H\"older regularity of the 2D dual semigeostrophic equations via analysis of linearized Monge-Amp\`ere equations

TL;DR

This paper establishes the H"older regularity of solutions to the 2D dual semigeostrophic equations and related maps, using new interior estimates for linearized Monge-Ampère equations, extending previous results.

Contribution

It provides the first interior H"older estimates for linearized Monge-Ampère equations in 2D with inhomogeneous terms, leading to improved regularity results for dual semigeostrophic equations.

Findings

Proved H"older regularity of the time derivative of solutions.

Established H"older regularity of polar factorization maps.

Extended regularity results to densities bounded away from zero and infinity.

Abstract

We obtain the H\"older regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity. Our main tool is an interior H\"older estimate in two dimensions for an inhomogeneous linearized Monge-Amp\`ere equation with right hand side being the divergence of a bounded vector field. As a further application of our H\"older estimate, we prove the H\"older regularity of the polar factorization for time-dependent maps in two dimensions with densities bounded away from zero and infinity. Our applications improve previous work by G. Loeper who considered the cases of densities sufficiently close to a positive constant.

Equations

$$\left\{\begin{array}[]{rll}\partial_{t}\rho_{t}(x)+\mbox{div}\,(\rho_{t}(x)U_{t}(x))&=0&(t,x)\in[0,\infty)\times{\mathbb{T}}^{2},\\ U_{t}(x)&=(x-\nabla P_{t}^{\ast}(x))^{\perp}&(t,x)\in[0,\infty)\times{\mathbb{T}}^{2},\\ \det D^{2}P_{t}^{\ast}(x)&=\rho_{t}(x)&(t,x)\in[0,\infty)\times{\mathbb{T}}^{2},\\ P_{t}^{\ast}(x)\text{ convex }&&(t,x)\in{\mathbb{T}}^{2},\\ \rho_{0}(x)&=\rho^{0}(x)&x\in{\mathbb{T}}^{2}\end{array}\right.$$

$$P_t^{\ast }-|x{|}^2/2\text{is}{\mathbb{Z}}^2-\text{periodic}.$$

$$\det D^2{P}_t^{\ast }={\rho }_t\text{ in }{\mathbb{T}}^2.$$

$$\int {\int }_{{\mathbb{T}}^2}\left\{{\partial }_t{\phi }_t(x)+\nabla {\phi }_t(x)\cdot U_t(x)\right\}{\rho }_t(x)dxdt+{\int }_{{\mathbb{T}}^2}{\phi }_0(x){\rho }^0(x)dx=0$$

$$P_t(x)=\underset{y\in {\mathbb{R}}^2}{\sup }(x\cdot y-P_t^{\ast }(y)).$$

$$\left\{\begin{array}[]{rl}p_{t}(x)&:=P_{t}(x)-|x|^{2}/2,\\ u_{t}(x)&:=(\partial_{t}\nabla P_{t}^{\ast})\circ\nabla P_{t}(x)+D^{2}P_{t}^{\ast}(\nabla P_{t}(x))\cdot(\nabla P_{t}(x)-x)^{\perp}.\end{array}\right.$$

$$\left\{\begin{array}[]{rll}\partial_{t}\nabla p_{t}(x)+(u_{t}(x)\cdot\nabla)\nabla p_{t}(x)-(\nabla p_{t}(x))^{\perp}+u_{t}(x)&=0&(t,x)\in[0,\infty)\times{\mathbb{T}}^{2},\\ \mbox{div}\,u_{t}(x)&=0&(t,x)\in[0,\infty)\times{\mathbb{T}}^{2},\\ p_{0}(x)&=p^{0}(x)&x\in{\mathbb{T}}^{2}.\end{array}\right.$$

$$\Vert {\partial }_t{P}_t^{\ast }{\Vert }_{L^{\infty }((0,\infty ),C^{\alpha }({\mathbb{T}}^2))}+\Vert {\partial }_t{P}_t{\Vert }_{L^{\infty }((0,\infty ),C^{\alpha }({\mathbb{T}}^2))}\le C.$$

$$\nabla \cdot (M_{P_t^{\ast }}(D^2{P}_t^{\ast })\nabla ({\partial }_t{P}_t^{\ast }))={\partial }_t{\rho }_t=\text{mbox}div(-{\rho }_t{U}_t):=\text{mbox}div{F}_t,$$

$$\lambda \le \det M=\det D^2{P}_t^{\ast }\le \Lambda .$$

$$\lambda \le \det D^2\phi \le \Lambda \text{in}\Omega$$

$$D^2\phi ={\left({\phi }_{ij}\right)}_{1\le i,j\le n}\equiv {\left(\frac{{\partial }^2\phi }{\partial x_i\partial x_j}\right)}_{1\le i,j\le n}.$$

$$\sum _{i,j=1}^n{\Phi }^{ij}{u}_{ij}=\text{mbox}divF$$

$$\sum _{i,j=1}^n{\partial }_j({\Phi }^{ij}{u}_i)\equiv \nabla \cdot (\Phi \nabla u)=\text{mbox}divF.$$

$$S_{\phi }(x,h):=\{y\in \Omega :\phi (y)<\phi (x)+\nabla \phi (x)\cdot (y-x)+h\}.$$

$${\Phi }^{ij}{u}_{ij}=\text{div}F$$

$$|u(x)-u(x_0)|\le C(p,\lambda ,\Lambda ,\text{diam}(\Omega ),h_0)\left(\Vert F{\Vert }_{L^{\infty }(S_{\phi }(x_0,2h_0))}+\Vert u{\Vert }_{L^p(S_{\phi }(x_0,2h_0))}\right)|x-x_0{|}^{\gamma }.$$

$$\left\{\begin{array}[]{rl}\Phi^{ij}u_{ij}&=\emph{div }F\quad\mbox{ in}\quad S_{\varphi}(x_{0},h),\\ u&=0\qquad\quad\,\mbox{on}\quad\partial S_{\varphi}(x_{0},h),\end{array}\right.$$

$$\underset{S_{\phi }(x_0,h)}{\sup }|u|\le C(\lambda ,\Lambda ,\text{diam}(\Omega ),h_0)\Vert F{\Vert }_{L^{\infty }(S_{\phi }(x_0,h))}{h}^{\delta }.$$

$${\Phi }^{ij}{u}_{ij}=\text{mbox}divF$$

$$\underset{S_{\phi }(x_0,h/2)}{\sup }|u|\le C(p,\lambda ,\Lambda ,\text{diam}(\Omega ))\left(\Vert F{\Vert }_{L^{\infty }(S_{\phi }(x_0,h))}+h^{-1/p}\Vert u{\Vert }_{L^p(S_{\phi }(x_0,h))}\right).$$

$${\Phi }^{ij}{u}_{ij}=0$$

$$\underset{S_{\phi }(x_0,h)}{\sup }u\le C(n,\lambda ,\Lambda )\underset{S_{\phi }(x_0,h)}{\inf }u.$$

$$\left\{\begin{array}[]{rl}L_{\varphi}g_{S}(\cdot,y)&=\delta_{y}\,\,\quad\mbox{ in}\quad S,\\ g_{S}(\cdot,y)&=0\qquad\mbox{on}\quad\partial S.\end{array}\right.$$

$${\left({\int }_S|{\nabla }_x{g}_S(x,y){|}^{1+\kappa }dx\right)}^{\frac{1}{1+\kappa }}\le C(\lambda ,\Lambda ,\text{diam}(\Omega ),h_0)h^{{\kappa }_1}.$$

$$\kappa =\frac{\epsilon }{2+\epsilon },{\kappa }_1=\frac{{\epsilon }_{\ast }-\epsilon }{2(1+{\epsilon }_{\ast })(1+\epsilon )}$$

$${\left({\int }_{S_{\phi }(x_0,1)}|w{|}^{p}dx\right)}^{1/p}\le K{\left({\int }_{S_{\phi }(x_0,1)}{\Phi }^{ij}{w}_i{w}_{j}dx\right)}^{1/2}\text{for all }w\in C_0^1(S_{\phi }(x_0,1)).$$

$${\int }_S{g}_S^p(x,y)dx\le C(\lambda ,\Lambda ,p)h\text{for all }y\in S.$$

$${\Phi }^{ij}{u}_{ij}=\text{mbox}divF$$

$$\text{mbox}div(\Phi \nabla u)=\text{mbox}divF.$$

Full text

The research of the author was supported in part by NSF grant DMS-1500400.

1. Introduction and statement of the main results

In this paper, we obtain the Hölder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity; see Theorem 1.2. Our main tool is an interior Hölder estimate in two dimensions for an inhomogeneous linearized Monge-Ampère equation with right hand side being the divergence of a bounded vector field when the Monge-Ampère measure is only assumed to be bounded between two positive constants; see Theorem 1.3. As a further application of our Hölder estimate, we prove the Hölder regularity of the polar factorization for time-dependent maps in two dimensions with densities bounded away from zero and infinity; see Theorems 6.1 and 6.2. Our applications improve previous work by Loeper [23] who considered the cases of densities sufficiently close to a positive constant.

1.1. The dual semigeostrophic equations on ${\mathbb{T}}^{2}$

The semigeostrophic equations are a simple model used in meteorology to describe large scale atmospheric flows. As explained for example in Benamou-Brenier [2, Section 2.2], Loeper [24, Section 1.1], and Cullen [10], the semigeostrophic equations can be derived from the three-dimensional incompressible Euler equations, with Boussinesq and hydrostatic approximations, subject to a strong Coriolis force. Since for large scale atmospheric flows the Coriolis force dominates the advection term, the flow is mostly bi-dimensional.

Here we focus on the dual semigeostrophic equations. Note that, using solutions to the dual equations together with the $W^{2,1}$ regularity for Aleksandrov solutions to the Monge-Ampère equations obtained by De Philippis and Figalli [13], Ambrosio-Colombo-De Philippis-Figalli [1] established global in time distributional solutions to the original semigeostrophic equations on the two dimensional torus. For more on the Monge-Ampère equations and Aleksandrov solutions, see the books by Figalli [15] and Gutiérrez [19].

The dual equations of the semigeostrophic equations on the two dimensional torus ${\mathbb{T}}^{2}={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$ are the following system of nonlinear transport equations

[TABLE]

for $(\rho_{t},P_{t}^{\ast})$ with the boundary condition

[TABLE]

Here the initial potential density $\rho^{0}$ is a probability measure on ${\mathbb{T}}^{2}$. Throughout, we use $w^{\perp}$ to denote the rotation by $\pi/2$ vector $(-w_{2},w_{1})$ for $w=(w_{1},w_{2})\in{\mathbb{R}}^{2}$ and $f_{t}(\cdot)$ to denote the function $f(t,\cdot)$.

Existence of global weak solutions for the (1.1)-(1.2) system has been established via time discretization in Benamou-Brenier [2] and Cullen-Gangbo [12]. To be precisely, in these cited papers, the proof is given in ${\mathbb{R}}^{3}$, but it can be rewritten verbatim on the two-dimensional torus by using the optimal transport maps; see [1, Theorems 2.1 and 3.1] for further details. When $\rho^{0}$ is Hölder continuous and bounded away from zero and infinity on ${\mathbb{T}}^{2}$, Loeper [24] showed that there is a unique, short-time, Hölder solution $\rho$ to (1.1)-(1.2); the time interval for this Hölder solution depends only on the bounds on $\rho^{0}$. However, when $\rho^{0}$ is only a general probability measure, the uniqueness of weak solutions is still an open question. Due to this lack of uniqueness and to avoid unnecessary confusions, we make the following definition on weak solutions (as already established in [2] and [12]) to (1.1)-(1.2) that we are going to use throughout the paper.

Definition 1.1**.**

By a weak solution to (1.1)-(1.2), we mean a pair of functions $(\rho_{t},P_{t}^{\ast})$ on ${\mathbb{R}}^{2}$ with the following properties:

• (i) $P_{t}^{\ast}$ is convex on ${\mathbb{R}}^{2}$ with $P_{t}^{\ast}-|x|^{2}/2$ being ${\mathbb{Z}}^{2}$ periodic; $\rho_{t}$ is ${\mathbb{Z}}^{2}$ periodic;

(ii) $P_{t}^{\ast}$ is an Aleksandrov solution to the Monge-Ampère equation

[TABLE]

(iii) $U_{t}(x)=(x-\nabla P_{t}^{\ast}(x))^{\perp}$ and $\rho_{t}$ satisfy the equations $\partial_{t}\rho_{t}(x)+\mbox{div}\,(\rho_{t}(x)U_{t}(x))=0$ and $\rho_{0}=\rho^{0}$ on ${\mathbb{T}}^{2}$ in the distributional sense, that is,

[TABLE]

for every $\varphi\in C_{0}^{\infty}([0,\infty)\times{\mathbb{R}}^{2})$ ${\mathbb{Z}}^{2}$-periodic in the space variable.

For completeness, we briefly indicate how to obtain distributional solutions to the original semigeostrophic equations from solutions $(\rho_{t},P_{t}^{\ast})$ of the dual equations (1.1)-(1.2); see [1] for a rigorous treatment. Let us denote by $P_{t}$ the Legendre transform of $P_{t}^{\ast}$, that is,

[TABLE]

Let $p^{0}(x)=P_{0}(x)-|x|^{2}/2$ and

[TABLE]

Then $(p_{t},u_{t})$ is a global Eulerian solution to the original semigeostrophic equations:

[TABLE]

In (1.4), the functions $u_{t}$ and $p_{t}$ represent respectively the velocity and the pressure. The quantity $u_{t}^{g}$ related to the system (1.4) defined by $u_{t}^{g}(x)=(\nabla p_{t}(x))^{\perp}$ is called the semi-geostrophic wind.

We now return to the regularity of solutions to (1.1)-(1.2) in the typical case where the initial density $\rho^{0}$ is bounded between two positive constants $\lambda$ and $\Lambda$. The space regularity of the solutions is now well understood thanks to regularity results for the Monge-Ampère equations which are mainly due to Caffarelli, De Philippis, Figalli, Savin, and Schmidt [4, 5, 6, 7, 13, 14, 28]. We will recall these results in Theorems 4.1 and 4.2.

Regarding the regularity with respect to time, to the best of our knowledge, the most refined result so far is due to Loeper [23] under the condition that $\lambda$ and $\Lambda$ are close. More precisely, Loeper shows that if the initial potential density $\rho^{0}$ is sufficiently close to a positive constant, say, $1-\varepsilon_{0}\leq\rho^{0}\leq 1+\varepsilon_{0}$ on ${\mathbb{T}}^{2}$ for some $\varepsilon_{0}>0$ small, then $\partial_{t}P_{t},\partial_{t}P_{t}^{\ast}\in L^{\infty}((0,\infty),C^{\alpha_{0}}({\mathbb{T}}^{2}))$ where $\alpha_{0}>0$ depends only on $\varepsilon_{0}$; see [23, Theorems 2.2, 2.3 and 9.2].

It is thus an interesting problem to study the Hölder continuity of $\partial_{t}P_{t}^{\ast}$ and $\partial_{t}P_{t}$ in the system (1.1)-(1.2) when the closeness of the density $\rho^{0}$ to $1$ is removed. This is precisely what we prove in the following theorem.

Theorem 1.2** (Hölder regularity of the two dimensional dual semigeostrophic equations).**

Let $\rho^{0}$ be a probability measure on ${\mathbb{T}}^{2}$. Suppose that that $\lambda\leq\rho^{0}\leq\Lambda$ in ${\mathbb{T}}^{2}$ for positive constants $\lambda$ and $\Lambda$. Let $(\rho_{t},P_{t}^{\ast})$ solve (1.1)-(1.2). Let $P_{t}$ be the Legendre transform of $P_{t}^{\ast}$. Then, there exist $\alpha=\alpha(\lambda,\Lambda)\in(0,1)$ and $C=C(\lambda,\Lambda)>0$ such that

[TABLE]

We will prove Theorem 1.2 in Section 4, using Theorem 1.3 and following the strategy in [23].

Let us briefly explain how to prove the Hölder continuity of $\partial_{t}P_{t}^{\ast}$ and $\partial_{t}P_{t}$ in (1.1)-(1.2). To simplify the presentation, we assume all functions involved are smooth but the estimates we wish to establish will depend only on $\lambda$ and $\Lambda$. Since $\mbox{div}\,U_{t}=0$, the $L^{\infty}({\mathbb{T}}^{2})$ norm of $\rho_{t}$ is preserved in time; see also [2, Proposition 5.2] and [23, Lemma 9.1]. Thus, for all $t\geq 0$, we have $\lambda\leq\rho_{t}\leq\Lambda$ in ${\mathbb{T}}^{2}$. Differentiating both sides of $\det D^{2}P_{t}^{\ast}=\rho_{t}$ with respect to $t$, and using the first and second equations of (1.1), we find that $\partial_{t}P_{t}^{\ast}$ solves the linearized Monge-Ampère equation

[TABLE]

where $M_{P_{t}^{\ast}}(D^{2}P_{t}^{\ast})$ represents the matrix of cofactors of the Hessian matrix $D^{2}P_{t}^{\ast}$.

With the bounds $\lambda\leq\rho_{t}\leq\Lambda$ on $\rho_{t}$, (1.5) is a degenerate elliptic equation because we only know that the coefficient matrix $M=M_{P_{t}^{\ast}}(D^{2}P_{t}^{\ast})$ in (1.5) is positive definite (due to the convexity of $P_{t}^{\ast}$) and satisfies

[TABLE]

Moreover, we can bound $F_{t}$ in $L^{\infty}({\mathbb{T}}^{2})$ and $\partial_{t}P_{t}^{\ast}$ in $L^{2}({\mathbb{T}}^{2})$, uniformly in $t$; see Theorem 4.1(i, ii, iii). The Hölder regularity of $\partial_{t}P_{t}^{\ast}$ hence relies on the Hölder regularity of solutions to equation of the type (1.5) given the $L^{p}$ bounds on the solutions, where $F_{t}$ is a bounded vector field.

At this point, Loeper’s approach and assumption on the initial potential density $\rho^{0}$ and ours differ.

The key tools used by Loeper [23] are global and local maximum principles for solutions of degenerate elliptic equations proved by Murthy and Stampacchia [27] and Trudinger [30], and a Harnack inequality of Caffarelli and Gutiérrez [8] for solutions of the homogeneous linearized Monge-Ampère equation; see Theorem 2.3. These results hold in all dimensions $n\geq 2$. The results of Murthy-Stampacchia and Trudinger, that we will recall in Theorems 2.8 and 2.9, require the high integrability of the coefficient matrix of the degenerate elliptic equations. In application to the dual semigeostrophic equations (1.1)-(1.2), this high integrability translates to the high integrability of the matrix $M_{P_{t}^{\ast}}(D^{2}P_{t}^{\ast})$ in (1.5), or equivalently, to the high integrability of $D^{2}P_{t}^{\ast}$. In view of Caffarelli’s $W^{2,p}$ estimates for the Monge-Ampère equation [4] and Wang’s counterexamples [31], the last point forces the closeness of the density $\rho^{0}$ to 1. This is exactly the assumption on $\rho^{0}$ in [23, Theorems 2.2, 2.3 and 9.2].

Our main tool in proving Theorem 1.2 is the Hölder estimate in Theorem 1.3 for the inhomogeneous linearized Monge-Ampère equation of the type (1.5) in two dimensions, *without relying on $\lambda$ and $\Lambda$ being close. *This is the topic of the next section.

1.2. Hölder estimates for inhomogeneous linearized Monge-Ampère equation

Let $\Omega\subset{\mathbb{R}}^{n}$ ($n\geq 2$) be a bounded convex set with nonempty interior and let $\varphi\in C^{2}(\Omega)$ be a convex function such that

[TABLE]

for some positive constants $\lambda$ and $\Lambda$.

Let $\Phi=(\Phi^{ij})_{1\leq i,j\leq n}=(\det D^{2}\varphi)(D^{2}\varphi)^{-1}$ denote the cofactor matrix of the Hessian matrix

[TABLE]

Note that, in terms of the notation of the previous Section 1.1, we have $\Phi=M_{\varphi}(D^{2}\varphi)$.

We are interested in obtaining interior Hölder estimates for solutions to the inhomogeneous linearized Monge-Ampère equation

[TABLE]

in terms of $L^{p}$ bounds on the solutions where $F:\Omega\rightarrow{\mathbb{R}}^{n}$ is a bounded vector field. Our motivation comes from the regularity of the semigeostrophic equations [1, 2, 12, 16] as mentioned in Section 1.1.

Since the matrix $\Phi$ is divergence free; that is, $\displaystyle\sum_{j=1}^{n}\partial_{j}\Phi^{ij}=0~{}\text{for all }i=1,\cdots n,$ the equation (1.7) can also be written in the divergence form

[TABLE]

When $F\equiv 0$, interior Hölder estimates for solutions to (1.7), under the condition (1.6) on the Monge-Ampère measure of $\varphi$, were established by Caffarelli and Gutiérrez in their fundamental work [8]. It is worth mentioning that one of the motivations of the work [8] was Lagrangian models of atmospheric and oceanic flows, including the dual semigeostrophic equations.

When $F\not\equiv 0$, we are able to obtain in this paper the Hölder estimates for solutions to (1.7) in two dimensions; see Theorem 1.3. The important point to note here is that our Hölder exponent depends only on the bounds $\lambda$ and $\Lambda$ of the Monge-Ampère measure of $\varphi$.

Besides its application to the semigeostrophic equations, Theorem 1.3 also applies to the Hölder regularity of the polar factorization for time dependent maps in two dimensions with densities bounded away from zero and infinity, improving previous results by Loeper [23]; see Section 6.

To state our estimates for (1.7), we recall the notion of sections of a convex function $\varphi\in C^{1}(\Omega)$. Given $x\in\Omega$ and $h>0$, the Monge-Ampère section of $\varphi$ centered at $x$ with height $h$ is defined by

[TABLE]

Our main Hölder estimate is contained in the following theorem.

Theorem 1.3** (Interior Hölder estimate for the inhomogeneous linearized Monge-Ampère equation in two dimensions).**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $F:\Omega\rightarrow{\mathbb{R}}^{n}$ is a bounded vector field. Given a section $S_{\varphi}(x_{0},4h_{0})\subset\subset\Omega$. Let $p\in(1,\infty)$. There exist a universal constant $\gamma>0$ depending only on $\lambda$ and $\Lambda$ and a constant $C>0$, depending only on $p$, $\lambda$, $\Lambda,h_{0}$ and $\text{diam}(\Omega)$ with the following property. For every solution $u$ to

[TABLE]

in $S_{\varphi}(x_{0},4h_{0})$, and for all $x\in S_{\varphi}(x_{0},h_{0})$, we have the Hölder estimate:

[TABLE]

We will prove Theorem 1.3 in Section 5. Our main technical tools, in addition to Caffarelli-Gutiérrez’s Harnack inequality for solutions to the homogeneous linearized Monge-Ampère equation in Theorem 2.3, are new $L^{\infty}$ interior and global estimates for solutions to the inhomogeneous linearized Monge-Ampère equation (1.7) in Theorems 2.1 and 2.2.

Caffarelli and Gutiérrez [8] proved Theorem 2.3 by using basically the non-divergence form of (1.7); while we prove Theorems 2.1 and 2.2 by exploiting the divergence form character of (1.7). They are related to fine properties of Green’s function $G_{\varphi}$ of the degenerate operator $-\partial_{i}(\Phi^{ij}\partial_{j})$. The crucial observation here (see also [21, 22]) is that Green’s function $G_{\varphi}$ has, in all dimensions, the same integrability as that of the Laplace operator $\displaystyle\Delta=\sum_{i=1}^{n}\partial_{ii}$ which corresponds to the case $\varphi(x)=\frac{|x|^{2}}{2}$. On the other hand, in two dimensions, the gradient of $G_{\varphi}$ has almost integrability as that of the Laplace operator. We do not know whether the last fact is true or not in higher dimensions. Thus, it is an open question if the Hölder estimate in Theorem 1.3 holds for dimensions $n\geq 3$.

The rest of the paper is organized as follows. In Section 2, we provide key global and local estimates in Theorem 2.1 and 2.2 for the inhomogeneous linearized Monge-Ampère equation and discuss related results by Murthy-Stampacchia and Trudinger. In Section 3, we recall several basics of the Monge-Ampère equation and its linearization. We present the proof of Theorem 1.2 in Section 4. We prove Theorems 1.3, 2.1 and 2.2 in Section 5. In Section 6, we apply Theorem 1.3 to the regularity of polar factorization of time dependent maps in two dimensions. The proofs of technical results concerning Green’s function that we use in the proofs of Theorems 2.1 and 2.2 are presented in Section 7. The proofs of rescaling properties of the Monge-Ampère equation and its linearization will be given in the final section, Section 8.

2. Estimates for linearized Monge-Ampère equations and related results

In this section, we state key global and local estimates for solutions to the inhomogeneous linearized Monge-Ampère equation $\Phi^{ij}u_{ij}=\mbox{div}\,F$ and discuss related results by Murthy-Stampacchia and Trudinger regarding solutions to degenerate elliptic equations.

2.1. Estimates for the equation $\Phi^{ij}u_{ij}=\mbox{div}\,F$

Our key estimates are the following theorems.

Theorem 2.1** (Global estimate for solutions to the Dirichlet problem in two dimensions).**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $F:\Omega\rightarrow{\mathbb{R}}^{n}$ is a bounded vector field. There exist a universal constant $\delta>0$ depending only on $\lambda$ and $\Lambda$ such that for every section $S_{\varphi}(x_{0},h)$ with $S_{\varphi}(x_{0},2h_{0})\subset\subset\Omega$ for $h_{0}\geq h$ and every solution $u$ to

[TABLE]

we have

[TABLE]

Theorem 2.2** (Interior estimate for the inhomogeneous linearized Monge-Ampère equation in two dimensions).**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $F:\Omega\rightarrow{\mathbb{R}}^{n}$ is a bounded vector field. Given $p\in(1,\infty)$, there exists a constant $C>0$, depending only on $p$, $\lambda$, $\Lambda$ and $\emph{diam}(\Omega)$ with the following property: Every solution $u$ of

[TABLE]

in a section $S_{\varphi}(x_{0},h)$ with $S_{\varphi}(x_{0},2h)\subset\subset\Omega$ satisfies

[TABLE]

We will prove Theorems 2.1 and 2.2 in Section 5.

Given Theorems 2.1 and 2.2, we can easily prove Theorem 1.3 by combining them with Caffarelli-Gutiérrez’s Harnack inequality [8, Theorem 5] for the homogeneous linearized Monge-Ampère equation. For completeness, we recall their result here.

Theorem 2.3** (Caffarelli-Gutiérrez’s Harnack inequality for the linearized Monge-Ampère equation).**

Assume $n\geq 2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $u\in W^{2,n}_{\text{loc}}(\Omega)$ be a nonnegative solution of the homogeneous linearized Monge-Ampère equation

[TABLE]

in a section $S_{\varphi}(x_{0},2h)\subset\subset\Omega$. Then

[TABLE]

The main technical tool in the proof of Theorem 2.1 is the $L^{1+\kappa}$ estimate $(\kappa>0)$ stated in Proposition 2.4 for Green’s function associated to the operator $-\partial_{j}(\Phi^{ij}\partial_{i})=-\Phi^{ij}\partial_{ij}$. We will prove Theorem 2.2 using the Moser iteration. The main technical tool is the Monge-Ampère Sobolev inequality stated in Proposition 2.6. We state these results in Section 2.2.

2.2. Integrability of Green’s function and its gradient and Monge-Ampère Sobolev inequality

Let $\Omega\subset{\mathbb{R}}^{2}$ be a bounded convex set with nonempty interior and let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $g_{S}(x,y)$ be Green’s function of the divergence form operator $\displaystyle L_{\varphi}:=-\sum_{i,j=1}^{2}\partial_{j}(\Phi^{ij}\partial_{i})$ on the section $S:=S_{\varphi}(x_{0},h)\subset\subset\Omega$; that is, for each $y\in S$, $g_{S}(\cdot,y)$ is a positive solution of

[TABLE]

Here $\delta_{y}$ is the Dirac measure centered at $y$. Due to the divergence free property of $\Phi$, we will also use interchangeably $L_{\varphi}=-\Phi^{ij}\partial_{ij}$ for simplicity. The main technical tool in the proof of Theorem 2.1 is the following global $L^{1+\kappa}$ estimates for $\nabla g_{S}$.

Proposition 2.4** ($L^{1+\kappa}$ estimates for gradient of Green’s function).**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Assume that $S_{\varphi}(x_{0},2h_{0})\subset\subset\Omega$. Let $g_{S}(x,y)$ be Green’s function of the operator $L_{\varphi}:=-\Phi^{ij}\partial_{ij}$ on $S:=S_{\varphi}(x_{0},h)$ where $h\leq h_{0}$, as in (2.5). There exist universal constants $\kappa,\kappa_{1}>0$ depending only on $\lambda$ and $\Lambda$ such that for every $y\in S$, we have

[TABLE]

Remark 2.5**.**

Let $\varepsilon_{\ast}=\varepsilon_{\ast}(\lambda,\Lambda)>0$ be the universal constant in De Philippis-Figalli-Savin and Schmidt’s $W^{2,1+\varepsilon}$ estimate for the Monge-Ampère equation (1.6); see [14, 15, 28] and (3.3). Then we can choose $\kappa$ and $\kappa_{1}$ in Proposition 2.4 as follows:

[TABLE]

where $\varepsilon$ is any fixed number in the interval $(0,\varepsilon_{\ast})$. In the case of $\varphi(x)=|x|^{2}/2$, $L_{\varphi}=-\Delta$, we have $\varepsilon_{\ast}=\infty$. Thus, in this case, $\kappa$ can be chosen to be any positive number less than $1$, which is optimal.

The main technical tool in the proof of Theorem 2.2 is the following Monge-Ampère Sobolev inequality; it is a two dimensional counterpart of the higher dimensional result in [29, Theorem 3.1].

Proposition 2.6** (Monge-Ampère Sobolev inequality).**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Suppose that $S_{\varphi}(x_{0},2)\subset\subset\Omega$ and $B_{1}(0)\subset S_{\varphi}(x_{0},1)\subset B_{2}(0)$. Then, for every $p\in(2,\infty)$ there exists a constant $K>0$, depending only on $\lambda,\Lambda$ and $p$, such that

[TABLE]

The proofs of Propositions 2.4 and 2.6 are based on the following high integrability of $g_{S}$ when $n=2$ whose proof is based on [21].

Proposition 2.7** (High integrability of Green’s function).**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $S:=S_{\varphi}(x_{0},h)$ where $S_{\varphi}(x_{0},2h)\subset\subset\Omega$. Let $g_{S}(x,y)$ be Green’s function of the operator $L_{\varphi}:=-\Phi^{ij}\partial_{ij}$ on $S$, as in (2.5). Then, for any $p\in(1,\infty)$, we have

[TABLE]

We will prove Propositions 2.4, 2.6 and 2.7 in Section 7.

2.3. Related results by Murthy-Stampacchia and Trudinger

Since the matrix $\Phi=(\Phi^{ij})$ in our Theorems 2.1 and 2.2 are divergence free, the equation

[TABLE]

can be written in the divergence form

[TABLE]

In this section, we discuss related results by Murthy-Stampacchia [27] and Trudinger [30] concerning the maximum principle, local and global estimates, local and global regularity for degenerate elliptic equations in the divergence form

[TABLE]

where $M=(M_{ij})_{1\leq i,j\leq n}$ is nonnegative symmetric matrix, and $V$ is a bounded vector field in ${\mathbb{R}}^{n}$.

Without any special structure on the matrix M, it is difficult to obtain the $L^{\infty}$ bound on the solution $u$ to (2.7) in terms of the $L^{\infty}$ bound on the vector field $V$ for equation (2.7) with Dirichlet boundary data, or in terms of the $L^{\infty}$ bound on the vector field $V$ and an integral bound on the solution $u$ in a larger domain. To the best of our knowledge, some of the strongest results in this generality are due to Murthy-Stampacchia [27] and Trudinger [30]. To obtain these results, they require high integrability of the matrix $M$ and its inverse. That is, the usual strict ellipticity condition in the classical De Giorgi-Nash-Moser theory (see, for example, Chapter 8 in Gilbarg-Trudinger [17])

[TABLE]

is replaced by the following condition:

[TABLE]

where $\lambda_{M,1}(x)$ and $\lambda_{M,2}(x)$ are the smallest and largest eigenvalues of $M(x)$.

We denote by $S_{n}^{+}$ the set of $n\times n$ nonnegative symmetric matrices. For reader’s convenience, we state the following well known results.

Theorem 2.8**.**

*(Bound for Dirichlet boundary data; see [27, Chapter 7] and [30, Theorem 4.2])

Let $M=(M_{ij})_{1\leq i,j\leq n}:\Omega\to S_{n}^{+}$ be such that $\lambda_{M,1}^{-1}$ is in $L^{p}_{\text{loc}}(\Omega;S_{n}^{+})$ for some $p>n$. Let $V$ be in $L^{\infty}(\Omega;{\mathbb{R}}^{n})$. If $u$ is a solution of (2.7) in $B_{R}(y)\subset\subset\Omega$ and $u=0$ on $\partial B_{R}(y)$, then*

[TABLE]

Theorem 2.9**.**

*(Bound without boundary data; see [27, Chapter 8] and [30, Corollary 5.4])

Let $M=(M_{ij})_{1\leq i,j\leq n}:\Omega\to S_{n}^{+}$ be such that $\lambda_{M,2},\lambda_{M,1}^{-1}$ are both in $L^{p}_{loc}(\Omega)$, with $p>n$. Let $V$ be in $L^{\infty}(\Omega;{\mathbb{R}}^{n})$. Let $u$ be a solution of (2.7) in $\Omega$. Then we have for any ball $B_{2R}(y)\subset\subset\Omega$ and $q>0$*

[TABLE]

where $C$ depends on $n,R,q,p,\|\lambda_{M,2}\|_{L^{p}(B_{2R}(y))}$ and $\|\lambda_{M,1}^{-1}\|_{L^{p}(B_{2R}(y))}$.

In our Theorems 2.1 and 2.2 in two dimensions, the matrix $\Phi$ belongs to $L^{1+\varepsilon_{\ast}}_{\text{loc}}(\Omega)$, by De Philippis-Figalli-Savin and Schmidt’s $W^{2,1+\varepsilon}$ estimates for the Monge-Ampère equation [14, 28]. Thus, the smallest and largest eigenvalues $\lambda_{\Phi,1}$ and $\lambda_{\Phi,2}$ of $\Phi$ satisfies $\lambda_{\Phi,1}^{-1},\lambda_{\Phi,2}\in L^{1+\varepsilon_{\ast}}_{\text{loc}}(\Omega)$. The exponent $\varepsilon_{\ast}=\varepsilon_{\ast}(\lambda,\Lambda)>0$ is small and can be taken to be arbitrary close to [math] when the ratio $\Lambda/\lambda$ is large, by Wang’s examples [31]. In particular, when $\Lambda/\lambda$ is large, and when $M=\Phi$, the assumptions in Theorems 2.8 and 2.9 on the eigenvalues of $M$ are not satisfied.

On the other hand, in any dimension, when we impose either the continuity or closeness to a positive constant of $\det D^{2}\varphi$, then by Caffarelli’s $W^{2,p}$ estimates for the Monge-Ampère equation [4], $\lambda_{\Phi,1}^{-1}$ and $\lambda_{\Phi,2}$ belong to $L^{p}_{\text{loc}}(\Omega)$ for any $p\in(1,\infty)$. Thus, we can apply Theorems 2.8 and 2.9 to (1.7). This is what Loeper used in his proofs of the Hölder regularity of the polar factorization for time-dependent maps and the semigeostrophic equations in [23, Theorems 2.2, 2.3 and 9.2].

3. Preliminaries on the Monge-Ampère equation and its linearization

Throughout this section we fix a bounded convex set with nonempty interior $\Omega\subset{\mathbb{R}}^{n}$ and assume that $\varphi\in C^{2}(\Omega)$ is a strictly convex function satisfying

[TABLE]

for some $0<\lambda\leq\Lambda$. The results in this section hold for all dimensions $n\geq 2$.

3.1. Basics of the Monge-Ampère equation

We recall in this section some well-known results on the Monge-Ampère equation that we will use in later sections of the paper.

Universal constants

Constants depending only on $\lambda$ and $\Lambda$ in (3.1) as well as on dimension $n$ will be called universal constants.

Monge-Ampère sections

Given $x\in\Omega$ and $h>0$, the Monge-Ampère section of $\varphi$ centered at $x$ and with height $h$ is defined as

[TABLE]

A section $S_{\varphi}(x,h)$ is said to be normalized if it satisfies the following inclusions

[TABLE]

where $B_{r}(0)$ denotes the $n$-dimensional ball centered at [math] and with radius $r>0$. Recall that, by John’s lemma, every open bounded convex set with non-empty interior can be normalized by affine transformations.

Volume estimates for sections

There exists a universal constant $C(n,\lambda,\Lambda)>0$ such that for every section $S_{\varphi}(x,h)\subset\subset\Omega$, we have the following volume estimates:

[TABLE]

see [19, Corollary 3.2.4].

$W^{2,1+\varepsilon}$ estimate

By De Philippis-Figalli-Savin and Schmidt’s $W^{2,1+\varepsilon}$ estimates for the Monge-Ampère equation [14, 28] (see also [15, Theorem 4.36]), there exists $\varepsilon_{\ast}=\varepsilon_{\ast}(n,\lambda,\Lambda)>0$ such that $D^{2}\varphi\in L^{1+\varepsilon_{\ast}}_{loc}(\Omega)$. More precisely, if $S_{\varphi}(x_{0},1)$ is a normalized section and $S_{\varphi}(x_{0},2)\subset\subset\Omega$ then

[TABLE]

In the following lemma, we estimate the $L^{1+\varepsilon_{\ast}}$ norm of $\Delta\varphi$ and the $C^{\alpha}$ norm of $D\varphi$ on a section $S_{\varphi}(x_{0},h)\subset\subset\Omega$. They will be applied to Theorem 1.3 and Proposition 2.4 for $h=h_{0}$.

Lemma 3.1**.**

Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $\varepsilon_{\ast}$ be as in (3.3). There exist positive universal constants $\alpha\in(0,1),\alpha_{1}$ and $\alpha_{2}$ depending only on $\lambda$, $\Lambda$ and $n$ such that the following statements hold. If $S_{\varphi}(x_{0},2h)\subset\subset\Omega$ then

• (i)

[TABLE]

(ii)

[TABLE]

The proof of Lemma 3.1 will be given in Section 8.

3.2. Rescaling properties for the equation $\Phi^{ij}u_{ij}=\mbox{div}\,F$

Here we record how the equation (1.7) changes with respect to normalization of a section $S_{\varphi}(x_{0},h)\subset\subset\Omega$ of $\varphi$.

By subtracting $\varphi(x_{0})+\nabla\varphi(x_{0})\cdot(x-x_{0})+h$ from $\varphi$, we may assume that $\varphi\mid_{\partial S_{\varphi}(x_{0},h)}=0$ and $\varphi$ achieves its minimum $-h$ at $x_{0}$. By John’s lemma, there exists an affine transformation $Tx=A_{h}x+b_{h}$ such that

[TABLE]

Introduce the following rescaled functions on $T^{-1}(S_{\varphi}(x_{0},h))$:

[TABLE]

Then, we have

[TABLE]

with $\tilde{\varphi}=0$ on $\partial T^{-1}(S_{\varphi}(x_{0},h))$ and

[TABLE]

where $\tilde{x}_{0}$ is the minimum point of $\tilde{\varphi}$ in $T^{-1}(S_{\varphi}(x_{0},h))$.

The following lemma records how the equation (1.7) changes with respect to the normalization (3.5) of a section $S_{\varphi}(x_{0},h)\subset\subset\Omega$ of $\varphi$.

Lemma 3.2**.**

Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $F:\Omega\rightarrow{\mathbb{R}}^{n}$ is a bounded vector field. Assume that $S_{\varphi}(x_{0},2h)\subset\subset\Omega$. Under the rescaling (3.5), the linearized Monge-Ampère equation

[TABLE]

becomes

[TABLE]

with

[TABLE]

and, for any $q>1$, we have

[TABLE]

The proof of Lemma 3.2 will be given in Section 8.

4. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. We first collect some regularity properties of weak solutions to (1.1)-(1.2) from previous work by Benamou-Brenier [2], Cullen-Feldman [11], Loeper [23]; see also the lecture notes by Figalli [16].

Theorem 4.1**.**

([2, 11, 16, 23]) Let $\rho^{0}$ be a probability measure on ${\mathbb{T}}^{2}$. Suppose that that $\lambda\leq\rho^{0}\leq\Lambda$ in ${\mathbb{T}}^{2}$ for positive constants $\lambda$ and $\Lambda$. Let $(\rho_{t},P_{t}^{\ast})$ solve (1.1)-(1.2). Let $P_{t}$ be the Legendre transform of $P_{t}^{\ast}$. Then:

• (i) $\lambda\leq\rho_{t}\leq\Lambda$ in ${\mathbb{T}}^{2}$ for all $t\geq 0$,

(ii) $\|U_{t}\|_{L^{\infty}({\mathbb{T}}^{2})}\leq\frac{\sqrt{2}}{2}$ for all $t\geq 0$,

(iii) There is a positive constant $\kappa=\kappa(\lambda,\Lambda)$ such that for all $t>0$, we have

[TABLE]

(iv) For all $t>0$, $P_{t}$ is an Aleksandrov solution to the Monge-Ampère equation

[TABLE]

Combining the previous theorem with the known regularity results for strictly convex Aleksandrov solutions to the Monge-Ampère equation, we have the following theorem.

Theorem 4.2**.**

*([5, 6, 7, 14, 28])

Let $\rho^{0}$ be a probability measure on ${\mathbb{T}}^{2}$. Suppose that that $\lambda\leq\rho^{0}\leq\Lambda$ in ${\mathbb{T}}^{2}$ for positive constants $\lambda$ and $\Lambda$. Let $(\rho_{t},P_{t}^{\ast})$ solve (1.1)-(1.2) with the normalization $\int_{{\mathbb{T}}^{2}}P_{t}^{\ast}(x)dx=0$. Let $P_{t}$ be the Legendre transform of $P_{t}^{\ast}$. Then:*

• (i) There exist universal constants $\beta=\beta(\lambda,\Lambda)\in(0,1)$ and $C=C(\lambda,\Lambda)>0$ such that

[TABLE]

(ii) $P_{t},P_{t}^{\ast}\in L^{\infty}([0,\infty),W^{2,1+\varepsilon_{\ast}}({\mathbb{T}}^{2}))$ for some $\varepsilon_{\ast}>0$ depending only on $\lambda$ and $\Lambda$.

Remark 4.3**.**

By [16, Theorem 4.5], the positive constants $\kappa$ in Theorem 4.1 and $\varepsilon_{\ast}$ in Theorem 4.2 are related by

[TABLE]

Moreover, the constant $\varepsilon_{\ast}$ in Theorem 4.2 can be chosen to be the same $\varepsilon_{\ast}$ in (3.3) when $n=2$.

Proof of Theorem 1.2.

By an approximation argument as in [1, 23], we only need to establish the bounds in $L^{\infty}((0,\infty);C^{\alpha}({\mathbb{T}}^{2}))$ for $\partial_{t}P_{t}^{\ast}$ and $\partial_{t}P_{t}$ when the solution $(\rho_{t},P_{t}^{\ast})$ is smooth as long as these bounds depend only on $\lambda$ and $\Lambda$. Thus, we can assume in what follows, $\rho_{t},U_{t}$, $P_{t}^{\ast}$ and $P_{t}$ are smooth.

We will use $C$ to denote a generic positive constant depending only on $\lambda$ and $\Lambda$; its value may change from line to line.

By Theorem 4.1(i), for all $t\geq 0$, we have

[TABLE]

Differentiating both sides of $\det D^{2}P_{t}^{\ast}=\rho_{t}$ with respect to $t$, we find that $\partial_{t}P_{t}^{\ast}$ solves the linearized Monge-Ampère equation

[TABLE]

where $M_{P_{t}^{\ast}}$ represents the matrix of cofactors of $D^{2}P_{t}^{\ast}$; that is, $M_{P_{t}^{\ast}}=(\det D^{2}P_{t}^{\ast})(D^{2}P_{t}^{\ast})^{-1}$.

Using the first and second equations of (1.1), we get

[TABLE]

where $F_{t}=-\rho_{t}U_{t}$ with the following bound obtained from Theorem 4.1(ii):

[TABLE]

By Theorem 4.1(i, iii), we have

[TABLE]

By subtracting a constant from $P_{t}^{\ast}$, we can assume that for each $t\in(0,\infty)$,

[TABLE]

Thus, we deduce from (4.4), (4.5) and the Sobolev imbedding theorem that

[TABLE]

From (4.5) and Caffarelli’s global $C^{1,\beta}$ estimates [7] (see also Theorem 4.2), we find universal constants $\beta=\beta(\lambda,\Lambda)\in(0,1)$ and $C=C(\lambda,\Lambda)>0$ such that

[TABLE]

Using (4.7) together with the ${\mathbb{Z}}^{2}$-periodicity of $P_{t}^{\ast}-|x|^{2}/2$, we can find positive constants $h_{0}(\lambda,\Lambda)$ and $R_{0}(\lambda,\Lambda)$ such that

[TABLE]

Again, using the ${\mathbb{Z}}^{2}$-periodicity of $P_{t}^{\ast}-|x|^{2}/2$ and $F_{t}$, we deduce from (4.3) and (4.6) that

[TABLE]

With (4.1), (4.8) and (4.9) in hand, we can apply Theorem 1.3 to (4.2) in each section $S_{P_{t}^{\ast}}(x_{0},4h_{0})$ with $p=2$ and $\Omega=B_{R_{0}}(0)$ to conclude that: For some $\gamma=\gamma(\lambda,\Lambda)\in(0,1)$, we have

[TABLE]

For the Hölder regularity of $\partial_{t}P_{t}$, we use the equation

[TABLE]

which follows from differentiating with respect to $t$ the equation

[TABLE]

Combining (4.10) with (4.7), we obtain from (4.11) the following Hölder estimate for $\partial_{t}P_{t}$:

[TABLE]

The proof of Theorem 1.2 is complete by setting $\alpha=\gamma\beta$. ∎

5. Proof of Theorems 1.3, 2.1, and 2.2

In this section, we prove Theorems 1.3, 2.1, and 2.2.

Proof of Theorem 1.3.

The conclusion of the theorem follows from Lemma 3.1 and the following oscillation estimate: For every $h\leq h_{0}$, we have

[TABLE]

where $\gamma_{0}\in(0,1)$ depends only on $\lambda$ and $\Lambda$. Here and what follows, we use the following notation for a function $f$ defined on a set $E$:

[TABLE]

Indeed, suppose (5.1) is established. For each $x\in S_{\varphi}(x_{0},h_{0})\backslash\{x_{0}\}$, we can find some $h\in(0,h_{0}]$ such that $x\in\partial S_{\varphi}(x_{0},h)$. By the mean value theorem, we can find some $z$ in the interval from $x_{0}$ to $x$ such that

[TABLE]

The $C^{1,\alpha}$ estimate in Lemma 3.1 applied to $z,x_{0}\in S_{\varphi}(x_{0},h_{0})$ then gives

[TABLE]

Using (5.1), we find that

[TABLE]

where $\gamma=\gamma_{0}(1+\alpha).$ The conclusion of Theorem 1.3 follows.

It remains to prove (5.1). On the section $S_{\varphi}(x_{0},h)$ with $h\leq h_{0}$, we split $u$ as $u=v+w$ where

[TABLE]

and

[TABLE]

By Theorem 2.1 applied to the equation for $v$, we can find a universal constant $\delta=\delta(\lambda,\Lambda)$ such that

[TABLE]

On the other hand, as a consequence of Theorem 2.3 (see also the Corollary in [8, p.455]), we have from the homogeneous linearized Monge-Ampère equation for $w$ that

[TABLE]

for some $\beta=\beta(\lambda,\Lambda)\in(0,1)$. Therefore,

[TABLE]

Using the maximum principle to $\Phi^{ij}w_{ij}=0$, we have

[TABLE]

Together with (5.3) and (5.2), we find for every $h\leq h_{0}$

[TABLE]

Hence, by a standard argument (see, for example, Han-Lin [20, Lemma 4.19]), we have for all $h\leq h_{0}$

[TABLE]

for a structural constant $\gamma_{0}=\gamma_{0}(\lambda,\Lambda)\in(0,1)$ and some constant $C_{1}=C_{1}(\lambda,\Lambda,\text{diam}(\Omega),h_{0})$.

By Theorem 2.2, we have

[TABLE]

The above estimate combined with (5.4) gives (5.1). The proof of Theorem 1.3 is complete. ∎

Proof of Theorem 2.1.

Let $g_{S}(x,y)$ be Green’s function of the operator $L_{\varphi}=-\partial_{j}(\Phi^{ij}\partial_{i})=-\Phi^{ij}\partial_{ij}$ on $S=S_{\varphi}(x_{0},h)$, that is, $g_{S}$ satisfies (2.5). Then, by using that $u$ solves $\Phi^{ij}u_{ij}=\nabla\cdot F$ with $u=0$ on $\partial S$, we get

[TABLE]

Using symmetry of Green’s function and integrating by parts, we obtain for all $y\in S$

[TABLE]

It follows that for all $y\in S$, we have

[TABLE]

From the $L^{1+\kappa}$-bound for $\nabla g_{S}$ in Proposition 2.4, we have

[TABLE]

Thus, by the volume estimates for sections in (3.2), we obtain the asserted $L^{\infty}(S)$ bound for $u$ from

[TABLE]

∎

The rest of this section is devoted to the proof of Theorem 2.2 using Moser’s iteration. The key step is to prove the theorem when the section $S_{\varphi}(x_{0},h)$ is normalized (that is, when it is comparable to the unit ball) and when we have a high integrability of the solution. This is the content of Proposition 5.1. After this, the theorem easily follows from a rescaling argument.

Proposition 5.1**.**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $F:\Omega\rightarrow{\mathbb{R}}^{n}$ is a bounded vector field. There exist universally large constants $C_{0}>1$ and $p_{0}>2$ depending only on $\lambda$ and $\Lambda$ such that for every solution $u$ of (1.7) in a section $S_{\varphi}(x_{0},2)\subset\subset\Omega$ with $B_{1}(0)\subset S_{\varphi}(x_{0},1)\subset B_{2}(0),$ we have

[TABLE]

Note that, by the volume estimates (3.2), any normalized section $S_{\varphi}(x_{0},h)\subset\subset\Omega$ has height $h$ with $c(\lambda,\Lambda)\leq h\leq C(\lambda,\Lambda)$. Our proof of Proposition 5.1 works for all these $h$. However, to simplify the presentation, we choose to work with $h=1$ in Proposition 5.1.

By combining Proposition 5.1 and Lemma 3.2 we immediately obtain:

Corollary 5.2**.**

Assume $n=2$. Let $\varphi\in C^{2}(\Omega)$ be a convex function satisfying (1.6). Let $F:\Omega\rightarrow{\mathbb{R}}^{2}$ is a bounded vector field. There exist a universal constant $p_{0}=p_{0}(\lambda,\Lambda)$ and a constant $C_{1}$ depending only on $\lambda,\Lambda$ and $\emph{diam}(\Omega)$ such that for every solution $u$ of (1.7) in a section $S_{\varphi}(x_{0},2h)\subset\subset\Omega$, we have

[TABLE]

Now, with Corollary 5.2, we are ready to give the proof of Theorem 2.2.

Proof of Theorem 2.2.

We show that (2.3) follows from (5.7). The proof is based on a simple rescaling argument as in the classical proof of the local boundedness of solutions to uniformly elliptic equations (see, for example, [20, Theorem 4.1]) with Euclidean balls replaced by sections of $\varphi$.

By [19, Theorem 3.3.10(i)], there exist universal constants $c_{1}>0$ and $\mu>0$ such that for every $\theta\in(0,1)$ and $y\in S_{\varphi}(x_{0},\theta h)$ we have the inclusion

[TABLE]

Then, by applying (5.7) to $u$ on the section $S_{\varphi}(y,c_{1}(1-\theta)^{\mu}h)$ we obtain

[TABLE]

Varying $y\in S_{\varphi}(x_{0},\theta h)$ and recalling (5.8), we obtain

[TABLE]

Now, given $p\in(1,p_{0})$ we obtain from (5.9) the estimate

[TABLE]

By Young’s inequality with two exponents $p_{0}/p$ and $p_{0}/(p_{0}-p)$, we have

[TABLE]

Hence, for every $\theta\in(0,1)$ we have for a constant $C_{2}$ depending only on $\lambda,\Lambda,\text{diam}(\Omega)$

[TABLE]

It is now standard (see [20, Lemma 4.3]) that for every $p\in(1,p_{0})$, we get

[TABLE]

for a constant $C_{3}$ depending only on $p,\lambda,\Lambda$ and $\text{diam}(\Omega)$. Theorem 2.2 follows from the above estimate by setting $\theta=1/2$. ∎

To complete the proof of Theorem 2.2, it remains to prove Proposition 5.1.

Proof of Proposition 5.1.

Let $\varepsilon=\varepsilon_{\ast}(\lambda,\Lambda)>0$ be the universal constant in De Philippis-Figalli-Savin and Schmidt’s $W^{2,1+\varepsilon}$ estimate; see [14, 28] and (3.3). Then, by the convexity of $\varphi$, we have in two dimensions

[TABLE]

We prove the proposition for a large constant $C_{0}$ depending only on $\lambda$ and $\Lambda$ and

[TABLE]

By the homogeneity of (1.7), we can assume that

[TABLE]

In order to prove (5.6), we then need to show that, for some universal constant $C_{0}>0$, we have

[TABLE]

We will use Moser’s iteration to prove the proposition. Given $r\in(0,1]$, let us put

[TABLE]

Let $\eta\in C^{1}_{0}(S)$ be a cut-off function to be determined later. Let $\beta\geq 0$. By testing (1.7) against $|u|^{\beta}u\eta^{2}$ using its divergence form (1.8), we get

[TABLE]

Next, the Cauchy-Schwarz inequality gives

[TABLE]

It follows from (5.13) that

[TABLE]

We now handle the right hand side $M$ of (5.15). First, using (5.11), we have

[TABLE]

Second, using $\det D^{2}\varphi\geq\lambda$ and the following inequality $\Phi^{ij}v_{i}(x)v_{j}(x)\geq\frac{\det D^{2}\varphi|\nabla v|^{2}}{\Delta\varphi}$ whose simple proof can be found in [8, Lemma 2.1], we deduce that

[TABLE]

Using the Cauchy-Schwartz inequality, we obtain

[TABLE]

It follows from (5.15) that

[TABLE]

Consider the quantity

[TABLE]

Using (5.14), we obtain

[TABLE]

By (5.16), we have

[TABLE]

Therefore, we have

[TABLE]

We will bound from above each term on the left hand side of (5.18). Using Hölder’s inequality and (5.10), we get

[TABLE]

and

[TABLE]

We now apply the Sobolev inequality in Proposition 2.6 to the function $w=|u|^{\beta/2}u\eta$ and the exponent $q=4\frac{\varepsilon+1}{\varepsilon}$. We then have from the definition of $Q$ in (5.17) that

[TABLE]

Thus, invoking (5.19) and (5.20), we obtain from (5.18) the estimate

[TABLE]

Let $\gamma:=(\beta+2)\frac{\varepsilon+1}{\varepsilon}.$ Then

[TABLE]

Now, it is time to select the cut-off function $\eta$ in (5.21). Assume that $0<r<R\leq 1$. Using the Aleksandrov maximum principle [19, Theorem 1.4.2], we find that

[TABLE]

Indeed, by subtracting $R+\varphi(x_{0})+\nabla\varphi(x_{0})\cdot(x-x_{0})$ from $\varphi(x)$, we can assume that $\varphi=0$ on $\partial S_{R}$. Thus, $\varphi=-(R-r)$ on $\partial S_{r}$. By the Aleksandrov maximum principle, we have for any $x\in\partial S_{r}$

[TABLE]

Therefore, we obtain (5.22) as claimed.

With (5.22), we can choose a cut-off function $\eta\equiv 1$ in $S_{r}$, $\eta=0$ outside $S_{R}$, $0\leq\eta\leq 1$ and

[TABLE]

It follows from (5.21) that

[TABLE]

Now, for a nonnegative integer $j$, set

[TABLE]

Then $r_{j}-r_{j+1}=\frac{1}{2^{j+1}}$. Applying the estimate (5.23) to $R=r_{j}$, $r=r_{j+1}$ and $\gamma=\gamma_{j}$, we get

[TABLE]

By iterating, we obtain for all nonnegative integer $j$

[TABLE]

Letting $j\rightarrow\infty$ in the above inequality, we obtain (5.12). The proof of Proposition 5.1 is complete. ∎

6. Regularity for Polar Factorization of time-dependent maps in two dimensions

In this section we use Theorem 1.3 to prove the local Hölder regularity for the polar factorization of time-dependent maps in two dimensions with densities bounded away from zero and infinity. Our applications improve previous work by Loeper who considered the cases of densities sufficiently close to a positive constant. Our presentation in this section closely follows [23].

Throughout, we use $|E|$ to denote the Lebesgue measure of a Lebesgue measurable set $E\subset{\mathbb{R}}^{n}$.

6.1. Polar factorization

Let us start with the polar factorization. The polar factorization of vector-valued mappings was introduced by Brenier in his influential paper [3]. He showed that given a bounded open set $\Omega$ of ${\mathbb{R}}^{n}$ (which we can assume that $0\in\Omega$) such that $|\partial\Omega|=0$, every Lebesgue measurable mapping $X\in L^{2}(\Omega;{\mathbb{R}}^{n})$ satisfying the non-degeneracy condition

[TABLE]

can be factorized into:

[TABLE]

where $P$ is a convex function defined uniquely up to an additive constant and $g:\Omega\rightarrow\Omega$ is a Lebesgue-measure preserving mapping of $\Omega$; that is,

[TABLE]

where $C_{b}$ is the set of bounded continuous functions.

If $\mathcal{L}_{\Omega}$ denotes the Lebesgue measure of $\Omega$, the push-forward of $\mathcal{L}_{\Omega}$ by $X$, that we denote $X\#\mathcal{L}_{\Omega}$, is the measure $\rho$ defined by

[TABLE]

One can see that the condition (6.1) is equivalent to the absolute continuity of $\rho$ with respect to the Lebesgue measure, or $\rho\in L^{1}({\mathbb{R}}^{n},dx)$.

By (6.2)-(6.4), $P$ is a Brenier solution to the Monge-Ampère equation

[TABLE]

that is,

[TABLE]

Moreover, $P$ satisfies the following second boundary condition

[TABLE]

where $\Omega^{\ast}$ is the support of $\rho$.

Let us denote by $P^{\ast}$ the Legendre transform of $P$; that is, $P^{\ast}$ is defined by

[TABLE]

Then $P^{\ast}$ is a Brenier solution to the Monge-Ampère equation

[TABLE]

that is,

[TABLE]

Moreover, $P^{\ast}$ satisfies the following second boundary condition

[TABLE]

Note that the Brenier solution to the Monge-Ampère equation is in general not the Aleksandrov solution. However, Caffarelli showed in [6] that if $\Omega^{\ast}$ is convex then $P$ is an Aleksandrov solution to $\rho(\nabla P(x))\det D^{2}P(x)=1.$

In [23], Loeper investigated the regularity of the polar factorization of time-dependent maps $X_{t}\in L^{2}(\Omega;{\mathbb{R}}^{n})$ where $t$ belongs to some open interval $I\subset{\mathbb{R}}$. The open, bounded set $\Omega\subset{\mathbb{R}}^{n}$ is now assumed further to be smooth, strictly convex and has Lebesgue measure one.

As above, we assume that for each $t\in I$, $X_{t}$ satisfies (6.1). For each $t\in I$, let $d\rho_{t}=X_{t}\#\mathcal{L}_{\Omega}$ be as in (6.4). Then, from $|\Omega|=1$, we find that $\rho_{t}$ is a probability measure on ${\mathbb{R}}^{n}$.

Let $P_{t}$ and $P_{t}^{\ast}$ be as in (6.5) and (6.8). Since $P_{t}$ is defined up to a constant, we impose the condition

[TABLE]

to guarantee uniqueness. Consider the function $g_{t}$ in the polar decomposition of $X_{t}$ as in (6.2), that is

[TABLE]

$g_{t}:\Omega\to\Omega$ is a Lebesgue-measure preserving mapping. For each $t\in I$, the convex function $P_{t}$ is a Brenier solution to the following Monge-Ampère equation in $\Omega$

[TABLE]

On the other hand, $P_{t}^{\ast}$ is a Brenier solution to the Monge-Ampère equation

[TABLE]

with the boundary condition $\nabla P_{t}^{\ast}(\Omega_{t}^{\ast})=\Omega$.

In [23], Loeper investigated the regularity of the curve $t\rightarrow(g_{t},P_{t},P_{t}^{\ast})$ under the assumptions:

• $\bullet$ $X_{t}$ and $\partial_{t}X_{t}$ belong to $L^{\infty}(I\times\Omega)$;

$\bullet$ $\rho_{t}$ belongs to $L^{\infty}(I\times{\mathbb{R}}^{n})$.

We note that in this case

[TABLE]

Several results were obtained in [23]. Among other results, Loeper proved (see [23, Theorems 2.1, 2.3 and 2.3]):

• 1. For a.e. $t\in I$, $\partial_{t}g_{t}$ and $\partial_{t}\nabla P_{t}$ are bounded measures in $\Omega$. In particular, letting $\mathcal{M}(\Omega)$ denote the set of vector-valued bounded measures on $\Omega$, we have

[TABLE]

2. The Hölder continuity of $\partial_{t}P_{t}^{\ast}$ under the additional assumption that the density $\rho_{t}$ is sufficiently close to a positive constant.

3. The Hölder continuity of $\partial_{t}P_{t}$ under the additional assumptions that $\Omega_{t}^{\ast}$ is convex and the density $\rho_{t}$ is sufficiently close to a positive constant.

In Theorem 6.1 below, we are able to obtain the local Hölder regularity for $\partial_{t}P_{t}^{\ast}$ and $\partial_{t}P_{t}$ in two dimensions without assuming the closeness to 1 of the density $\rho_{t}$. Instead, we just assume it to be bounded away from zero and infinity, that is, for some positive constants $\lambda$ and $\Lambda$, we have

[TABLE]

Theorem 6.1** **(Hölder regularity of polar

factorization of time-dependent maps in two dimensions).

Let $n=2$. Let $\Omega$ be a smooth, strictly convex set in ${\mathbb{R}}^{2}$ with $|\Omega|=1$. Let $I\subset{\mathbb{R}}$ be an open interval. Assume that $X_{t}$ satisfies (6.1), $X_{t}$ and $\partial_{t}X_{t}$ belong to $L^{\infty}(I\times\Omega)$ with $R^{\ast}=\|X_{t}\|_{L^{\infty}(I\times\Omega)}$. Let $d\rho_{t}=X_{t}\#\mathcal{L}_{\Omega}$. Assume that the density $\rho_{t}$ satisfies (6.14). Let $P_{t}$ and $P_{t}^{\ast}$ be as in (6.5) and (6.8). Assume (6.10) holds. Then

• (i) For any $\omega^{\ast}\subset\subset\Omega_{t}^{\ast}$, $\partial_{t}P_{t}^{\ast}\in C^{\alpha}(\omega^{*})$ for some constant $\alpha=\alpha(\lambda,\Lambda)\in(0,1)$ with

[TABLE]

where $\delta=\delta(\lambda,\Lambda)\in(0,1)$. Here $\hat{w}$ is the set of points in $\Omega_{t}^{\ast}$ of distance at least $\frac{1}{2}(\omega^{\ast},\partial\Omega_{t}^{\ast})$ from $\partial\Omega_{t}^{\ast}$.

(ii) If $\Omega_{t}^{\ast}$ is convex then for any $\omega\subset\subset\Omega$, $\partial_{t}P_{t}\in C^{\beta}(\omega)$ for some constant $\beta=\beta(\lambda,\Lambda)\in(0,1)$ with

[TABLE]

Proof of Theorem 6.1.

Since $P_{t}^{\ast}$ is a Brenier solution to (6.13) on $\Omega_{t}^{\ast}$ with the boundary condition $\nabla P_{t}^{\ast}(\Omega_{t}^{\ast})=\Omega$ where $\Omega$ is convex, we deduce from (6.14) and Caffarelli’s regularity results for the Monge-Ampère equation [5, 6] that $P_{t}^{\ast}$ is locally $C^{1,\delta}$ with $\delta\in(0,1)$ depends only on $\Lambda/\lambda$. Note that $P_{t}$ is not $C^{1}$ in general.

(i) In [23, Section 4], Loeper constructed an adequate smooth approximation of the polar factorization problem for time-dependent maps when $X_{t}$ and $\partial_{t}X_{t}$ belong to $L^{\infty}(I\times\Omega)$ and $\rho_{t}$ belongs to $L^{\infty}(I\times{\mathbb{R}}^{n})$. Thus, we will assume in what follows, all functions $P_{t}$ and $P_{t}^{\ast}$ are smooth. However, our estimates will not depend on the smoothness.

Differentiating both sides of (6.13) with respect to t, we obtain the following linearized Monge-Ampère for $\partial_{t}P_{t}^{\ast}$

[TABLE]

where $M_{P_{t}^{\ast}}$ represents the matrix of cofactors of $D^{2}P_{t}^{\ast}$; that is, $M_{P_{t}^{\ast}}=(\det D^{2}P_{t}^{\ast})(D^{2}P_{t}^{\ast})^{-1}$.

Loeper’s important insight (see [23, Section 4]) is that $\rho_{t}$ satisfies a continuity equation of the form

[TABLE]

where $v_{t}$ is a smooth vector field on ${\mathbb{R}}^{2}$ and

[TABLE]

In fact, $v_{t}$ can be computed explicitly via $g_{t}$, $P_{t}^{\ast}$ and $P_{t}$ by the formula (see, [23, p. 345] )

[TABLE]

Therefore, $\partial_{t}P_{t}^{\ast}$ satisfies the linearized Monge-Ampère equation

[TABLE]

We now divide the proof into several steps.

Step 1: Given a section $S_{P_{t}^{\ast}}(x_{0},4h_{0})\subset\subset\Omega_{t}^{*}$ and $x\in S_{P_{t}^{\ast}}(x_{0},h_{0})$, by Theorem 1.3 applied to (6.16), we can find a constant $\gamma\in(0,1)$ depending only on $\lambda$ and $\Lambda$ such that

[TABLE]

Since $P_{t}$ is the Legendre transform of $P_{t}^{\ast}$, we have

[TABLE]

Differentiating both sides of the above equation with respect to $t$, and using $\nabla P_{t}^{\ast}(\nabla P_{t}(x))=x$, we obtain

[TABLE]

By changing variables $y:=\nabla P_{t}(x)$ we have from (6.12) that

[TABLE]

From the condition (6.10) and the Poincaré-Sobolev embedding theorem $W^{1,1}(\Omega)\hookrightarrow L^{2}(\Omega)$ for the convex set $\Omega\subset{\mathbb{R}}^{2}$, we can find a positive constant $C(\Omega)>0$ depending only on $\Omega$ such that

[TABLE]

By [23, Theorem 2.1], we have

[TABLE]

Therefore, by combining (6.19) and (6.20), we obtain

[TABLE]

Putting (6.15), (6.21) and (6.17) all together, we have

[TABLE]

Step 2: Suppose now $\omega^{\ast}\subset\subset\Omega^{\ast}_{t}$. Let $\hat{w}$ be the set of points in $\Omega_{t}^{\ast}$ of distance at least $\frac{1}{2}(\omega^{\ast},\partial\Omega_{t}^{\ast})$ from $\partial\Omega_{t}^{\ast}$. Then, there is $h_{0}>0$ depending only on $\lambda,\Lambda,\text{diam}(\Omega)$, $\|\nabla P_{t}^{\ast}\|_{C^{\delta}(\hat{\omega})}$ and $\text{dist}(\omega^{\ast},\partial\Omega_{t}^{\ast})$ such that

[TABLE]

Indeed, the strict convexity of $P_{t}^{\ast}$ implies the existence of $h_{0}>0$ such that $S_{P_{t}^{\ast}}(x_{0},4h_{0})\subset\subset\hat{\omega}.$ For any of these sections, we first use the $C^{1,\delta}$ property of $P_{t}^{\ast}$ to deduce that

[TABLE]

The volume estimates (3.2) give $|S_{P_{t}^{\ast}}(x_{0},4h_{0})|\leq C(\lambda,\Lambda)h_{0}.$ Using the convexity of $S_{P_{t}^{\ast}}(x_{0},4h_{0})$, we easily find that

[TABLE]

Thus, $h_{0}>0$ can be chosen to depend only on $\lambda,\Lambda,\text{diam}(\Omega)$, $\|\nabla P_{t}^{\ast}\|_{C^{\delta}(\hat{\omega})}$ and $\text{dist}(\omega^{\ast},\partial\Omega_{t}^{\ast})$ so that $S_{P_{t}^{\ast}}(x_{0},4h_{0})\subset\subset\hat{\omega}\subset\subset\Omega_{t}^{\ast}$.

Now, we can conclude from Step 1 and Step 2 the Hölder continuity of $\partial_{t}P_{t}^{\ast}$ as asserted in (i).

(ii) Because $P_{t}$ is the Brenier solution to the Monge-Ampère equation (6.12) with the second boundary condition $\nabla P_{t}(\Omega)\subset\Omega_{t}^{\ast}$, and $\Omega_{t}^{\ast}$ is convex, it is also the Aleksandrov solution as proved by Caffarelli [6]. The hypothesis (6.14) yields that, in the sense of Aleksandrov,

[TABLE]

Hence, Caffarelli’s global regularity result [7] yields $P_{t}\in C^{1,\delta}(\overline{\Omega})$ and $P^{\ast}_{t}\in C^{1,\delta}(\overline{\Omega_{t}^{\ast}})$. These combined with (6.18) and (i) give the conclusion of (ii) with $\beta=\gamma\delta$. The proof of Theorem 6.1 is complete. ∎

6.2. Polar factorization of time-dependent maps on the torus

The polar factorization of maps on general Riemannian manifolds has been treated by McCann [26], and also in the particular case of the flat torus ${\mathbb{T}}^{n}={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$ by Cordero-Erausquin [9]. Given a mapping $X$ from ${\mathbb{T}}^{n}$ into itself, then under the non-degeneracy condition (6.1), there is a unique pair $(P,g)$ such that

• 1. $X=\nabla P\circ g,$

2. $g:{\mathbb{T}}^{n}\to{\mathbb{T}}^{n}$ is a Lebesgue-measure preserving map,

3. $P:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is convex and $P-\frac{|x|^{2}}{2}$ is ${\mathbb{Z}}^{n}$-periodic.

The analogue of Theorem 6.1 for the regularity of polar factorization of time-dependent maps on the two-dimensional torus is the following theorem.

Theorem 6.2** **(Hölder regularity of polar

factorization of time-dependent maps on the 2D torus).

Let $I\subset{\mathbb{R}}$ be an open interval. Suppose that $X_{t},\partial_{t}X_{t}\in L^{\infty}(I\times{\mathbb{T}}^{2})$ where $X_{t}:{\mathbb{T}}^{2}\rightarrow{\mathbb{T}}^{2}$ satisfies (6.1) for all $t\in I$. Let $d\rho_{t}=X_{t}\#\mathcal{L}_{{\mathbb{T}}^{2}}$. Suppose that $\lambda\leq\rho_{t}\leq\Lambda$ on ${\mathbb{T}}^{2}$ for some positive constants $\lambda,\Lambda$ and all $t\in I$. Let $P_{t}$ and $P_{t}^{\ast}$ be as in (6.5) and (6.8) where $\Omega$ is now replaced by ${\mathbb{T}}^{2}$. Then, there exist $\alpha=\alpha(\lambda,\Lambda)\in(0,1)$ and $C=C(\lambda,\Lambda)>0$ such that

[TABLE]

Proof of Theorem 6.2.

As in the proof of Theorem 6.1, the density $\rho_{t}$ satisfies the continuity equation $\partial_{t}\rho_{t}+\mbox{div}\,(\rho_{t}v_{t})=0$ where $v_{t}$ is a bounded vector field with $\left\|v_{t}\right\|_{L^{\infty}({\mathbb{T}}^{2})}\leq\left\|\partial_{t}X_{t}\right\|_{L^{\infty}({\mathbb{T}}^{2})}.$ This vector field $v_{t}$ is similar to the vector field $U_{t}$ in Theorem 1.2 where the only information we used is its uniform boundedness in $t$. Moreover, for all $t>0$, we still have (see [16, Theorem 4.5])

[TABLE]

Thus, Theorem 6.2 follows from the same arguments as in the proof of Theorem 1.2. ∎

7. Green’s function and The Monge-Ampère Sobolev inequality

In this section, we prove Propositions 2.4, 2.6 and 2.7. Recall that in these propositions and this section, $\Omega\subset{\mathbb{R}}^{2}$ is a bounded convex set with nonempty interior and $\varphi\in C^{2}(\Omega)$ is a convex function such that

[TABLE]

for some positive constants $\lambda$ and $\Lambda$.

Given a section $S=S_{\varphi}(x_{0},h)\subset\subset\Omega$ of $\varphi$, we let $g_{S}(x,y)$ be Green’s function of the operator $L_{\varphi}:=-\partial_{j}(\Phi^{ij}\partial_{i})=-\Phi^{ij}\partial_{ij}$ on $S$, as in (2.5).

To prove Proposition 2.7, we recall the following fact regarding Green’s function in 2D.

Lemma 7.1**.**

Suppose that $S=S_{\varphi}(x_{0},h)\subset\subset\Omega$ and $S_{\varphi}(y,\eta h)\subset S_{\varphi}(x_{0},h)$ for some $\eta\in(0,1)$. Then

• (i) (**[22*, Section 6]**) For all $x\in\partial S_{\varphi}(y,\frac{\eta}{2}h)$, we have *$g_{S}(x,y)\leq C(\lambda,\Lambda,\eta).$

(ii) (**[21, Lemma 3.2]**) For all $0<h_{1}<\frac{\eta}{4}h$, we have

[TABLE]

For reader’s convenience, we explain how to derive Lemma 7.1(i) from [22]. By Lemma 6.1 in [22] and the volume estimates (3.2), we obtain

[TABLE]

Applying Lemma 6.2 in [22] to $g_{S}(\cdot,y)$ in $S_{\varphi}(y,\eta h)$, we obtain for all $x\in\partial S_{\varphi}(y,\frac{\eta}{2}h)$ the estimate

[TABLE]

Lemma 7.1 implies the following lemma:

Lemma 7.2**.**

Assume that $S=S_{\varphi}(x_{0},h)\subset\subset\Omega$ and $S_{\varphi}(y,\eta h)\subset S_{\varphi}(x_{0},h)$ for some $\eta\in(0,1)$. Let $\tau_{0}=C(\lambda,\Lambda,\eta)+C(\lambda,\Lambda)$ where $C(\lambda,\Lambda,\eta)$ and $C(\lambda,\Lambda)$ are as in Lemma 7.1. Then for all $\tau>\tau_{0}$, we have

[TABLE]

Proof of Lemma 7.2.

If $\tau>\tau_{0}$ then by the maximum principle, we can find $h_{1}<\eta h/2$ such that

[TABLE]

Let $m$ be a positive integer such that $\eta h/2\leq 2^{m}h_{1}<\eta h$. Iterating Lemma 7.1(ii), we find that

[TABLE]

The maximum principle and Lemma 7.1(i) give

[TABLE]

Hence,

[TABLE]

Thus, we obtain from (7.1) the estimate $\tau\leq\tau_{0}\log_{2}(\eta h/h_{1}).$ It follows that $h_{1}\leq\eta h2^{-\tau/\tau_{0}}.$ Lemma 7.2 now follows from (7.1). ∎

Now we are ready to give the proof of Proposition 2.7.

Proof of Proposition 2.7.

Recall that $S=S_{\varphi}(x_{0},h)$ with $S_{\varphi}(x_{0},2h)\subset\subset\Omega$.

Step 1: Special case. We first prove

[TABLE]

in the special case that $S_{\varphi}(y,\eta h)\subset S_{\varphi}(x_{0},h)$ for some universal constant $\eta\in(0,1)$ depending only on $\lambda$ and $\Lambda$. In this case, by Lemma 7.2, we find that for all $\tau>\tau_{0}(\lambda,\Lambda)$,

[TABLE]

Using the upper bound on the volume of sections in (3.2), we find that

[TABLE]

It follows from the layer cake representation and the volume estimate $|S|\leq C(\lambda,\Lambda)h$ that

[TABLE]

Hence (7.2) is proved.

Step 2: General case. It remains to prove the proposition for the general case $y\in S_{\varphi}(x_{0},h)$. By [19, Theorem 3.3.10(i)], there is a universal constant $\eta\in(0,1)$ depending only on $\lambda$ and $\Lambda$ such that for all $y\in S_{\varphi}(x_{0},h)$, we have

[TABLE]

Thus, for any $x\in S_{\varphi}(x_{0},h)$, we have $g_{S_{\varphi}(x_{0},\frac{3}{2}h)}(x,y)\geq g_{S_{\varphi}(x_{0},h)}(x,y)$ by the maximum principle. It follows that

[TABLE]

In the last inequality, we applied the estimate (7.2) to the section $S_{\varphi}(x_{0},\frac{3}{2}h)$ and the point $y$ in $S_{\varphi}(x_{0},\frac{3}{2}h)$ that satisfies (7.3). The proof of Proposition 2.7 is now complete. ∎

Note that, Proposition 2.4 is closely related to [21, Theorem 1.1(iii)]. For reader’s convenience we provide the detailed proof of Proposition 2.4 here.

Proof of Proposition 2.4.

Recall that $S=S_{\varphi}(x_{0},h)$. Fix $y\in S$ and set

[TABLE]

Then $v\geq 1$ in $S$, $v=1$ on $\partial S$ and $\partial_{i}(\Phi^{ij}v_{j})=-\delta_{y}$ in $S$. Let $\hat{S}=S_{\varphi}(x_{0},2h)$ and extend $v$ to be $1$ in $\hat{S}\backslash S$. Then $v\geq 1$ in $\hat{S}$. Using the divergence-free property of $(\Phi^{ij})$, we have $\partial_{i}(\Phi^{ij}v_{j})=-\delta_{y}$ in $\hat{S}$.

Step 1: Integral bound for $\log v$. We show that

[TABLE]

Indeed, given $w\in C^{1}_{0}(\hat{S})$, multiply the inequality $\partial_{i}(\Phi^{ij}v_{j})\leq 0$ by $\frac{w^{2}}{v}\geq 0$ and then integrate by parts to get

[TABLE]

By the Cauchy-Schwarz inequality, we obtain

[TABLE]

and therefore

[TABLE]

By choosing a suitable $0\leq w\leq 1$ as in the proof of Theorem 6.2 in Maldonado [25], and using the volume estimates in (3.2), we obtain (7.4). For completeness, we include a construction of $w$.

By subtracting $\varphi(x_{0})+\nabla\varphi(x_{0})\cdot(x-x_{0})$ from $\varphi$, we can assume that $\varphi(x_{0})=0$ and $\nabla\varphi(x_{0})=0$. Therefore $\varphi\geq 0$ on $\hat{S}=S_{\varphi}(x_{0},2h)$. Let $\gamma:{\mathbb{R}}\rightarrow[0,1]$ be a smooth function supported in $[0,2]$ with $\gamma\equiv 1$ on $[0,1]$ and $\|\gamma^{\prime}\|_{L^{\infty}({\mathbb{R}})}\leq 10$. Let $w(x):=\gamma(\varphi(x)/h)$. Then $w\in C^{1}_{0}(\hat{S})$ with $w\equiv 1$ on $S_{\varphi}(x_{0},h)=S$ and

[TABLE]

Therefore,

[TABLE]

Integrating by parts the last term and using $\sum_{i=1}^{2}\partial_{i}\Phi^{ij}=0$ for all $j=1,2$, we obtain

[TABLE]

In the last inequality, we used the upper bound on volume of sections in (3.2) to get $|\hat{S}|\leq C(\lambda,\Lambda)h$. Therefore, (7.4) now follows from (7.5) and the above inequalities.

Step 2: $L^{1+\kappa}$ estimate for $v$. By Proposition 2.7 and the inequality $v^{q}(x)\leq C(q)(g^{q}_{S}(x,y)+1),$ together with the volume bound on $S$, we find that $v\in L^{q}(S)$ for all $q<\infty$ with the bound

[TABLE]

Next, we use the following inequality $\Phi^{ij}v_{i}(x)v_{j}(x)\geq\frac{\det D^{2}\varphi|\nabla v|^{2}}{\Delta\varphi}$ whose simple proof can be found in [8, Lemma 2.1]. It follows from (7.4) that

[TABLE]

Now, for all $1<p<2$, using the Holder inequality to $|\nabla v|^{p}=\frac{|\nabla v|^{p}}{(\Delta\varphi)^{\frac{p}{2}}}\frac{1}{v^{p}}\left((\Delta\varphi)^{\frac{p}{2}}v^{p}\right)$ with exponents $\frac{2}{p}$ and $\frac{2}{2-p}$, we have

[TABLE]

Let $\varepsilon_{\ast}=\varepsilon_{\ast}(\lambda,\Lambda)>0$ be as in (3.3). Let us fix any $0<\varepsilon<\varepsilon_{\ast}$ and

[TABLE]

Thus, recalling $h_{0}\geq h$, Lemma 3.1 and (7.6), we obtain

[TABLE]

The proof of Proposition 2.4 is complete by choosing $\kappa=p-1=\frac{\varepsilon}{2+\varepsilon}$ and $\kappa_{1}=\frac{\varepsilon_{\ast}-\varepsilon}{2(1+\varepsilon_{\ast})(1+\varepsilon)}.$ ∎

Proof of Proposition 2.6.

Suppose that $S_{\varphi}(x_{0},2)\subset\subset\Omega$ and $S_{\varphi}(x_{0},1)$ is normalized. Set $S:=S_{\varphi}(x_{0},1)$. Let $g_{S}(x,y)$ be Green’s function of $S$ with respect to $L_{\varphi}:=-\partial_{j}(\Phi^{ij}\partial_{i})=-\Phi^{ij}\partial_{ij}$ with pole $y\in S$, that is, $g_{S}(\cdot,y)$ is a positive solution of (2.5). By Proposition 2.7, for any $q>1$, there exists a constant $K>0$, depending on $q$, $\lambda$ and $\Lambda$, such that for every $y\in S$ we have

[TABLE]

As the operator $L_{\varphi}$ can be written in the divergence form with symmetric coefficients, we infer from Grüter-Widman [18, Theorem 1.3] that $g_{S}(x,y)=g_{S}(y,x)$ for all $x,y\in S$. This together with (7.7) allows us to deduce that, for every $x\in S$, there holds

[TABLE]

Then, one can use Lemma 2.1 in Tian-Wang [29] to conclude (2.6). ∎

8. Proofs of Lemmas 3.1 and 3.2

Proof of Lemma 3.2.

For $x\in\tilde{S}:=T^{-1}(S_{\varphi}(x_{0},h))$, we have

[TABLE]

and

[TABLE]

In the variables $y:=Tx$ and $x\in\tilde{S}$, we have the relation $\nabla_{x}=A_{h}^{t}\nabla_{y}.$ Thus, letting $\langle,\rangle$ denote the inner product on ${\mathbb{R}}^{n}$, we have

[TABLE]

The cofactor matrix $\tilde{\Phi}=(\tilde{\Phi}^{ij}(x))_{1\leq i,j\leq n}$ of $D^{2}\tilde{\varphi}(x)$ is related to $\Phi(Tx)$ and $A_{h}$ by

[TABLE]

Therefore,

[TABLE]

and hence, recalling (1.7) and (8.1),

[TABLE]

Thus, we get (3.7) as asserted.

Next, we claim that

[TABLE]

Indeed, let $\hat{\varphi}=\varphi-h$. Then, by our assumption that $\varphi|_{\partial S_{\varphi}(x_{0},h)}=0$, we have $\hat{\varphi}=0$ on $\partial S_{\varphi}(x_{0},2h)$. Applying the Aleksandrov maximum principle (see [19, Theorem 1.4.2]) to $\hat{\varphi}$ on $S_{\varphi}(x_{0},2h)$, we have for any $x\in S_{\varphi}(x_{0},h)$,

[TABLE]

where in the last inequality we used the volume estimates in (3.2). Thus, we obtain (8.3) as claimed.

Using (8.3) and the convexity of $\varphi$, we find that

[TABLE]

It follows that $S_{\varphi}(x_{0},h)\supset B(x_{0},c_{1}h^{\frac{n}{2}})$ for some constant $c_{1}=c_{1}(n,\lambda,\Lambda,\text{diam}(\Omega))$, which combined with (3.4) implies that

[TABLE]

On the other hand, by means of the volume estimates in (3.2), we find from (3.4) that

[TABLE]

Hence (3.8) follows from (3.5), (8.5) and (8.6).

Finally, using

[TABLE]

together with (8.6), we obtain (3.9). ∎

Proof of Lemma 3.1.

(i) Rescaling as in (3.5), we have for all $x\in S_{\varphi}(x_{0},h)$

[TABLE]

Using the inequality $\text{trace}(AB)\leq\|A\|\text{trace}(B)$ for nonnegative definite matrices $A,B$, we thus have

[TABLE]

By the $W^{2,1+\varepsilon}$ estimate (3.3) applied to $\tilde{\varphi}$ and its normalized section $\tilde{S}=T^{-1}(S_{\varphi}(x_{0},h))$, we have

[TABLE]

for some $\varepsilon_{\ast}=\varepsilon_{\ast}(n,\lambda,\Lambda)>0$ depending only on $n,\lambda$ and $\Lambda$. Using (8.5) and (8.6), we find that

[TABLE]

where

[TABLE]

(ii) Rescaling as in (3.5), we have for $x\in S_{\varphi}(x_{0},h)$

[TABLE]

Suppose that $x,y\in S_{\varphi}(x_{0},h/2)$. Then $T^{-1}x,T^{-1}y\in S_{\tilde{\varphi}}(\tilde{x}_{0},(\det A_{h})^{-2/n}h)$. Applying the $C^{1,\alpha}$ estimate for the Monge-Ampère equation, due to Caffarelli [5], to $\tilde{\varphi}$, we have

[TABLE]

where $\alpha=\alpha(n,\lambda,\Lambda)\in(0,1)$. In terms of the function $\varphi$, we infer from (8.7) and (8.8) that

[TABLE]

Using the volume estimates (3.2), we obtain from (8.9) and (8.5) the following estimate:

[TABLE]

where $\alpha_{1}=-1+\frac{n}{2}(\alpha+1)>0.$ ∎

Acknowledgments. The author would like to thank the anonymous referee for the pertinent comments and the careful reading of the paper together with his/her constructive criticisms.