Strict convexity and $C^{1}$ regularity of solutions to generated Jacobian equations in dimension two

Thanks to Neil Trudinger who suggested extending the proof in [20] to generated Jacobian equations.

The structure of a particular generated Jacobian equation derives from a generating function. We denote this function by $g$, and require it satisfy the following assumptions.
A0. $g\in C^{4}(\overline{\Gamma})$ where $\Gamma\subset\mathbf{R}^{n}\times\mathbf{R}^{n}\times\mathbf{R}$ is a bounded domain satisfying that the projections

are open intervals. Moreover we assume there are domains $\Omega,\Omega^{*}\subset\mathbf{R}^{n}$ such that for all $x\in\overline{\Omega},y\in\overline{\Omega^{*}}$ we have that $I_{x,y}$ is nonempty.
A1. For all $(x,u,p)\in\mathcal{U}$ defined as

there is a unique $(x,y,z)\in\Gamma$ such that

A2. On $\overline{\Gamma}$ there holds $g_{z}<0$ and the matrix E with entries

satisfies $\det E\neq 0$.

Assumption A1 allows us to define mappings $Y:\mathcal{U}\rightarrow\mathbf{R}^{n}$ and $Z:\mathcal{U}\rightarrow\mathbf{R}$ by requiring they solve

We call a PDE of the form

a generated Jacobian equation provided $Y$ derives from solving (4) and (5) for some generating function.

Generated Jacobian equations may be rewritten as Monge–Ampère type equations. Suppose $u\in C^{2}(\Omega)$ satisfies (1). Then differentiating (4) and (5) evaluated at $(x,Y(x,u,Du),Z(x,u,Du))$ yields

Since the first equation with (5) implies

the second can be solved for $DY$ by which we have

with $E$ as in A2. Thus $C^{2}$ solutions of (1) solve

for

This PDE is elliptic when $D^{2}u>g_{xx}(Y(\cdot,u,Du),Z(\cdot,u,Du))$ as matrices. A necessary condition for ellipticity is $B>0$.

We introduce an analogue of convexity theory in which the generating function plays the role of a supporting hyperplane. We say a function $u:\Omega\rightarrow\mathbf{R}$ is $g$-convex provided for all $x_{0}\in\Omega$ there is $y_{0},z_{0}$ such that

In this case $g(\cdot,y_{0},z_{0})$ is called a $g$-support at $x_{0}$. We say $u$ is strictly $g$-convex if the inequality in (10) is strict, that is, any given $g$-support only touches $u$ at a single point. Note when $u$ is differentiable, since $u(\cdot)-g(\cdot,y_{0},z_{0})$ has a minimum at $x_{0}$, we have $Du(x_{0})=g_{x}(x_{0},y_{0},z_{0})$. This combined with (9) is equivalent to (5) and (4) so that $y_{0}=Y(x_{0},u(x_{0}),Du(x_{0}))$ and $z_{0}=Z(x_{0},u(x_{0}),Du(x_{0}))$. Moreover, if $u$ is $C^{2}$, again since $u(\cdot)-g(\cdot,y_{0},z_{0})$ has a minimum at $x_{0}$, we have $D^{2}u(x_{0})\geq g_{xx}(x_{0},y_{0},z_{0})$ and (8) is degenerate elliptic. However when $u$ is not differentiable (9) and (10) may hold for more than one $y_{0},z_{0}$. The set of $y_{0}$ for which there is $z_{0}$ such that (9) and (10) hold is denoted $Y_{u}(x_{0})$. Similarly for $Z_{u}(x_{0})$. When both are a singleton we identify these sets with their single element.

The mapping $Y_{u}$ allows us to define a notion of Alexandrov solution. A $g$-convex function $u$ is called an Alexandrov solution of (1) on $\Omega$ provided for every Borel $E\subset\Omega$

Since $g$-convex functions are locally semi-convex and thus differentiable almost everywhere the integrand on the right-hand side is well defined almost everywhere. We see, via the change of variables formula, that when $u$ is $C^{2}$ and $Y(\cdot,u,Du)$ is a $C^{1}$ diffeomorphism Alexandrov solutions are classical solutions. Moreover in the case of inequality (2) the notion of Alexandrov solution is that for every $E\subset\Omega$

Here we call the measure $\mu$ defined on Borel $E\subset\Omega$ by $\mu(E)=|Y_{u}(E)|$ the $g$-Monge–Ampère measure.

With the goal of introducing a notion of domain convexity we introduce a generalisation of line segments. A collection of points $\{x_{\theta}\}_{\theta\in[0,1]}$ is called a $g$-segment with respect to $y\in\Omega^{*},z\in\cap_{\theta}I_{x_{\theta},y}$ joining $x_{0}$ to $x_{1}$ provided

Using this we say a set $\Omega$ is $g$-convex with respect to $(y,z)$ provided for each $x_{0},x_{1}\in\Omega$ the above $g$-segment is contained in $\Omega$. The $g$-segment is unique via condition A1^∗ in Section 2.7. If $A\subset\mathbf{R}^{n}$ and $B\subset\mathbf{R}$ we say $\Omega$ is $g$-convex with respect to $A\times B$ if $\Omega$ is $g$-convex with respect to every $(y,z)\in A\times B.$

Domain convexity and a Monge–Ampère measure bounded below is necessary and sufficient for strict convexity in 2D in the Monge–Ampère case. In the optimal transport case we need the now well known A3w condition. This condition was introduced for optimal transport by Ma, Trudinger, and Wang [16] and generalised to GJE by Trudinger. For GJE we use an additional condition A4w (due to Trudinger [18]).

A3w. A generating function $g$ is said to satisfy the condition A3w provided for all $\xi,\eta\in\mathbf{R}^{n}$ with $\xi\cdot\eta=0$ and $(x,u,p)\in\mathcal{U}$ there holds

A4w. The matrix $A$ is non-decreasing in $u$, in the sense that for any $\xi\in\mathbf{R}^{n}$

In light of Lemma 1 (§3) we regard A3w and A4w as tools for controlling how $g$-convex functions separate from their supporting hyperplanes.

Strictly convex functions have $C^{1}$ Legendre transforms. This provides a useful technique for proving solutions of Monge–Ampère inequalities are $C^{1}$. The same technique in the $g$-convex case requires the $g^{*}$-transform which is defined in terms of the dual generating function. We introduce the dual generating function here and the $g^{*}$ transform in Section 5. We set

The dual generating function, $g^{*}$, is the unique function defined on $\Gamma^{*}$ by

It follows that if $(x,y,z)\in\Gamma$ then

Further, if $u$ is $g$-convex and $y\in Y_{u}(x_{0})$ then the corresponding support is $g(\cdot,y_{0},g^{*}(x_{0},y_{0},u(x_{0})))$.

We introduce dual conditions on $g^{*}$.
A1^∗. For all $(y,z,q)\in\mathcal{V}$ defined as

there is a unique $(x,y,u)\in\Gamma^{*}$ such that

Moreover we define mapping $X(y,z,q),U(y,z,q)$ as the unique $x,u$ that satisfy these equations (cf. (4) and (5)). As remarked in Section 2.5 A1^∗ implies uniqueness of the $g$-segment between two points. For this note by differentiating (13) we obtain $\frac{g_{y}}{g_{z}}(x,y,z)=-g^{*}_{y}(x,y,g(x,y,z))$. The right hand side is injective in $x$ for fixed $y,z$.

We define dual objects by swapping the roles of $x$ and $y$ as well as $z$ and $u$. In particular $g^{*}$-convex functions are those defined on $\Omega^{*}$ with $g^{*}$ supports, and for a $g^{*}$-convex $v$ we have the mappings $X_{v}(\cdot),U_{v}(\cdot)$ which are analogues of $Y_{u},Z_{u}$. Similarly $g^{*}$-segments are used to define $g^{*}$-convex sets. We also define dual conditions A2^∗, A3^∗, A4w^∗. However Trudinger [18] showed the conditions A2^∗ and A3w^∗ are satisfied provided A2 and A3w are.

Note conditions A4w and A4w^∗ are always satisfied in the optimal transport case. Indeed Corollary 1 implies the $C^{1}$ differentiability of potentials for the optimal transport problem (and subsequently continuity of the optimal transport map) whenever the right hand side of the associated Monge–Ampère type equation is bounded from above, the target is $c^{*}$-convex, and the A3w condition is satisfied. These conditions are all known to be necessary. Necessity of the $c^{*}$-convexity is due to Ma, Trudinger, and Wang [16] whilst necessity of A3w is due to Loeper [15].

The proof of Theorem 1 is given in Section 4. Finally the proof of Corollary 1 (which follows from Theorem 1) is given in Section 5.