.2 by following the same arguments as in [4]. In Sect. 3, we will also discuss the application of Theorem 1.1 to global second derivative estimates for the Dirichlet and second boundary value problems for optimal transportation and generated prescribed Jacobian equations, yielding improvements of the corresponding results in [4, 13, 14, 20]. In Sect. 4 we focus on applications to optimal transportation and near field geometric optics. In the optimal transportation case, we obtain a different and more direct proof of the improved global regularity result in [15]. Then we conclude by treating some examples of generating functions in geometric optics satisfying our hypotheses, which arise from the reflection and refraction of parallel beams.
To conclude this introduction we should also emphasise that while we have considered the more general case of generated prescribed Jacobian equations our results are also new for optimal transportation. Further we note that when we strengthen condition A3w to the strict inequality A3, the corresponding second derivative estimates as treated in [11], (see also [20] and [18]), are much simpler to prove although even here the constructions in Sect. 2 are still new.
2 Barrier constructions
Our first lemma shows the existence of a uniformly elliptic function under conditions A1, A2 and A1*.
Lemma 2.1
Let be a generating function satisfying conditions A1, A2 and A1*, and suppose is a -affine function on , where is a bounded domain in . Then there exists a function such that , in and
for some positive constant depending on and .
Proof
Suppose for some , let be a ball of radius and centre in , and consider the function in , given by
Our desired function is now defined as the -transform of , in accordance with the notion of duality introduced in [16], namely
Note that is well defined for for sufficiently small to ensure the set
Furthermore, the supremum in (2.3) will be taken on at a unique point . To prove this assertion, we fix a point and set for some . Since in , we then have, by the mean value theorem,
for positive constants and . Consequently, choosing , we have
and hence the supremum in (2.3) cannot occur on . By differentiation, we then have at an interior maximum point ,
and hence by condition A1*, there exists a unique corresponding to . From our construction (2.3) the function is -convex in and its -normal mapping consists of those maximizing (2.3). Here we recall from [16] that a function is -convex in if for each point , there exists and , such that
for all and the -normal mapping of at is given by
and moreover, . From (2.4), we also have
whence for some constant . To show is uniquely determined by , we need to prove is a single point. For this we invoke the duality considerations in [16]. Letting
we let denote the dual generating function of , given by
Also denoting
the dual matrix, corresponding to , is given by
where is the unique mapping, given by A1*, satisfying
Note that the mapping constructed above is given by
Since the graph of is a hemisphere of radius and hence has constant curvature , we have, on
for some fixed constant , provided is taken sufficiently small, which implies in particular that is locally strictly -convex, that is for any , we have
for all , near . Clearly, by again taking sufficiently small, we infer that the function is strictly globally -convex, that is inequality (2.10) holds for all . Consequently is uniquely determined by , whence the mapping is invertible with inverse given by
and is differentiable with . Since is smooth so also is with Jacobian matrix
Using the relation
we then have . Since
and , , we thus obtain (2.1) from condition A2. Finally we observe that
by using in the last inequality. We then obtain as asserted and the proof is complete.
Remark 2.1
For a matrix , not arising from optimal transportation or generated Jacobian mappings, Lemma 2.1 may not be true even if satisfies the regularity condition A3w in a strong way. To see this, suppose for some constant . Then the uniform ellipticity condition, , , implies is convex in a convex domain with
By integration, we obtain
which can only be satisfied for sufficiently small.
We now use Lemma 2.1 to construct a barrier for the linearized operator of Eq. (1.1), in accordance with Remark 2.4 in [5]. Letting be elliptic for Eq. (1.1) with we set and define an associated linear operator by
where and denotes the inverse matrix of . It is known that is a concave operator with respect to for elliptic . We then have the following improvement and extension of the fundamental lemma, Lemma 2.1 in [4] in the optimal transportation case.
Lemma 2.2
Let be a generating function satisfying conditions A1, A2, A1*, A3w, A4w and let be elliptic with respect to , and in for some -affine function, , on . Then
holds in , for sufficiently large positive constant , uniform positive constants , depending on and .
Proof
By Lemma 2.1, is a uniformly elliptic function satisfying , in and
where is the positive constant in (2.1). For any , the perturbation is still a uniformly elliptic function with , for sufficiently small , satisfying
for some constant . From our construction of , we can also choose sufficiently small such that in .
Setting , and , we have . By detailed calculation, we have
By the concavity of with respect to , we have
Since in , we then have, by the mean value theorem and condition A4w
where for some . Note that from our conditions on , we have , for all so that is well defined in (2.13), and is well defined in (2.15). Using the Taylor expansion, we have
where for some . Here we are also using the convexity of the set for each which ensures the point so that the term in (2.16) is also well defined.
By (2.14), (2.15), (2.16), we have from (2.13)
Next we choose a finite family of balls , with centres at , , covering and with fixed radii , where . Then we select for some such that . Accordingly, we have for a fixed small positive ,
holds for . We see that (2.18) holds in all balls , , with a fixed positive constant . Then by the finite covering, (2.18) holds in with a uniform positive constant .
Without loss of generality, assume that at a given point in , we get
where condition A3w is used in the second inequality.
Since the matrix is positive definite, any diagonal minor has positive determinant. By the Cauchy’s inequality, we have
and
for any positive constant .
Thus, we have
By first choosing small such that , and then choosing large such that , we obtain
By choosing and , the conclusion of this lemma is proved.
Remark 2.2
When we apply the operator to the function , the second derivatives of appear in the coefficients of which are controlled by using the concavity property of . Accordingly in (2.12) depends on the one jet but not on . We note that the constant in (2.12) has the additional dependence on , which does involve second derivatives of , but when we apply Lemma 2.2 to elliptic solutions and uniformly elliptic function , such a dependence is reduced. For example, when we consider the Eq. (1.1), namely , the additional dependence of in (2.12) is just .
Remark 2.3
When we assume the reverse monotonicity A4*w and , then by modifying appropriately and using (2.15), we still obtain the key barrier inequality (2.12).
3 Applications to second derivative estimates
Theorems 1.1 and 1.2 follow from Lemma 2.2 and the proofs of Theorem 3.1 in [20] and Theorem 1.1 in [9]. These latter results are proved under an additional hypothesis that is -bounded on , that is there exists a function satisfying
in . However, as pointed out in [4], we need only assume the weaker inequality
for some constant to carry out the same proofs. By virtue of Lemma 2.2, the barrier inequality (3.2) is satisfied by the function
We will first treat the global estimates for boundary value problems, and subsequently the interior estimate, Theorem 1.2, as we will need to modify slightly the proof in [9] to accommodate the dependance of on .
Taking account of the above remarks we may extend the statement of Theorem 3.1 in [20] as follows.
Lemma 3.1
Let be an elliptic solution of Eq. (1.1) in with . Suppose that is regular on , that is
in , for all such that , and that satisfies (3.2) for some , with . Then we have the estimate
where the constant depends on and .
Clearly, Theorem 1.1 follows immediately from Lemmas 2.2 and 3.1. As an immediate consequence of Theorem 1.1, the hypothesis of the existence of a subsolution in Theorem 1.1 of [4] can be removed altogether in the optimal transportation case, while for the Dirichlet problem in Theorem 1.2 of [4], we need only assume the existence of a subsolution in a neighbourhood of the boundary, whose boundary trace is the prescribed boundary function. However, the natural boundary condition for optimal transportation and the more general prescribed Jacobian equations is the prescription of the image of the mapping, , that is
for some given target domain . Global estimates for solutions of Eq. (1.1) subject to (3.6) now follow from Theorem 1.1 under appropriate convexity hypotheses on and . As with (3.1) and (3.4), we will express these initially in terms of . Accordingly, we assume the domain is uniformly -convex with respect to in the sense that there exists a defining function satisfying , on together with the “uniform convexity” condition (3.1) in a neighbourhood of , while for the domain we assume the dual condition, that is uniformly -convex with respect to , namely that there exists a defining function satisfying , on together with the “dual uniform convexity” condition:
in a neighbourhood of , where is given by
for each . Taking account of (1.2) we see that these conditions can be formulated for general prescribed Jacobian equations where satisfies , and are not necessarily restricted to those determined through generating functions. Furthermore, writing
condition (3.7) may be expressed in the form
for a positive constant , whenever and moreover the boundary condition (3.6) implies a nonlinear oblique boundary condition,
on , for an elliptic solution of Eq. (1.1) [14]. Building on the optimal transportation case in [20], an obliqueness estimate
on , for a positive constant , where denotes the unit outer normal to , is derived in [10, 14] for elliptic solutions of (1.1) (3.6), in , under the above hypotheses that and are respectively uniformly -convex and uniformly -convex with respect to .
To proceed from (3.12) to global second derivative estimates for the second boundary value problem, we need a boundary estimate for second derivatives, which we can extract from [20], Sect. 4, as a refinement of the Monge–Ampère case in [21].
Lemma 3.2
Let be an elliptic solution of Eq. (1.1) in subject to a nonlinear oblique boundary condition (3.11), (3.12) on with . Suppose that is regular on , is uniformly -convex with respect to , and satisfies the uniform convexity condition (3.10). Then we have the estimate,
where the constant C depends on and .
Using the estimates, (3.12) and (3.13), we can thus conclude from Theorem 1.1, the following global second derivative estimate for generated prescribed Jacobian equations.
Theorem 3.1
Let be an elliptic solution of the second boundary value problem (1.1), (3.6) in , where is given by (1.2) with generating function , satisfying conditions A1, A2, A1*, A3w and A4w (or A4*w), and the domains are respectively uniformly -convex and uniformly -convex with respect to . Suppose also (or ) in for some -affine function on . Then we have the estimate,
where the constant depends on and .
We may express the domain convexity hypotheses in Theorem 3.1 in terms of boundary data depending on the solution or more specifically, as in [10], intervals containing the range of . For this purpose we will say that is uniformly -convex with respect to if
for all , , unit outer normal and unit tangent vector , for some constant . Similarly the target domain is uniformly -convex with respect to if
for all , , unit outer normal and unit tangent vector , for some constant . It then follows that the uniform convexity assumptions in Theorem 3.1 may be replaced by , being respectively uniformly -convex, -convex with respect to , . Moreover these assumptions are equivalent to the convexity notions defined in terms of the generating function in [16]. Assuming and are connected and is a diffeomorphism, then is uniformly -convex with respect to if and only if is uniformly -convex with respect to , that is the images are uniformly convex for all , where is the -transform of on , given by
for , while is uniformly -convex with respect to if and only if is uniformly -convex with respect to , that is the images are uniformly convex for all , where for ; see [16] for more details.
Returning to the Dirichlet problem, we state the following analogue of Theorem 3.1, which follows from Theorem 1.1 and our boundary estimates in [4].
Theorem 3.2
Let be an elliptic solution of (1.1) in , satisfying on , where is given by (1.2) with generating function , satisfying conditions A1, A2, A1*, A3w and A4w (or A4*w), , and the domain is uniformly -convex with respect to . Suppose also (or ) in for some -affine function on . Then we have the estimate,
where the constant depends on and .
To reduce the proof of Theorem 3.2 to our boundary estimates in Section 3 of [4], we replace by in inequality (3.5) there and use the barrier in place of , for sufficiently large constant , in the rest of the proof, where is a defining function in the definition of -convexity. Note that under the assumption of uniform -convexity we do not need Lemma 2.2 to obtain the second derivative boundary estimate, which is already asserted in [13]. Note also that in these arguments the dependence of the coefficients on can be removed by replacing by .
To conclude this section we indicate how Theorem 1.2 follows from Lemma 2.2 and [9]. In the optimal transportation case (1.3), when is independent of , we simply use (3.3) in the proof of Theorem 1.1 in [9] instead of (3.1), with replaced by . In the general case we proceed the same way, first noting that the monotonicity condition A4w ensures in , by virtue of the strong maximum principle [2] and the fact that is a degenerate elliptic solution of the homogeneous equation, (). We then consider as in [9], the corresponding auxiliary function
where are positive constants to be determined and estimate from below at an interior maximum point , where
As in Section 3 of [20], the additional terms arising from the the dependence, in the differentiations of (1.1), are absorbed in the lower bounds (18) and (19) in [9] so it remains to extend the lower bound for in inequalities (20) and (21) in [9], where now . This is done similarly to the argument in the proof of Lemma 2.2, as follows. By calculation, we have
where for some , condition A4w is used in the second inequality and Taylor’s formula is used in the last equality. By assuming is diagonal, we have from (3.20)
where condition A3w is used in the second inequality and the constant depends on and . We remark that the particular form of the estimate (3.21) is actually crucial in the ensuing estimations in [9].
Accordingly we obtain from [9], an estimate,
for positive constants and , corresponding to the estimate (9) in [9]. Our desired estimate (1.5) follows, noting also that the estimate (28) in [9] for from below extends automatically when we substitute for in and .
4 Optimal transportation and geometric optics
4.1 Optimal transportation
For our applications to optimal transportation when reduces to a cost function through (1.3), it will suffice to assume where the domains and are respectively uniformly and -convex with respect to each other, that is the images and are uniformly convex in , for each , .
Conditions A1, A1*, A2 and A3w can then be written as:
-
A1: The mappings are one-to-one for each ;
-
A2: det in ;
-
A3w: , for all such that and such that .
Here we also have the simpler formulae,
We then conclude from Theorem 3.2 the following global second derivative estimate which improves Theorem 1.1 in [20] and Theorem 2.1 in [15].
Corollary 4.1
Let be an elliptic solution of the second boundary value problem (1.1), (3.6) in , where is given by (4.1) with cost function satisfying conditions A1, A2 and A3w and and the domains are respectively uniformly -convex and uniformly -convex with respect to each other. Then we have the estimate,
where the constant depends on and .
For application to optimal transportation, the function has the form
where and are positive densities defined on and respectively, satisfying the mass balance condition
which is a necessary condition for a solution for which the mapping is a diffeomorphism. From Corollary 4.1, we obtain following [20] the existence of an elliptic solution which is unique up to additive constants, as formulated in Theorem 1.1 in [15]. However our proof of Corollary 4.1 in this case is more direct and more general than the approach in [15] which depends on a strict convexity regularity result of Figalli et al. [1]. From the existence we obtain the existence of a smooth optimal mapping solving the transportation problem as formulated in Corollary 1.2 in [15].
Examples of cost functions which satisfy condition A3w and not A3 are given in [20] and [9]. However for these examples the global second derivative bounds in Theorems 1.1 and are proved in [20] from other structures such as -boundedness or duality so that Corollary 4.1 is not new in these cases. This is not the case though for the interior estimate, Theorem 1.2, which for example is new for the cost function . For explicit calculations in this case the reader is referred to [8], as well as [20].
4.2 Geometric optics
Keeping to domains in Euclidean space, we will just consider here parallel beams in , directed in the direction, through a domain , illuminating targets which are graphs over domains in . The formulae are easily adjusted to cover more general situations and we may also consider point sources of light as in [6]. The special cases where the targets are themselves domains in yield strictly regular matrix functions and global estimates for the second boundary value problem are already proved in [10, 14]. Consequently our main interest here is with situations where the resultant matrices satisfy A3w and not necessarily A3, although as we indicated in Sect. 1 our lemmas in Sect. 2 are still new in this case.
-
(i)
Reflection. We modify slightly the example in [16] to allow for non-flat targets; see also [7] for a thorough description of the geometric picture. Let be a domain in , containing , and consider the generating function:
(4.5)defined for and where is a smooth function on . By differentiation we have
(4.6)so that if there are mappings , satisfying A1, we must have and from (4.5),
(4.7)By differentiation we then obtain,
(4.8)provided
that is,
(4.9)Accordingly conditions A1 and A2 will be satisfied for , where
(4.10)and , given implicitly by (4.7), satisfies . Now from (4.6), we see that the matrix is given by
(4.11)and hence conditions A3w and A4w are both satisfied whenever the dual function is concave in the gradient variables. It also follows that condition A1* is also satisfied when (4.9) holds, that is the mapping given by
is one to one in , for , which can be proved for example from the quadratic structure of the equation, . Furthermore by taking sufficiently small, we see that there exist arbitrarily large -affine functions and hence all the hypotheses of Theorems 1.1 and 1.2 are satisfied by the generating function if is concave in . Note that when is constant, then and the strict condition A3 holds, [16]. By direct calculation, we have an explicit formula for the matrix ,
and its determinant
when (4.9) holds. By the Sherman–Morrison formula, we also have a formula for the inverse of ,
-
(ii)
Refraction. We consider refraction from media to media with respective refraction indices and set . For , we consider now generating functions,
(4.12)where again , for , for and is a smooth function on . For more details about the geometric and physical aspects of this model, see for example [12]. We will now restrict attention to the case , in which case by setting and rescaling , , , we can write
(4.13)By differentiation we now have
(4.14)so that if there are mappings , satisfying A1, we must have
so from (4.13),
(4.15)By differentiation we then obtain,
(4.16)provided
that is,
(4.17)Accordingly conditions A1 and A2 will be satisfied for , where , if and is given by
(4.18)if . Furthermore the mapping is given by and , given implicitly by (4.15), satisfies . From (4.14), we now have
(4.19)It follows then that conditions A3w and A4w are both satisfied whenever the function is convex. Again the condition A1* is also satisfied when (4.17) holds so that the hypotheses of Theorems 1.1 and 1.2 are fulfilled provided in the case of Theorem 1.1 the solution has a -affine majorant, which is satisfied automatically from the boundary condition, on in the case of Theorem 1.2. Note that when is constant, we obtain
so again the strict condition A3 holds. By direct calculation, we now have an explicit formula for the matrix ,
and its determinant
when (4.17) holds. We also obtain the following formula for the inverse of ,
In the case , the monotonicity is reversed in the case = constant. For general and taking now , we have after rescaling , , ,
(4.20)and hence we obtain in place of (4.14) to (4.16),
(4.21)provided
However now we have in place of (4.19),
(4.22)so that we obtain A4*w instead of A4w. Hence for A4w we need to assume the reverse inequality, , that is , where now is given by
(4.23)which implies at least that and are disjoint and excludes the case constant. However with , Theorems 1.1, and are still applicable provided the function is convex as there exist arbitrarily small -affine functions. Also then the case of constant is embraced with
By direct calculation, we again have an explicit formula for the matrix ,
and its determinant
when . By the Sherman-Morrison formula, we also obtain the inverse of ,
As in the previous cases the condition A1* will hold if either , (that is ), or , (that is , which again follows from the equivalent quadratic structure of the equation .
-
(iii)
Further remarks. It is interesting to note that when we pass to the dual functions in the above examples the monotonicity of is reversed but, as shown in general in [16], the condition A3w is preserved. In particular for the dual in the reflection case (4.5) we have from [16],
(4.24)while in the refraction cases (4.12), a simple calculation gives
(4.25)for , () respectively. Note that by replacing by in (4.5), the dual function in (4.24) is the limit as of the case in (4.25).
Also we note that when we consider instead the complementary ellipticity condition, , in Eq. (1.1), condition A3w is replaced by substituting for but conditions A4w and A4*w are maintained. Thus in the above examples we need to replace by in the convexity conditions, (and interchange the inequalities ), to fulfil our hypotheses.
As well as the monotonicity properties, another interesting ingredient of the above examples is that we cannot infer, as in the optimal transportation examples mentioned in the previous section, that bounded domains are -bounded so second derivative estimates do not follow from the relevant arguments in [14, 20].
4.3 Final remarks
The existence of locally smooth solutions to the second boundary value problem for generated prescribed Jacobian equations is treated in [16] under conditions A1, A2, A1*, A3 and A4w. We remark that there that the montonicity condition A4w may also be replaced by condition A4* for local regularity. In [17] and ensuing work the monotonicity condition A4w is relaxed and local boundary regularity is also considered, yielding an alternative approach to that in [10] which deals with estimates and existence of globally smooth solutions. We remark that the strict monotonicity of , in our examples in Sect. 4.2, also follows from a general formula in [17],
In a follow up work [3] we will use our estimates here to extend the global theory when A3 is weakened to A3w, without -boundedness conditions, as well as further develop the optics examples. We also develop an alternative duality approach to second derivative estimates when the matrix depends only on and . As already noted in [20], this situation is much simpler in two dimensions.
Finally we wish to thank the anonymous referee of this paper for their careful checking and useful comments.
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Communicated by S.K. Jain.
Research supported by Australian Research Council Grant, National Natural Science Foundation of China (No. 11401306) and the Jiangsu Natural Science Foundation of China (BK20140126)
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Jiang, F., Trudinger, N.S. On Pogorelov estimates in optimal transportation and geometric optics. Bull. Math. Sci. 4, 407–431 (2014). https://doi.org/10.1007/s13373-014-0055-5
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DOI: https://doi.org/10.1007/s13373-014-0055-5