On the Three-Dimensional Shape of a Crystal

Abstract

The purpose of this paper is to investigate the Almgren problem in ${\mathbb{R}}^3$ under generic conditions on the potential and tension functions which define the free energy. This problem appears in classical thermodynamics when one seeks to understand whether minimizing the free energy with convex potential in the class of sets of finite perimeter under a mass constraint generates a convex minimizer representing a crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize convexity and a stability theorem when the mass is small, as well as a first-variation partial differential equation along with a new maximum principle approach.

1. Introduction

A fundamental problem in thermodynamics is proving the convexity of minimizers to the free-energy minimization with mass constraint. The free energy $\mathcal{E}\left(E\right)$ of a set of finite perimeter $E\subset {\mathbb{R}}^n$ with a reduced boundary ${\mathrm{\partial }}^{\ast }E$ is defined via the surface energy

$$\mathcal{F}\left(E\right)={\int }_{{\mathrm{\partial }}^{\ast }E}f\left({\nu }_E\right)d{\mathcal{H}}^{n-1},$$

and the potential energy

$$\mathcal{G}\left(E\right)={\int }_{E}g\left(x\right)dx,$$

where f (i.e., the surface tension) is a convex positively 1-homogeneous $f:{\mathbb{R}}^n\to [0,\infty )$ with $f\left(x\right)>0$ if $\left|x\right|>0$, $g\ge 0$, $g\left(0\right)=0$:

$$\mathcal{E}\left(E\right)=\mathcal{F}\left(E\right)+\mathcal{G}\left(E\right).$$

The following problem is historically attributed to Almgren.

Problem: If the potential g is convex (or, more generally, if the sub-level sets $\{g<t\}$ are convex), are the minimizers convex or, at least, connected? [1] (p. 146).

Mathematically, the central purpose is to investigate the minimization problem

$$\inf \left\{\mathcal{E}\right(E):|E|=m\}.$$

Historically, this is one of the most complex problems to analyze and one of the most important problems in physics. The physical principle connecting minimizers $E_m$ to crystals was independently discovered by Gibbs in 1878 [2] and Curie in 1885 [3]. When the energy minimization is the surface-area minimization, the solution is the convex set

$$K=\bigcap _{v\in {\mathbb{S}}^{n-1}}\{x\in {\mathbb{R}}^n:\langle x,v\rangle <f\left(v\right)\}$$

called the Wulff shape. Only a handful of convexity results exist with a potential energy for all $m>0$, even in two dimensions. In two recent papers, Indrei (i) proved that in one dimension, assuming solely that the sub-level sets $\{g<t\}$ are convex, there exist minimizers for all masses $m>0$ and all minimizers are intervals [4]; (ii) proved that if $n\ge 2$, there are convex functions $g\ge 0$, $g\left(0\right)=0$, so that there are no minimizers for $m>0$ [5] (the figure in the paper illustrates the main idea: the potential energy is reduced via translation). Observe the general partition of the convexity problem into coercive (e.g., the monotone radial potential) and non-coercive potentials (e.g., the gravitational potential). Supposing $n=2$, under additional assumptions, the first author proved convexity for all $m>0$ [5] (cf. [6]). The tools include geometric results in the plane related to convex hulls, the first variation formula for the anisotropic surface energy, a perturbation technique, and density estimates. In the argument, the planar context is crucial; some of the theorems do not have higher dimensional extensions. Recently, the authors proved a sharp quantitative inequality for the isotropic radial Almgren problem ($f\left(x\right)=\left|x\right|,g=g\left(\right|x\left|\right)$) in ${\mathbb{R}}^n$ [7]. One thus has additional information on the geometry of almost-minimizers. The theorem we obtained in [7] is the first positive result for all $m>0$ on the stability and convexity for a large class of potentials in higher dimension.

For $g=0$, the stability appeared in [8] with an explicit modulus; in [9] for $g=0$ and the isotropic case with a semi-explicit modulus; and, in [5] for m small with a semi-explicit modulus and a locally bounded potential.

Naturally, in physics, the most important dimension is $n=3$. We introduce a new method to prove:

Theorem 1.

If $g\in C^{2,{\alpha }_{\ast }}$ is convex, $f\in C^4({\mathbb{R}}^3\setminus \left\{0\right\})$ is $\lambda -$elliptic, and $f,g$ admit minimizers $E_m\subset B_{R\left(m\right)}$ with $R\in L_{loc}^{\infty }\left({\mathbb{R}}^{+}\right)$, then exactly one of the following is true:

(i) 

$E_m$ is convex for all $m\in (0,\infty )$;

(ii) 

there exists $\mathcal{M}>0$ & for all $m\in (0,\mathcal{M})$, $E_m$ is convex and there is a modulus $w_m\left(0^{+}\right)=0$ and two constants ${ϵ}_0,\gamma >0$, such that for all $ϵ\le {ϵ}_0$,

$$\underset{m\to {\mathcal{M}}^{-}}{\lim \inf }{\displaystyle \frac{\gamma (\mathcal{M}-m)}{w_m\left(ϵ\right)}}\ge 1.$$

Our theorem implies convexity for a large collection of potentials; our argument is also inclusive of non-convex potentials. The main element of this involves estimating the modulus. One class of $(f,g)$ where the convexity holds is: $f\left(x\right)=\left|x\right|$, $g\left(x\right)=h\left(\right|x\left|\right)$, $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is increasing, $h\left(0\right)=0$. The main reason is that then for any $m_1>m_0>0$, $ϵ>0$,

$$\underset{m_1\ge m\ge m_0}{\inf }{w}_m\left(ϵ\right)>0$$

ref. [7] and therefore, for all $\mathcal{M},ϵ,\gamma >0$,

$$\underset{m\to {\mathcal{M}}^{-}}{\lim \inf }{\displaystyle \frac{\gamma (\mathcal{M}-m)}{w_m\left(ϵ\right)}}=0$$

which precludes (ii) in Theorem 1. Therefore, if $f\left(x\right)=\left|x\right|$, $g\left(x\right)=h\left(\right|x\left|\right)$, $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is increasing, $h\left(0\right)=0$, then (i) is true.

Our new idea for the three-dimensional Almgren problem is to utilize a stability theorem when m is small [5] (see Theorem A1), convexity when m is small, and the first variation in the free-energy PDE with a new maximum principle approach. The maximum principle argument is to show that if u is a convex function which encodes a convex minimizer in a neighborhood and $\det D^{2}u\left(x_0\right)=0$ for some $x_0\in \Omega$, then $\det D^{2}u\left(x\right)=0$ for all $x\in \Omega$. The first variation PDE (2) connects the function u with a partial differential equation via the potential and mean curvature.

Assuming g is coercive, the existence of a minimizer $E_m$ is true. Nevertheless, in certain configurations, one may prove the existence result for non-coercive potentials, e.g., the gravitational potential. In a gravitational field, the equilibrium for liquids was studied by Laplace [10]. Supposing the surface tension is isotropic, uniqueness and convexity were obtained by Finn [11] and Wente [12]. Assuming $n=2$, under a wetting condition, the anisotropic tension was investigated by Avron, Taylor, and Zia, employing quadrature [13]. The research was motivated by low-temperature experiments on helium crystals in equilibrium with a superfluid [14]. Also, various phase-transition experiments were conducted in [15].

The stability result contains an invariance collection. Define

$$\begin{array}{c}{\mathcal{A}}_m={\mathcal{A}}_{f,g,m}=\{A:Ax=A_{a}x+x_a,x_a\in {\mathbb{R}}^3,\mathcal{E}\left(A_{a}E\right)=\mathcal{E}\left(E\right),\\ \hspace{1em}|A_{a}E|=|E|=m\mathrm{for}\mathrm{some}\mathrm{minimizer}E\}.\end{array}$$

An invariance map of the free energy is a transformation $A\in {\mathcal{A}}_m$. The uniqueness of minimizers can only be true mod sets of measure zero and an invariance map generated by the mass, potential, and tension. In many classes of potentials, assuming m is small, $A\in {\mathcal{A}}_m$ is a translation $Ax=x+z$, $z\in {\mathbb{R}}^3$. For example, suppose g is zero on a ball B. If m is small, note that uniqueness can only be shown up to a translation: $A_a=I_{3\times 3}$, $x_a\in {\mathbb{R}}^3$ is such that $K_m+x_a\subset \{g=0\}$ when $K_m\subset B$ ($K_m$ is the Wulff shape, such that $|K_m|=m$ [5]). The three transformations, reflection, rotation, and translation, always satisfy closure under convexity: $AE$ is convex if E is convex.

2. Proof of Theorem 1

Define

$$\begin{array}{c}{\mathcal{A}}_a=\{m:E_{\overline{m}}\mathrm{is}\mathrm{unique}\&\mathrm{convex}\mathrm{for}\mathrm{all}0<\overline{m}\le m\}\\ \mathcal{M}=\sup {\mathcal{A}}_a.\end{array}$$

Theorem A1 and Theorem 2 in Figalli and Maggi [1] imply $(0,m_a)\subset {\mathcal{A}}_a$. Hence, $\mathcal{M}>0$. In addition, one may assume the invariance maps are closed under convexity. If $\mathcal{M}<\infty$, for $m\in (0,\mathcal{M})$, $E_m$ is unique and convex. Therefore, either (a) there exists a non-convex minimizer with the mass $\mathcal{M}$; (b) there exist two convex minimizers not mod an invariance map equal, with the mass $\mathcal{M}$; or (c) for all $m\in (0,\mathcal{M}]$, $E_m$ is unique, convex, and for $m>\mathcal{M}$, there exists $a<m$ such that either convexity or uniqueness fails for minimizers with mass a. If $m_k<\mathcal{M}$, $m_k\to \mathcal{M}$, along a subsequence, $E_{m_k}\to T_{\mathcal{M}}$, with $|T_{\mathcal{M}}|=\mathcal{M}$, $T_{\mathcal{M}}$ as a convex minimizer. Set

$$ϵ={\displaystyle \frac{1}{5}}\underset{R}{\inf }{\displaystyle \frac{|R{E}_{\mathcal{M}}∆T_{\mathcal{M}}|}{|E_{\mathcal{M}}|}}>0,$$

where if (a) is valid, $E_{\mathcal{M}}$ is the non-convex minimizer, and if (b) is true, $E_{\mathcal{M}}$ is a convex minimizer not (mod invariance transformations) equal to $T_{\mathcal{M}}$. If $m\in (0,\mathcal{M})$, the uniqueness of convex minimizers implies that there exists $w_m\left(ϵ\right)>0$ such that for all $ϵ>0$, if $\left|E\right|=|E_m|$, $E\subset B_R$, and

$$|\mathcal{E}\left(E\right)-\mathcal{E}\left(E_m\right)|<w_m\left(ϵ\right),$$

then, there exists R such that

$${\displaystyle \frac{|E_m∆RE|}{|E_m|}}<ϵ.$$

Let $\left\{m_k\right\}$ be the sequence such that

$$\underset{m\to {\mathcal{M}}^{-}}{\lim \inf }{\displaystyle \frac{{\mathcal{M}}^{\frac{2}{3}}-m^{\frac{2}{3}}}{w_m\left(ϵ\right)}}=\underset{k\to \infty }{\lim }{\displaystyle \frac{{\mathcal{M}}^{\frac{2}{3}}-m_k^{\frac{2}{3}}}{w_{m_k}\left(ϵ\right)}},$$

and define ${\gamma }_k$ via $|{\gamma }_k{E}_{\mathcal{M}}|=|E_{m_k}|$, i.e., ${\gamma }_k={({\displaystyle \frac{m_k}{\mathcal{M}}})}^{{\displaystyle \frac{1}{3}}}$. Note

$$\begin{array}{cc}\hfill |\mathcal{E}\left({\gamma }_k{E}_{\mathcal{M}}\right)-\mathcal{E}\left(E_{m_k}\right)|& \le |\mathcal{E}\left({\gamma }_k{E}_{\mathcal{M}}\right)-\mathcal{E}\left(E_{\mathcal{M}}\right)|+|\mathcal{E}\left(T_{\mathcal{M}}\right)-\mathcal{E}\left(E_{m_k}\right)|\hfill \\ \hfill & \le \mathcal{F}\left(E_{\mathcal{M}}\right)(1-{\gamma }_k^2)+(\underset{B_R}{\sup }g)|E_{\mathcal{M}}∆\left({\gamma }_k{E}_{\mathcal{M}}\right)|\hfill \\ \hfill & +|\mathcal{E}\left(T_{\mathcal{M}}\right)-\mathcal{E}\left(E_{m_k}\right)|.\hfill \end{array}$$

Moreover,

$$\begin{array}{cc}\hfill \mathcal{E}\left(T_{\mathcal{M}}\right)& \le \mathcal{E}({\displaystyle \frac{1}{{\gamma }_k}}{E}_{m_k})\hfill \\ \hfill & ={\displaystyle \frac{1}{{\gamma }_k^2}}\mathcal{F}\left(E_{m_k}\right)+{\int }_{{\displaystyle \frac{1}{{\gamma }_k}}{E}_{m_k}}g\left(x\right)dx\hfill \\ \hfill & \le ({\displaystyle \frac{1}{{\gamma }_k^2}}-1)\mathcal{F}\left(E_{m_k}\right)+(\underset{B_R}{\sup }g)|{\displaystyle \frac{1}{{\gamma }_k}}{E}_{m_k}∆E_{m_k}|+\mathcal{E}\left(E_{m_k}\right)\hfill \end{array}$$

and similarly, thanks to $|{\displaystyle \frac{1}{{\gamma }_k}}{E}_{m_k}∆E_{m_k}|\le a({\displaystyle \frac{1}{{\gamma }_k}}-1)$, (e.g, via [1] (Lemma 4)) this implies

$$|\mathcal{E}\left(T_{\mathcal{M}}\right)-\mathcal{E}\left(E_{m_k}\right)|\le {\alpha }_p({\displaystyle \frac{1}{{\gamma }_k^2}}-1)=\alpha ({\mathcal{M}}^{{\displaystyle \frac{2}{3}}}-m_k^{{\displaystyle \frac{2}{3}}}),$$

$m_k\approx \mathcal{M}$.

In particular,

$$|\mathcal{E}\left({\gamma }_k{E}_{\mathcal{M}}\right)-\mathcal{E}\left(E_{m_k}\right)|\le {\gamma }_1({\mathcal{M}}^{{\displaystyle \frac{2}{3}}}-m_k^{{\displaystyle \frac{2}{3}}})$$

where ${\gamma }_1={\gamma }_1\left(\mathcal{M}\right)$.

Suppose

$$\underset{m\to {\mathcal{M}}^{-}}{\lim \inf }{\displaystyle \frac{{\mathcal{M}}^{\frac{2}{3}}-m^{\frac{2}{3}}}{w_m\left(ϵ\right)}}<{\displaystyle \frac{1}{{\gamma }_1}},$$

(1)

then, for k large

$$|\mathcal{E}\left({\gamma }_k{E}_{\mathcal{M}}\right)-\mathcal{E}\left(E_{m_k}\right)|\le {\displaystyle \frac{{\gamma }_1({\mathcal{M}}^{\frac{2}{3}}-m_k^{\frac{2}{3}})}{w_{m_k}\left(ϵ\right)}}{w}_{m_k}\left(ϵ\right)<w_{m_k}\left(ϵ\right)$$

and this implies the existence of $R_k$ such that

$${\displaystyle \frac{|E_{m_k}∆R_k\left({\gamma }_k{E}_{\mathcal{M}}\right)|}{|E_{m_k}|}}<ϵ.$$

However, if k is large, ${\gamma }_k\approx 1$, which implies

$$\begin{array}{cc}\hfill {\displaystyle \frac{|\left(E_{m_k}\right)∆R_k\left({\gamma }_k{E}_{\mathcal{M}}\right)|}{|E_{m_k}|}}& \approx {\displaystyle \frac{|T_{\mathcal{M}}∆R_k\left(E_{\mathcal{M}}\right)|}{|E_{\mathcal{M}}|}}\hfill \\ \hfill & \ge \underset{R}{\inf }{\displaystyle \frac{|R{E}_{\mathcal{M}}∆T_{\mathcal{M}}|}{|E_{\mathcal{M}}|}}=5ϵ,\hfill \end{array}$$

a contradiction. Therefore, (1) is not true and

$$\underset{m\to {\mathcal{M}}^{-}}{\lim \inf }{\displaystyle \frac{{\mathcal{M}}^{\frac{2}{3}}-m_k^{\frac{2}{3}}}{w_m\left(ϵ\right)}}\ge {\displaystyle \frac{1}{{\gamma }_1}},$$

for

$$ϵ\le {ϵ}_0:={\displaystyle \frac{1}{5}}\underset{R}{\inf }{\displaystyle \frac{|R{E}_{\mathcal{M}}∆T_{\mathcal{M}}|}{|E_{\mathcal{M}}|}}.$$

Thus, this yields $\gamma =\gamma \left(\mathcal{M}\right)>0$,

$$\underset{m\to {\mathcal{M}}^{-}}{\lim \inf }{\displaystyle \frac{\mathcal{M}-m_k}{w_m\left(ϵ\right)}}\ge {\displaystyle \frac{1}{\gamma }};$$

observe that the bound in (ii) is proven. One can consider general invariance maps closed under convexity to preclude (b). The last part is to preclude (c).

Claim 1: A convex minimizer at mass $\mathcal{M}$ is uniformly convex.

Proof of Claim 1. 

The anisotropic mean curvature is

$$H_f=\mathrm{trace}\left(D^{2}fA\right),$$

where $D^{2}f$ is the matrix of second tangential derivatives and A is the second fundamental form. The formula for the first variation implies

$$H_f=\mu -g,$$

(2)

where

$$\mu ={\displaystyle \frac{2\mathcal{F}\left(E_{\mathcal{M}}\right)+{\int }_{{\mathrm{\partial }}^{\ast }{E}_{\mathcal{M}}}g\langle x,{\nu }_{E_{\mathcal{M}}}\rangle d{\mathcal{H}}^2}{n|E_{\mathcal{M}}|}}.$$

The convexity of $E_{\mathcal{M}}$ and (2) imply that, locally, there is a convex function $u\in C^{2.1}(\Omega ),\Omega \subset {\mathbb{R}}^2$, so that

$$a_{ij}(\nabla u)u_{ij}=\mu -g(x,u),$$

where $a_{ij}(\nabla u),i,j\in \{1,2\}$, is a uniformly elliptic matrix given in terms of the second-order derivatives of f and depending on $\nabla u$, with g being a convex function of $(x,u)\in {\mathbb{R}}^3$ and ∇ the gradient; see Chapter 16.4 [16]. Recall that for the classical case $f\left(\xi \right)=\left|\xi \right|$, we have

$$a_{ij}(\nabla u)={\displaystyle \frac{1}{\sqrt{1+{|\nabla u|}^2}}}\left({\delta }_{ij}-{\displaystyle \frac{u_i{u}_j}{1+{|\nabla u|}^2}}\right).$$

Note

$$\mu -g>0\mathrm{on}\mathrm{\partial }{E}_{\mathcal{M}}.$$

(3)

Indeed, let us choose a smoothly changing coordinate system so that $D^{2}u$ is diagonal. Then, the mean curvature takes the form $H_f=a_{11}{u}_{11}+a_{22}{u}_{22}.$ After differentiating, we obtain

$${\left(H_f\right)}_{ss}=-{\nabla }_{{\mathbb{R}}^3}^{2}g(x,u){\mathrm{\partial }}_s\left(\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right){\mathrm{\partial }}_s\left(\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right)-g_u{u}_{ss},s=1,2.$$

Then,

$$a_{ss}{\left(H_f\right)}_{ss}\le -g_u{H}_f,$$

and consequently

$$a_{ss}{\left(H_f\right)}_{ss}-{\left(g_u\right)}^{-}{H}_f\le 0,$$

where ${\left(g_u\right)}^{-}$ is the negative part of $g_u$. Hence, the result follows from the strong minimum principle. □

Subclaim: If $\det D^{2}u\left(x_0\right)=0$ for some $x_0\in \Omega$, then $\det D^{2}u\left(x\right)=0$ for all $x\in \Omega$.

Proof of Subclaim. 

Observe that under our assumptions $u\in C^{3,1}(\Omega )$, thanks to Corollary 16.7 [16]. The proof is based on the observation that $w:=\det D^{2}u\left(x\right)$ satisfies an inequality of the form $a_{ij}{w}_{ij}-cw+b·\nabla w\le 0$ near $x_0$, with $c\ge 0$.

Let us write the equation in the form $\mathcal{F}(D^{2}u,\nabla u)=\sum a_{ij}{u}_{ij}$ and let $f=\mu -g$; then, the equation takes the form

$$\mathcal{F}=f.$$

Differentiate twice in $x_s,x_t,1\le s,t\le 2$ to obtain

$$\begin{array}{c}\hfill {\mathcal{F}}_s=f_s,\\ \hfill {\mathcal{F}}_{st}=f_{st}.\end{array}$$

Now, we have that

$$\begin{array}{c}\hfill w_s=u^{ij}{u}_{ijs},w_{st}=u^{ij,kl}{u}_{ijs}{u}_{klt}+u^{ij}{u}_{ijst},\end{array}$$

where $u^{ij}$ is the cofactor matrix.

On the other hand,

$$\begin{array}{ccc}\hfill {\mathcal{F}}_s& =& {\nabla }_p{a}_{lm}·\nabla u_s{u}_{lm}+a_{lm}{u}_{lms},\hfill \\ \hfill {\mathcal{F}}_{st}& =& ({\nabla }_{pp}^2{a}_{lm}\nabla u_t)·\nabla u_s{u}_{lm}\hfill \\ & & +{\nabla }_p{a}_{lm}·\nabla u_{st}{u}_{lm}+{\nabla }_p{a}_{lm}·\nabla u_s{u}_{lmt}+{\nabla }_p{a}_{lm}·\nabla u_t{u}_{lms}\hfill \\ & & +a_{lm}{u}_{lmst}\hfill \\ & :=& {\mathcal{F}}^{\left(2\right)}+{\mathcal{F}}^{\left(3\right)}+a_{lm}{u}_{lmst},\hfill \end{array}$$

where we use the notation with dummy variable $p:=\nabla u$.

Since the Weingarten mapping is self-adjoint, then at each point x, near $x_0$, we have

$$D^{2}u\left(x\right)=diag[{\lambda }_1,{\lambda }_2]$$

(4)

in a continuously changing coordinate system. Moreover, ${\lambda }_2\ge {\lambda }_1\ge 0$. By (3), ${\lambda }_1+{\lambda }_2>0$. Suppose $w\left(x_0\right)=0$, then $u_{11}\left(x_0\right)u_{22}\left(x_0\right)=0$ and without loss of generality

$$u_{11}\left(x\right)>10\delta and{u}_{22}\left(x\right)<\delta$$

(5)

for $\delta >0$, in some neighborhoods $x\in B_{r_0}\left(x_0\right)$, $r_0>0$ is small. Using these observations, we can make the following explicit computations

$$\begin{array}{ccc}{w}_s& =& u_{11s}{u}_{22}+u_{11}{u}_{22s},\end{array}$$

(6)

$$\begin{array}{ccc}{w}_{st}& =& u_{11st}{u}_{22}+u_{11s}{u}_{22t}+u_{11t}{u}_{22s}+u_{11}{u}_{22st}-2u_{12t}{u}_{12s}.\end{array}$$

(7)

The second-order derivatives appearing in ${\mathcal{F}}_{st}$, after contracting with the cofactor matrix $u^{ij}=diag(u_{22},u_{11})$, and using (4), can be simplified as follows

$$\begin{array}{ccc}\hfill u^{st}{\mathcal{F}}_{st}^{\left(2\right)}& :=& ({\nabla }_{pp}^2{a}_{lm}\nabla u_t)·\nabla u_s{u}_{lm}{u}^{st}\hfill \\ & =& ({\nabla }_{pp}^2{a}_{lm}\nabla u_1)·\nabla u_1{u}_{lm}{u}^{11}+({\nabla }_{pp}^2{a}_{lm}\nabla u_2)·\nabla u_2{u}_{lm}{u}^{22}\hfill \\ & =& ({\nabla }_{pp}^2{a}_{lm}\nabla u_1)·\nabla u_1{u}_{lm}{u}_{22}+({\nabla }_{pp}^2{a}_{lm}\nabla u_2)·\nabla u_2{u}_{lm}{u}_{11}\hfill \\ & =& ({\nabla }_{pp}^2{a}_{11}\nabla u_1)·\nabla u_1{u}_{11}{u}_{22}+({\nabla }_{pp}^2{a}_{22}\nabla u_1)·\nabla u_1{u}_{22}{u}_{22}\hfill \\ & & +({\nabla }_{pp}^2{a}_{11}\nabla u_2)·\nabla u_2{u}_{11}{u}_{11}+({\nabla }_{pp}^2{a}_{22}\nabla u_2)·\nabla u_2{u}_{22}{u}_{11}\hfill \\ & =& \left(({\nabla }_{pp}^2{a}_{11}\nabla u_1)·\nabla u_1+({\nabla }_{pp}^2{a}_{22}\nabla u_2)·\nabla u_2\right)w\hfill \\ & & +({\mathrm{\partial }}_{p_1{p}_1}{a}_{22}+{\mathrm{\partial }}_{p_2{p}_2}{a}_{11})w^2.\hfill \end{array}$$

Consequently,

$$u^{st}{\mathcal{F}}_{st}^{\left(2\right)}=O(cw+b·\nabla w),$$

(8)

for some fixed $c>0$ and $b\in {\mathbb{R}}^2$.

Next, let us compute the expression

$$\begin{array}{ccc}\hfill a_{lm}{w}_{lm}& =& u^{st}{a}_{lm}{u}_{imst}+a_{lm}{u}^{st,ij}{u}_{stl}{u}_{ijm}\hfill \\ \hfill & =& -u^{st}{\mathcal{F}}_{st}^{\left(2\right)}-u^{st}{\mathcal{F}}_{st}^{\left(3\right)}+u^{st}{f}_{st}+a_{lm}(u_{11l}{u}_{22m}+u_{11m}{u}_{22l}-2u_{12m}{u}_{12l})\hfill \\ \hfill & =& -u^{st}{\mathcal{F}}_{st}^{\left(2\right)}-u^{st}{\mathcal{F}}_{st}^{\left(3\right)}+u^{st}{f}_{st}+J.\hfill \end{array}$$

(9)

We need to simplify the last term $J:=a_{lm}(u_{11l}{u}_{22m}+u_{11m}{u}_{22l}-2u_{12m}{u}_{12l})$. It can be written in a more explicit form, as follows

$$\begin{array}{ccc}\hfill J& =& a_{11}(u_{111}{u}_{221}+u_{111}{u}_{221}-2u_{121}^2)+a_{12}(u_{111}{u}_{222}+u_{112}{u}_{221}-2u_{122}{u}_{121})\hfill \\ & & +a_{21}(u_{112}{u}_{221}+u_{111}{u}_{222}-2u_{121}{u}_{122})+a_{22}(u_{112}{u}_{222}+u_{112}{u}_{222}-2u_{122}^2)\hfill \\ & =& 2\left(a_{11}(u_{111}{u}_{221}-u_{121}^2)+a_{12}(u_{111}{u}_{222}-u_{122}{u}_{121})+a_{22}(u_{112}{u}_{222}-u_{122}^2)\right).\hfill \end{array}$$

Using the explicit forms of $w_s,{\mathcal{F}}_s$, we obtain

$$\begin{array}{c}\hfill a_{11}{u}_{11s}+2a_{12}{u}_{12s}+a_{22}{u}_{22s}=f_s-\left({\mathrm{\partial }}_{p_s}{a}_{ll}\right)u_{ss}{u}_{ll},\end{array}$$

(10)

$$\begin{array}{c}\hfill u_{11s}{u}_{22}+u_{11}{u}_{22s}=w_s,\end{array}$$

(11)

since

$${\nabla }_p{a}_{lm}·\nabla u_s{u}_{lm}=\left({\mathrm{\partial }}_{p_s}{a}_{ll}\right)u_{ss}{u}_{ll}.$$

From (11)

$$u_{22s}={\displaystyle \frac{w_s-u_{11s}{u}_{22}}{u_{11}}},$$

(12)

plugging this into (11) yields

$$\begin{array}{ccc}\hfill f_s-\left({\mathrm{\partial }}_{p_s}{a}_{ll}\right)u_{ss}{u}_{ll}& =& a_{11}{u}_{11s}+2a_{12}{u}_{12s}+a_{22}{\displaystyle \frac{w_s-u_{11s}{u}_{22}}{u_{11}}}\hfill \\ & =& u_{11s}{\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}+a_{22}{\displaystyle \frac{w_s}{u_{11}}}+2a_{12}{u}_{12s}.\hfill \end{array}$$

If $s=1$, then $u_{12s}=u_{121}=u_{112}$, and from the above computation

$$\begin{array}{c}\hfill f_1-\left({\mathrm{\partial }}_{p_1}{a}_{ll}\right)u_{11}{u}_{ll}=u_{111}{\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}+a_{22}{\displaystyle \frac{w_1}{u_{11}}}+2a_{12}{u}_{112}.\end{array}$$

Similarly, for $s=2$, we obtain

$$\begin{array}{c}\hfill f_2-\left({\mathrm{\partial }}_{p_2}{a}_{ll}\right)u_{22}{u}_{ll}=u_{112}{\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}+a_{22}{\displaystyle \frac{w_2}{u_{11}}}+2a_{12}{\displaystyle \frac{w_1-u_{111}{u}_{22}}{u_{11}}}.\end{array}$$

Combining the last two equations, we obtain a system of equations for the remaining third-order derivatives $u_{111}$ and $u_{112}$;

$$\begin{array}{c}\hfill f_1-\left({\mathrm{\partial }}_{p_1}{a}_{ll}\right)u_{11}{u}_{ll}-a_{22}{\displaystyle \frac{w_1}{u_{11}}}=u_{111}{\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}+2a_{12}{u}_{112},\\ \hfill f_2-\left({\mathrm{\partial }}_{p_2}{a}_{ll}\right)u_{22}{u}_{ll}-a_{22}{\displaystyle \frac{w_2}{u_{11}}}-2a_{12}{\displaystyle \frac{w_1}{u_{11}}}=-2a_{12}{\displaystyle \frac{u_{22}}{u_{11}}}{u}_{111}+u_{112}{\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}.\end{array}$$

Note that the determinant of the coefficient matrix is

$$\mathcal{D}:={\displaystyle \frac{{(a_{11}{u}_{11}-a_{22}{u}_{22})}^2}{u_{11}^2}}+4a_{12}^2{\displaystyle \frac{u_{22}}{u_{11}}}>0,$$

and, moreover,

$${\displaystyle \frac{1}{\mathcal{D}}}={\displaystyle \frac{u_{11}^2}{{(a_{11}{u}_{11}-a_{22}{u}_{22})}^2}}\left(1+4a_{12}^2{\displaystyle \frac{u_{22}}{u_{11}}}h\right)$$

(13)

for some bounded function h in view of (5).

Solving the system, we find

$$\begin{array}{ccc}\hfill u_{111}& =& {\displaystyle \frac{1}{\mathcal{D}}}({\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}(f_1-\left({\mathrm{\partial }}_{p_1}{a}_{ll}\right)u_{11}{u}_{ll}-a_{22}{\displaystyle \frac{w_1}{u_{11}}})\hfill \\ & & -2a_{12}(f_2-\left({\mathrm{\partial }}_{p_2}{a}_{ll}\right)u_{22}{u}_{ll}-a_{22}{\displaystyle \frac{w_2}{u_{11}}}-2a_{12}{\displaystyle \frac{w_1}{u_{11}}}))\hfill \\ & =& {\displaystyle \frac{1}{\mathcal{D}}}\left({\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}(f_1-\left({\mathrm{\partial }}_{p_1}{a}_{11}\right)u_{11}^2)-2a_{12}(f_2-\left({\mathrm{\partial }}_{p_2}{a}_{22}\right)u_{22}^2)\right)+O(cw+b·\nabla w)\hfill \\ & =& {\displaystyle \frac{1}{\mathcal{D}}}\left({\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}(f_1-\left({\mathrm{\partial }}_{p_1}{a}_{11}\right)u_{11}^2)-2a_{12}\left(f_2\right)\right)+O(cw+b·\nabla w)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill u_{112}& =& {\displaystyle \frac{1}{\mathcal{D}}}({\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}(f_2-\left({\mathrm{\partial }}_{p_2}{a}_{ll}\right)u_{22}{u}_{ll}-a_{22}{\displaystyle \frac{w_2}{u_{11}}}-2a_{12}{\displaystyle \frac{w_1}{u_{11}}})\hfill \\ & & +2a_{12}{\displaystyle \frac{u_{22}}{u_{11}}}(f_1-\left({\mathrm{\partial }}_{p_1}{a}_{ll}\right)u_{11}{u}_{ll}-a_{22}{\displaystyle \frac{w_1}{u_{11}}}))\hfill \\ & =& {\displaystyle \frac{1}{\mathcal{D}}}\left({\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}(f_2-\left({\mathrm{\partial }}_{p_2}{a}_{22}\right)u_{22}^2)+2a_{12}{\displaystyle \frac{u_{22}}{u_{11}}}(f_1-\left({\mathrm{\partial }}_{p_1}{a}_{11}\right)u_{11}^2)\right)+O(cw+b·\nabla w)\hfill \\ & =& {\displaystyle \frac{1}{\mathcal{D}}}\left({\displaystyle \frac{a_{11}{u}_{11}-a_{22}{u}_{22}}{u_{11}}}{f}_2\right)+O(cw+b·\nabla w)\hfill \\ & =& {\displaystyle \frac{u_{11}}{a_{11}{u}_{11}-a_{22}{u}_{22}}}\left(1+4a_{12}^2{\displaystyle \frac{u_{22}}{u_{11}}}h\right)f_2+O(cw+b·\nabla w)\hfill \\ & =& {\displaystyle \frac{u_{11}}{a_{11}{u}_{11}-a_{22}{u}_{22}}}{f}_2+O(cw+b·\nabla w).\hfill \end{array}$$

Therefore, combining this with (13) and (5), we infer that $u_{112}$ and $u_{112}$ can be estimated in terms of the lower-order derivatives of u; hence, we conclude that

$$u_{111},u_{112}=O(cw+b·\nabla w).$$

(14)

Returning to

$$\begin{array}{ccc}\hfill J& =& 2\{a_{11}\left(u_{111}{\displaystyle \frac{w_1-u_{111}{u}_{22}}{u_{11}}}-u_{112}^2\right)\hfill \\ & & +a_{12}\left(u_{111}{\displaystyle \frac{w_2-u_{112}{u}_{22}}{u_{11}}}-{\displaystyle \frac{w_1-u_{111}{u}_{22}}{u_{11}}}{u}_{112}\right)\hfill \\ & & +a_{22}\left(u_{112}{\displaystyle \frac{w_2-u_{112}{u}_{22}}{u_{11}}}-{\left({\displaystyle \frac{w_1-u_{111}{u}_{22}}{u_{11}}}\right)}^2\right)\}\hfill \\ & =& 2\{u_{111}^2(-a_{11}{\displaystyle \frac{u_{22}}{u_{11}}}-a_{22}{\displaystyle \frac{u_{22}^2}{u_{11}^2}})+u_{112}^2(-a_{11}-a_{22}{\displaystyle \frac{u_{22}}{u_{11}}})\hfill \\ & & +u_{111}(w_1{\displaystyle \frac{a_{11}}{u_{11}}}+w_2{\displaystyle \frac{a_{22}}{u_{11}}}+2w_1{\displaystyle \frac{a_{22}{u}_{22}}{u_{11}}})\hfill \\ & & +u_{112}(-w_1{\displaystyle \frac{a_{12}}{u_{11}}}+w_2{\displaystyle \frac{a_{22}}{u_{11}}})-a_{22}{\displaystyle \frac{w_1^2}{u_{11}^2}}\}\hfill \\ & \le & u_{111}(w_1{\displaystyle \frac{a_{11}}{u_{11}}}+w_2{\displaystyle \frac{a_{22}}{u_{11}}}+2w_1{\displaystyle \frac{a_{22}{u}_{22}}{u_{11}}})\hfill \\ & & +u_{112}(-w_1{\displaystyle \frac{a_{12}}{u_{11}}}+w_2{\displaystyle \frac{a_{22}}{u_{11}}})\hfill \\ & =& O(cw+b·\nabla w),\hfill \end{array}$$

where the last line follows from (14) and (5).

For the third-order derivatives in ${\mathcal{F}}_{st}$, after a contraction with $u^{ij}$, we have

$$\begin{array}{ccc}\hfill u^{st}{\mathcal{F}}_{st}^{\left(3\right)}& :=& \left({\nabla }_p{a}_{lm}·\nabla u_{st}{u}_{lm}+{\nabla }_p{a}_{lm}·\nabla u_s{u}_{lmt}+{\nabla }_p{a}_{lm}·\nabla u_t{u}_{lms}\right)u^{st}\hfill \\ & =& \sum _{s=1}^2\sum _{l,m}\left({\nabla }_p{a}_{lm}·\nabla u_{ss}{u}_{lm}{u}^{ss}+2{\nabla }_p{a}_{lm}·\nabla u_s{u}_{lms}{u}^{ss}\right)\hfill \\ & =& \sum _{s=1}^2\left({\nabla }_p{a}_{11}·\nabla u_{ss}{u}_{11}{u}^{ss}+{\nabla }_p{a}_{22}·\nabla u_{ss}{u}_{22}{u}^{ss}+2\sum _{l,m}{\nabla }_p{a}_{lm}·\nabla u_s{u}_{lms}{u}^{ss}\right)\hfill \\ & =& \sum _{s=1}^2\left({\mathrm{\partial }}_{p_1}{a}_{11}{u}_{1ss}{u}_{11}{u}^{ss}+{\mathrm{\partial }}_{p_2}{a}_{11}{u}_{2ss}{u}_{11}{u}^{ss}\right)\hfill \\ & & +\sum _{s=1}^2\left({\mathrm{\partial }}_{p_1}{a}_{22}{u}_{1ss}{u}_{22}{u}^{ss}+{\mathrm{\partial }}_{p_2}{a}_{22}{u}_{2ss}{u}_{22}{u}^{ss}\right)\hfill \\ & & +2\sum _{s=1}^2\sum _{l,m}\left({\mathrm{\partial }}_{p_1}{a}_{lm}{u}_{1s}{u}_{lms}{u}^{ss}+{\mathrm{\partial }}_{p_2}{a}_{lm}{u}_{2s}{u}_{lms}{u}^{ss}\right)\hfill \\ & =& \left({\mathrm{\partial }}_{p_1}{a}_{11}{u}_{11}\right)w_1+\left({\mathrm{\partial }}_{p_2}{a}_{11}{u}_{11}\right)w_2\hfill \\ & & +\left({\mathrm{\partial }}_{p_1}{a}_{22}{u}_{22}\right)w_1+\left({\mathrm{\partial }}_{p_2}{a}_{22}{u}_{22}\right)w_2\hfill \\ & & +2\sum _{l,m}\left({\mathrm{\partial }}_{p_1}{a}_{lm}{u}_{lm1}+{\mathrm{\partial }}_{p_2}{a}_{lm}{u}_{lm2}\right)w.\hfill \end{array}$$

From here and our estimates for the third-order derivatives, we conclude that

$$u^{st}{\mathcal{F}}_{st}^{\left(3\right)}=O(cw+b·\nabla w),$$

(15)

for some fixed $c>0$ and $b\in {\mathbb{R}}^2$.

Using this and (9), we obtain

$$\begin{array}{c}\hfill a_{lm}{w}_{lm}\le u^{st}{f}_{st}+O(cw+b·\nabla w).\end{array}$$

(16)

To finish the proof, note that

$$\begin{array}{ccc}\hfill u^{st}{f}_{st}& =& u^{st}{\nabla }_{{\mathbb{R}}^3}f(x,u)·\left(\begin{array}{c}0\\ 0\\ u_{st}\end{array}\right)+u^{st}{\nabla }_{{\mathbb{R}}^3}^{2}f(x,u){\mathrm{\partial }}_s\left(\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right){\mathrm{\partial }}_t\left(\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right)\hfill \\ & =& f_{u}w-u^{ss}{\nabla }_{{\mathbb{R}}^3}^{2}g(x,u){\mathrm{\partial }}_s\left(\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right){\mathrm{\partial }}_s\left(\begin{array}{c}{x}_1\\ x_2\\ u\end{array}\right)\hfill \\ & \le & f_{u}w\hfill \end{array}$$

since we assume that g is convex. Summarizing, it follows from the last inequality and (16) that

$$a_{lm}{w}_{lm}+cw+b·\nabla w\le 0.$$

Writing $c=c^{+}-c^{-},c^{\pm }\ge 0$, and using $w\ge 0$, we obtain

$$a_{lm}{w}_{lm}-c^{-}w+b·\nabla w\le 0.$$

(17)

Applying the strong minimum principle, we see that $w=0$ in $B_{r_0}\left(x_0\right)$. Therefore, the proof of the Subclaim is finished. □

Next, we prove Claim 1: if the Gauss curvature of $\mathrm{\partial }{E}_{\mathcal{M}}$ vanishes at some point, $w\left(x_0\right)=0$, then the Gauss curvature is zero everywhere on $\mathrm{\partial }{E}_{\mathcal{M}}$ (Subclaim). By Theorem 2.8 [17], u is the lower boundary of the convex hull of the set of points $(x,u{|}_{\mathrm{\partial }\Omega }),$ for any strictly convex $\Omega$. For such $\Omega$, if we pick a point $x\in \Omega$, then there is a line segment passing through x. These line segments cannot intersect, since otherwise that mean the curvature vanishes at the intersection. Thus, the graph of u over $\Omega$ is a ruled surface. If we take a hyperplane perpendicular to the one containing the domain $\Omega$; then, for ${\Omega }^{\perp }$ lying on this hyperplane, the same conclusion will hold. However, the line segments generated by $\Omega$ and ${\Omega }^{\perp }$ must intersect, which will contradict the $C^3$ regularity of the surface. This yields the proof of Claim 1.

Suppose for $m_k>\mathcal{M}$, there is $m_{j_k}<m_k$ with $E_{m_{j_k}}$ a non-convex minimizer. Via Claim 1, $E_{\mathcal{M}}$ is uniformly convex. In particular, the two curvatures are uniformly positive. Via the smoothness, up to a subsequence, $E_{m_{j_k}}\to E_{\mathcal{M}}$ in $C^2$. Observe that for sufficiently large k, the regularity implies that the principal curvatures of $E_{m_{j_k}}$ are near the ones of $E_{\mathcal{M}}$, and thus this contradicts non-convexity. In particular,

$$\mathrm{if}m>\mathcal{M}\mathrm{is}\mathrm{near}\mathcal{M},\mathrm{then}{E}_m\mathrm{is}\mathrm{uniformly}\mathrm{convex}.$$

(18)

To show uniqueness, the next fact is sufficient:

The uniqueness fact: There exists $m_0>0$ and a modulus of continuity $a(m,0^{+})=0$ such that for all $m<\mathcal{M}+m_0$ there exists ${ϵ}_0>0$, such that for all $0<ϵ<{ϵ}_0$ & for all minimizers $E_m\subset B_R$, $E\subset B_R$, $\left|E\right|=\left|E_m\right|=m<\mathcal{M}+m_0$, if

$$|\mathcal{E}\left(E_m\right)-\mathcal{E}\left(E\right)|<a(m,ϵ),$$

there exists an invariance map A, such that

$${\displaystyle \frac{|E∆A{E}_m|}{|E_m|}}<ϵ.$$

Assume the uniqueness is false. Then, for all $m_0>0$, for all moduli q, there exists $m<\mathcal{M}+m_0$ such that for a fixed ${ϵ}_0>0$ there exists $ϵ<{ϵ}_0$ &, and there exist $E_{m,{ϵ}_0},E_{m,{ϵ}_0}^{\prime }\subset B_R$, $|E_{m,{ϵ}_0}|=|E_{m,{ϵ}_0}^{\prime }|=m$ such that

$$|\mathcal{E}\left(E_{m,{ϵ}_0}\right)-\mathcal{E}\left(E_{m,{ϵ}_0}^{\prime }\right)|<q_m\left(ϵ\right),$$

and

$$\underset{A}{\inf }{\displaystyle \frac{|E_{m,{ϵ}_0}^{\prime }∆A{E}_{m,{ϵ}_0}|}{|E_{m,{ϵ}_0}|}}\ge ϵ>0.$$

(19)

Let $m_0={\displaystyle \frac{1}{k}}$, $w_k\to 0^{+}$, $\widehat{q}$ be a modulus of continuity and define

$$q_k=w_k\widehat{q}\left(ϵ\right),$$

(20)

hence, there exists $m_k<\mathcal{M}+{\displaystyle \frac{1}{k}}$ such that for a fixed ${ϵ}_0>0$ there exists $ϵ<{ϵ}_0$ &, and there exist minimizers $E_{m_k,{ϵ}_0}$; in addition, some sets $E_{m_k,{ϵ}_0}^{\prime }\subset B_R$, $\left|E_{m_k,{ϵ}_0}\right|=\left|E_{m_k,{ϵ}_0}^{\prime }\right|=m_k<\mathcal{M}+{\displaystyle \frac{1}{k}}$ such that

$$|\mathcal{E}\left(E_{m_k,{ϵ}_0}\right)-\mathcal{E}\left(E_{m_k,{ϵ}_0}^{\prime }\right)|<q_k,$$

and

$$\underset{A}{\inf }{\displaystyle \frac{|E_{m_k,{ϵ}_0}^{\prime }∆A{E}_{m_k,{ϵ}_0}|}{|E_{m_k,{ϵ}_0}|}}\ge ϵ>0.$$

(21)

Set $E_{m_k}=E_{m_k,{ϵ}_0}$, $E_{m_k}^{\prime }=E_{m_k,{ϵ}_0}^{\prime }$. Also, define ${\gamma }_k={\left({\displaystyle \frac{\mathcal{M}}{m_k}}\right)}^{{\displaystyle \frac{1}{3}}}$ such that

$$\left|{\gamma }_k{E}_{m_k}\right|=\left|E_{\mathcal{M}}\right|.$$

Next, observe that thanks to the compactness, $E_{m_k,{ϵ}_0}\to E$, this yields $\mathcal{E}\left(E_{m_k,{ϵ}_0}\right)\to \mathcal{E}\left(E\right)$, where E is a minimizer, $\left|E\right|=\mathcal{M}$. In addition,

$$|\mathcal{E}\left(E_{m_k,{ϵ}_0}\right)-\mathcal{E}\left(E_{m_k,{ϵ}_0}^{\prime }\right)|<q_k\to 0$$

also implies that, along a subsequence,

$$E_{m_k,{ϵ}_0}^{\prime }\to \widehat{E},$$

$\left|\widehat{E}\right|=\mathcal{M},$ $\widehat{E}$ is a minimizer. The aforementioned

$$\underset{A}{\inf }{\displaystyle \frac{|E_{m_k,{ϵ}_0}^{\prime }∆A{E}_{m_k,{ϵ}_0}|}{|E_{m_k,{ϵ}_0}|}}\ge ϵ>0$$

(22)

therefore yields a contradiction: initially, the uniqueness at mass $\mathcal{M}$ yields $A_1$, so that $A_{1}E=\widehat{E}$; thus

$$A_1{E}_{m_k,{ϵ}_0}\to A_{1}E,$$

$$E_{m_k,{ϵ}_0}^{\prime }\to \widehat{E},$$

$${\displaystyle \frac{|E_{m_k,{ϵ}_0}^{\prime }∆A_1{E}_{m_k,{ϵ}_0}|}{|E_{m_k,{ϵ}_0}|}}\to 0.$$

Hence, (18), together with uniqueness, precludes (c).