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

Abstract

We present a proof of strict g-convexity in 2D for solutions of generated Jacobian equations with a g-Monge–Ampère measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a g-Monge–Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge–Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the g-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge–Ampère case our key assumptions, namely A3w and domain convexity, are necessary.

Proof of main lemma

In this appendix we provide the proof of Lemma 1.

Proof

We first compute a differentiation formula for second derivatives along g-segments. We suppose

$$\begin{aligned} \frac{g_y}{g_z}\left( x_\theta ,y_0,z_0\right) = \theta q_1+(1-\theta )q_0, \end{aligned}$$

(31)

and set $q = q_1-q_0$. We begin with a formula for first derivatives. Since

$$\begin{aligned} \frac{d}{d\theta } = \left( \dot{x_\theta }\right) _iD_{x_i} \end{aligned}$$

(32)

we need to compute $(\dot{x_\theta })_i$. Differentiate (31) with respect to $\theta$ and obtain¹

$$\begin{aligned} \left[ \frac{g_{i,m}}{g_z}-\frac{g_{i,z}g_{,m}}{g_z^2}\right] \left( \dot{x_\theta }\right) _i = q_m, \end{aligned}$$

from which it follows that

$$\begin{aligned} \left( \dot{x_\theta }\right) _i = g_zE^{m,i}q_m, \end{aligned}$$

where $E^{m,i}$ denotes the m, i${\text {th}}$ entry of $E^{-1}$. Thus (32) becomes

$$\begin{aligned} \frac{d}{d\theta } = g_zE^{m,i}q_mD_{x_i}. \end{aligned}$$

(33)

Using this expression to compute second derivatives we have

$$\begin{aligned} \frac{d^2}{d\theta ^2}&= g_zE^{n,j}D_{x_j}\left( g_zE^{m,i}D_{x_i}\right) q_mq_n\\&= g_{z}^2E^{n,j}E^{m,i}q_mq_nD_{x_ix_j} +g_z^2q_mq_nE^{n,j}D_{x_j}\left( E^{m,i}\right) D_{x_i}\\&\quad + g_zg_{j,z}E^{n,j}E^{m,i}q_mq_nD_{x_i}. \end{aligned}$$

The formula for differentiating an inverse yields

$$\begin{aligned} \frac{d^2}{d\theta ^2}&= \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j D_{x_ix_j} -g_z^2q_mq_nE^{n,j}E^{m,a}D_{x_j}\left( E_{ab}\right) E^{b,i}D_{x_i}\nonumber \\&\quad + g_zg_{j,z}E^{n,j}E^{m,i}q_mq_nD_{x_i}. \end{aligned}$$

(34)

Now compute

$$\begin{aligned} D_{x_j}\left( E_{ab}\right)&= D_{x_j}\left[ g_{a,b}-\frac{g_{a,z}g_{,b}}{g_z}\right] \nonumber \\&= g_{aj,b}-\frac{g_{aj,z}g_{,b}}{g_z}-\frac{g_{a,z}g_{j,b}}{g_z}+\frac{g_{j,z}g_{a,z}g_{,b}}{g_z^2}\nonumber \\&= -\frac{g_{a,z}}{g_z}E_{jb} +E_{l,b}D_{p_l}g_{aj} . \end{aligned}$$

(35)

Here we have used that

$$\begin{aligned} E_{l,b}D_{p_l}g_{aj}(\cdot ,Y(\cdot ,u,p),Z(\cdot ,u,p)) = g_{aj,b}-\frac{g_{aj,z}g_{,b}}{g_z}, \end{aligned}$$

which follows by computing $D_{p_l}g_{aj}$, differentiating (4) with respect to p to express $Z_p$ in terms of $Y_p$, and employing $E^{i,j} = D_{p_j}Y^i$ (which is obtained via calculations similar to those for (7)).

Substitute (35) into (34) to obtain

$$\begin{aligned} \frac{d^2}{d\theta ^2}&= \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j D_{x_ix_j} -g_z^2q_mq_nE^{n,j}E^{m,a}E_{l,b}D_{p_l}g_{aj}E^{b,i}D_{x_i}\\&\quad +\left[ g_zg_{a,z}E^{n,j}E^{m,a}E_{j,b}E^{b,i}D_{x_i+}g_zg_{j,z}E^{n,j}E^{m,i}D_{x_i}\right] q_mq_n\\&= \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j D_{x_ix_j} -g_z^2q_mq_nE^{n,j}E^{m,a}D_{p_i}g_{aj}D_{x_i}\\&\quad \quad +\left[ g_zg_{a,z}E^{n,i}E^{m,a}D_{x_i+}g_zg_{j,z}E^{n,j}E^{m,i}D_{x_i}\right] q_mq_n\\&= \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j\left( D_{x_i,x_j}-D_{p_k}g_{ij}D_{x_k}\right) \\&\quad +g_{j,z}\left( E^{m,j}q_m\frac{d}{d\theta }+E^{n,j}q_n\frac{d}{d\theta }\right) , \end{aligned}$$

where in the last equality we swapped the dummy indices i and a on the second term to allow us to collect like terms and also used (33).

Now let’s use this identity to compute $h''(\theta )$. We have

$$\begin{aligned} h''(\theta )&= \left[ D_{ij}u\left( x_{\theta }\right) -g_{ij}\left( x_{\theta },y_0,z_0\right) \right. \\&\left. \quad -D_{p_k}g_{ij}\left( x_{\theta },y_0,z_0\right) \left( D_ku\left( x_\theta \right) -D_kg\left( x_{\theta },y_0,z_0\right) \right) \right] \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j \\&\quad +g_{j,z}\left( E^{m,j}q_mh'+E^{n,j}q_nh'\right) . \end{aligned}$$

Terms on the final line are bounded below by $-K|h'(\theta )|$. Now after adding and subtracting $g_{ij}(x_{\theta },y,z)$ for $y=Y_u(x_\theta ),z=Z_u(x_\theta )$ we have

$$\begin{aligned} h''(\theta )&\ge \left[ D_{ij}u\left( x_{\theta }\right) -g_{ij}\left( x_{\theta },y,z\right) \right] \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j + \left[ g_{ij}\left( x_{\theta },y,z\right) -g_{ij}\left( x_{\theta },y_0,z_0\right) \right. \\&\left. \quad -D_{p_k}g_{ij}\left( x_{\theta },y_0,z_0\right) \left( D_ku\left( x_\theta \right) -D_kg\left( x_{\theta },y_0,z_0\right) \right) \right] \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j \\&\quad -K|h'(\theta )|. \end{aligned}$$

Set $u_0 = g(x_{\theta },y_0,z_0), \ u_1=u(x_\theta )$ $p_0 = g_x(x_{\theta },y_0,z_0),$ and $p_1 = Du(x_\theta )$. Then rewriting in terms of the matrix A we have

$$\begin{aligned} h''(\theta )&\ge \big [D_{ij}u(x_{\theta }) -g_{ij}(x_{\theta },y,z)\big ](\dot{x_\theta })_i(\dot{x_\theta })_j +\big [A_{ij}(x_{\theta },u_1,p_1)-A_{ij}(x_{\theta },u_0,p_0)\\&\quad -D_{p_k}A_{ij}(x_{\theta },u_0,p_0)(p_1-p_0)\big ](\dot{x_\theta })_i(\dot{x_\theta })_j -K|h'(\theta )|\\&= \big [D_{ij}u(x_{\theta }) -g_{ij}(x_{\theta },y,z)\big ](\dot{x_\theta })_i(\dot{x_\theta })_j + A_{ij,u}(x_\theta ,u_\tau ,p)(u_1-u_0)(\dot{x_\theta })_i(\dot{x_\theta })_j\\&\quad +\big [A_{ij}(x_{\theta },u_0,p_1)-A_{ij}(x_{\theta },u_0,p_0)\\&\quad -D_{p_k}A_{ij}(x_{\theta },u_0,p_0)(p_1-p_0)\big ](\dot{x_\theta })_i(\dot{x_\theta })_j-K|h'(\theta )| \end{aligned}$$

Here $u_\tau = \tau u +(1-\tau )u_0$ results from a Taylor series. Another Taylor series for $f(t):= A_{ij}(x_{\theta },u_0,t p_1+(1-t)p_0)$ and we obtain

$$\begin{aligned} h''(\theta )&\ge \left[ D_{ij}u\left( x_{\theta }\right) -g_{ij}\left( x_{\theta },y,z\right) \right] \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j + A_{ij,u}\left( u_1-u_0\right) \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j\\&-K|h'(\theta )| + D_{p_kp_l}A_{ij}\left( x_{\theta },u_0,p_t\right) \left( \dot{x_\theta }\right) _i\left( \dot{x_\theta }\right) _j\left( p_1-p_0\right) _k\left( p_1-p_0\right) _l. \end{aligned}$$

This is the desired formula. $\square$