A note on second derivative estimates for Monge-Ampère-type equations

Theorem 2.1

Let u ∈ C 4 ( Ω ) ∩ C 0 , 1 ( Ω ¯ ) be an elliptic solution of (1.1) in Ω , and let u 0 ∈ C 2 ( Ω ) ∩ C 0 , 1 ( Ω ¯ ) be a degenerate elliptic supersolution such that J 1 [ u ] , J 1 [ u 0 ] ⊂ U 0 ⊂ ⊂ U and u = u 0 on ∂ Ω , u < u 0 in Ω . Suppose there exists a barrier function ϕ ≥ 0 , ∈ C 2 ( Ω ¯ ) satisfying

in Ω , for some constant C ′ ≥ 0 . Then there exist positive constants β , δ 1 , δ 2 , and C depending on n , U , U 0 , ∣ A ∣ 2 , U 0 , ∣ D log B ∣ 1 , U 0 , C ′ , and ∣ ϕ ∣ 1 ; Ω such that, if either (i) D u A > − δ 1 I in U 0 or (ii) diam Ω < δ 2 , then

Proof

First we note that Case (ii) is proved in [18], Lemma 3.3 and the barrier hypothesis (2.2) is automatically satisfied with C ′ = 0 , for suitable δ 2 , by taking ϕ ( x ) = const . ∣ x − x 1 ∣ 2 for some point x 1 ∈ Ω . Accordingly, we will concentrate on Case (i). Adapting the proofs of Lemma 3.3 in [18], Theorem 1.2 in [2], and Theorem 2.1 in [7], we first consider an auxiliary function,

where ∣ ξ ∣ = 1 and τ , κ , β are positive constants to be chosen. Then, using inequality (3.8) in [18] combined with condition (i), we obtain in place of inequality (3.9) in [18], at a maximum point x 0 and vector ξ = e 1 of v in Ω ,

provided τ ≥ C and κ ≥ C ( τ + β δ 1 ) , where C is a constant depending on the same quantities as in the estimate (2.3) and L = ℒ − D p k log B ( ⋅ , u , D u ) D k , η = u 0 − u .

From the condition D v ( x 0 ) = 0 , we then have, for each i = 1 , … , n ,

so that for each i > 1 ,

Assuming η w 11 ( x 0 ) ≥ β and combining (2.5), (2.6), (2.7), we then obtain L v ( x 0 ) > 0 , by choosing τ ≥ C , κ ≥ C τ , β ≥ C ( τ 2 + e κ max ϕ ) , and δ 1 ≤ 1 β , for sufficiently large constant C depending on n , U , U 0 , ∣ A ∣ 2 , U 0 , ∣ D log B ∣ 1 , U 0 , and ϕ and hence conclude the estimate (2.3), from the corresponding estimate for w .□

Theorem 2.2

Let u ∈ C 4 ( Ω ) ∩ C 0 , 1 ( Ω ¯ ) be an elliptic solution of a generated Jacobian equation (1.1) in the domain Ω , with J 1 [ u ] ⊂ U 0 ⊂ ⊂ U . Then, for any strictly contained subdomain Ω ′ , we have the estimate

where C depends on n , U , U 0 , g , B , R 0 , and ω [ u ] ( Ω ′ , Ω ) .

Corollary 2.1

Let u ∈ C 4 ( Ω ) ∩ C 2 ( Ω ¯ ) be an elliptic solution of a generated Jacobian equation (1.1) in the domain Ω , J 1 [ u ] ⊂ U 0 ⊂ ⊂ U , and suppose N ′ = Ω − Ω ¯ ′ is A -bounded, with respect to u, for some subdomain Ω ′ ⊂ ⊂ Ω , with barrier ϕ . Then we have the estimate,

where C depends on n , U , U 0 , g , B , R 0 , ω [ u ] ( Ω ′ , Ω ) , and ϕ .

Theorem 3.1

Let u ∈ C 4 ( Ω ¯ ) , with J 1 [ u ] ⊂ U 0 ⊂ ⊂ U , u ( Ω ) ⊂ J 0 , for some open interval J 0 ⊂ J , be an elliptic solution of the second boundary value problem (1.1), (3.3) in Ω , where A ∈ C 2 ( U ) is given by (2.9) with generating function g ∈ C 4 ( Γ ) , satisfying conditions A1, A2, A1*, and A3w, B > 0 , ∈ C 2 ( U ) and the domains Ω , Ω ∗ ∈ C 4 are, respectively, uniformly Y-convex and uniformly Y ∗ -convex with respect to Ω ∗ × J 0 and Ω × J 0 . Suppose additionally that (i) Ω ⊂ ⊂ Ω 0 for a domain Ω 0 , also satisfying Ω 0 × Ω ∗ × J ⊂ ⊂ Γ ∗ , which is g -convex with respect to y ∈ Ω ∗ and z = g ∗ ( x , y , u 0 ) for all x ∈ Ω , u 0 ∈ J 0 and (ii) T u is one-to-one and u is g-convex in Ω , with any g-support, g 0 , satisfying g 0 ( Ω ) ⊂ J . Then we have the estimate,

where the constant C depends on n , U , U 0 , g , B , Ω , Ω ∗ , Ω 0 , J 0 , and J .

Proof

Since the proof is a straightforward extension of that of optimal transportation case in Theorem 2.1 in [16], we just describe it briefly here. From Lemma 3.2 in [2] and Theorem 2.2, it is enough to prove an estimate from below for the modulus of strict g -convexity of of u over any subdomain Ω ′ ⊂ ⊂ Ω , as formulated in (2.10). Defining ψ through equation (3.2), we first note that a g -convex, elliptic solution u ∈ C 2 ( Ω ) ∩ C 1 ( Ω ¯ ) of the second boundary value problem (1.1), (3.3), for which T u is one-to-one, will be a generalized solution, as defined in Section 4 of [17], with density f = ∣ ψ ∣ ( ⋅ , u , D u ) , satisfying ∫ Ω f = ∣ Ω ∗ ∣ , (and target density f ∗ ≡ 1 ). Furthermore, there exist positive constants c 0 and c 1 , depending on ψ , U , U 0 such that c 0 ≤ f ≤ c 1 . Consequently, if there does not exist a positive lower bound for ω [ u ] ( Ω ′ , Ω ) , by virtue of the weak continuity of the associated measures and the Radon-Nikodym theorem, there would exist a sequence of such solutions converging uniformly to a generalized solution u , with L ∞ density f satisfying the same bounds, which is not strictly g -convex at some point x 0 ∈ Ω ¯ ′ , thereby contradicting Theorem 1 in [14] and Theorem 2.3 in [1].

Alternatively, we remark that we can obtain an explicit estimate for ω [ u ] from the Hölder gradient estimate in Section 8 of [1] corresponding to Theorem 2.4 there, by using duality.□

Theorem 4.1

Let g ∈ C 4 ( Γ ) be a generating function satisfying conditions A1, A2, A1*, A3w, and A5, for C 4 bounded domains Ω , Ω ∗ in R n , with Ω uniformly g-convex with respect to y ∈ Ω ¯ ∗ , z ∈ g ∗ ( Ω ¯ , y , J ¯ 0 ) , and Ω ∗ uniformly g ∗ -convex with respect to x ∈ Ω ¯ , u ∈ J ¯ 0 . Also assume there exists a g-affine function, g 0 = g ( ⋅ , y 0 , z 0 ) , on Ω ¯ satisfying T g 0 = y 0 ∈ Ω ∗ ,

and

where d = diam ( Ω ) . Suppose also the function ψ satisfies (4.1), (4.2). Then for any x 0 ∈ Ω , there exists a unique g-convex elliptic solution u ∈ C 3 ( Ω ¯ ) of the second boundary value problem (3.1), (3.3), satisfying u ( x 0 ) = g 0 ( x 0 ) . Furthermore, the mapping T u is a C 2 smooth diffeomorphism from Ω ¯ to Ω ¯ ∗ .

Remark 4.1

To avoid possible confusion, we point out that here, as in condition (2.3) in [2] and Theorem 1.1 in [18], we are using the following meaning of the diameter of a domain Ω ⊂ R n ,

where d Ω ( x , y ) = dist Ω ( x , y ) is the geodesic distance in Ω between x and y , that is, the infimum of the lengths of C 1 curves in Ω joining the points x and y . Moreover, we may also refine condition (4.3) by replacing d by d ′ ≔ diam ( Ω ′ ) , where Ω ′ is any larger domain than Ω , also satisfying condition A5. If Ω ′ = Ω ˆ is the convex hull of Ω , we obtain the usual notion of diameter in R n . In fact, the above confusion goes back to the statement of the existence result for generalized solutions in Theorem 4.2 in [17], where the corresponding geodesic distance in Ω is also intended.

Proof

Substituting the second derivative estimate in Theorem 3.1 for that in Theorem 3.1 in [2], we would infer from the proof of Theorem 1.1 and Remark 3.2 in [3], the existence of a solution u whose graph intersects that of g 0 . For this we need to observe that the assumed uniform g -convexity conditions on Ω would imply the corresponding uniform Y -convexity of Ω as well as the supplementary condition (i) and the g -convexity of an elliptic solution u , from Lemma 2.1 in [18]. Then the rest of the supplementary condition (ii) now follows from the mass balance condition (4.2) and our assumed condition (4.4). To obtain the full strength of Theorem 4.1 we then need to adjust the homotopy family (3.4) in [3], as done by Rankin in equation (93) in [14], to conclude the existence of a classical solution u , with graph intersecting that of g 0 at x 0 , the uniqueness of which then follows from [13].□

Remark 4.2

We can write a cleaner but slightly weaker version of Theorem 4.1 by replacing the uniform g -convexity condition on Ω by uniform Y -convexity as in Theorem 1.1 of [3] with our condition on g 0 , (4.3), strengthened to

which also implies condition (4.4). As mentioned above, we also have a messier, more general statement if we replace the uniform g -convexity of Ω by uniform Y -convexity, but still assume the corresponding g -convexity of Ω and supplementary condition (i) in Theorem 3.1. Note that when we use Theorem 3.1 in [2] in the proof of Theorem 4.1, we do not need this last condition when any of the conditions A3, A4w, and A4 ∗ w hold.

Corollary 4.1

Let u be a g-convex generalized solution of the second boundary value problem, (3.1), (3.3), where g , Ω , Ω ∗ and ψ satisfy the hypotheses of Theorem 4.1, with g 0 a g-support of u at some point x 0 ∈ Ω . Then u ∈ C 3 ( Ω ¯ ) is a classical elliptic solution of (3.1), (3.3) and T u a C 2 smooth diffeomorphism from Ω ¯ to Ω ¯ ∗ .

Remark 4.3

The modified hypotheses of Theorem 4.1 in Remark 4.2 are also applicable to Corollary 4.1. Moreover, if any of conditions A3, A4w, and A4 ∗ w hold, we need to only assume, in accordance with Theorem 2 in [14], that (4.4) only holds for any g -support of u . By adapting the uniqueness argument for global optimal transportation regularity in Section 6 of [19], we may remove condition (4.4) completely in these cases if also

The overall proof is also much simpler in that we do not need to use the strict convexity of generalized solutions and their local regularity. Using condition (4.6), we then obtain, from Theorem 4.1, a classical solution v of (3.1), (3.3) whose graph touches that of the generalized solution u from above so that the uniqueness argument of [19] is applicable in a sufficiently small ball B ⊂ Ω ′ ≔ { u < v } , with boundary ∂ B intersecting ∂ Ω ′ .

A similar remark applies to the general case in Corollary 4.1, except we still need condition (4.4) for the strict convexity control used in our proof of the second derivative estimates in Theorem 3.1.