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I'm an undergraduate student and I'm currently reading a classical paper for my final project for the course differential geometry on the Bernstein problem of minimal surfaces, namely, the paper:

Bombieri, Enrico, E. De Giorgi, and Enrico Giusti, "Minimal Cones and the Bernstein Problem" Inventiones Mathematicae 7.3 (1969): 243-268.

In equation \eqref{1}, the authors considered the folowing Dirichlet problem for the minimal surface equation: $$ \begin{cases} \sum_{i=1}^{n} \left( D_i \left( \dfrac{D_i f}{\sqrt{1+\vert D f \vert^2}} \right) \right) = 0, \qquad f\in C^2(B_R), \\ f=f_1\quad \text{in} \quad \partial B_R \end{cases}\label{1}\tag{25} $$

where $B_R$ is the unit ball in $\mathbb{R}^8$. (NOT in three-dimensional Euclidean space)
We have known the existence and uniqueness of the solution of such a boundary problem. Denote its solution by $f^{(R)}(x)$.
Similarly, we consider the same boundary problem with function $f_2$ on $\partial B_R$.
With some assumptions and calculations mentioned in the paper before, we have obtained that $$ f_1(x) \leq f^{(R)}(x) \leq f_2(x) $$ on the boundary $\partial(B_R \cap D_1)$. Here comes my question which has been puzzling me for a long time:

The authors claims that: "by the well-known maximum principle for solutions of the Dirichlet problem and equation \eqref{2} and \eqref{3}" (listed below), we obtained that $$ f_1(x) \leq f^{(R)}(x) \leq f_2(x) \, \text{for} \, x \in \bar{B}_R\cap\bar{D_1}. $$

I'm confused at the "maximum principle mentioned there. I have learned the strong maximum principle and the Hopf maximum principle for Laplacian equations (with corresponding boundary conditions), but I have no idea how to apply these here. Or, it there any maximum principle stated for the minimal surface equation in the above contexts? I tried but find no reference for such a theorem. (For example, the book on elliptic PDEs by David Gilbarg, et.al). Moreover, I have no idea on the role played by the equation \eqref{2} and \eqref{3}.

P.S. I list here equations \eqref{2} and \eqref{3}: $$ \int_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{1}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{1}\right|^{2}\right)^{-\frac{1}{2}} dx \leq 0\label{2}\tag{23} $$ and $$ \int_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{2}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{2}\right|^{2}\right)^{-\frac{1}{2}} dx \geq 0\label{3}\tag{24} $$ where

  • the region $D_1$ is defined as $D_1 = \{ x \in \mathbb{R}^8 \vert 0 \leq v \leq u \}$,
  • $u=\left(x_{1}^{2}+\cdots+x_{4}^{2}\right)^{\frac{1}{2}}$ and
  • $v=\left(x_{5}^{2}+\cdots+x_{8}^{2}\right)^{\frac{1}{2}}$.

Thank you in advance! It is my first time to ask a question on MSE, and I am sincerely sorry for any possible mistakes and rudeness in this question.

Thank you!

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1 Answer 1

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In my opinion, the maximum principle to which Bombieri, De Giorgi and Giusti refer is the same elementary maximum principle described (again without giving a precise reference or demonstration) by Miranda in [2] (§1, Theorem 1.2, pp. 667-668). And I think they all refer to the so called weak maximum principle for the area functional, as described by Giusti in [1], so in this answer I'll closely follow this reference, which also uses the notation in the OP.

The "elementary maximum principle" for area minimizers

Let $\Omega$ a bounded domain with Lipschitz continuous boundary $\partial\Omega$. Consider the following subsets of the space of Lipschitz continuous functions $$ \newcommand{\Lip}{\operatorname{Lip}} \begin{eqnarray} \Lip_k(\Omega) &=&\{ f\in C^{0,1}(\Omega): |f|_\Omega\le k\} & \quad k>0 \\ \Lip_k(\Omega,\eta) &=&\{ f\in \Lip_k(\Omega): f|_{\partial\Omega}=\eta|_{\partial\Omega}\} & \quad \eta \in C^{0,1}(\Omega)\\ \end{eqnarray} $$ and let $$ \mathscr{A}(f,\Omega)=\int\limits_\Omega \sqrt{1+|Df|^2}\mathrm{d} x\label{4}\tag{1} $$ be the area functional, which is strictly convex, i.e. $$ \mathscr{A}\left(\frac{u+v}{2},\Omega\right)<\frac{1}{2}\big[\mathscr{A}(u,\Omega) + \mathscr{A}(v,\Omega)\big]\label{5}\tag{2} $$ for each $u, v \in C^{0,1}(\Omega)$ such that $Du\neq Dv$ in $\Omega$. Then we have the following

Weak Maximum principle ([1], Lemma 12.5, p. 139). Let $f_1, f_2 \in \Lip_k(\Omega)$ respectively be a subsolution and a supersolution of the area functional minimization problem, i.e. $$ \begin{eqnarray} \mathscr{A}(f_1,\Omega)\le \mathscr{A}(v,\Omega)&\quad\forall v\in\Lip_k(\Omega)\text{ such that }f_1 \ge v\\ \mathscr{A}(f_2,\Omega)\le \mathscr{A}(v,\Omega)&\quad\forall v\in\Lip_k(\Omega)\text{ such that }f_2 \le v \end{eqnarray} $$ If $f_1\le f_2$ on $\partial\Omega$, then $f_1\le f_2$ on $\bar\Omega$ (the closure of $\Omega$).
Proof. Let's prove the result by contradiction and thus suppose that $$ K=\{x\in\Omega:f_1(x)>f_2(x)\}\neq\emptyset. $$ Define $\overline{f}=\max\{f_1, f_2\}$: obviously $\overline{f}\in \Lip_k(\Omega,f_2)$ and $\overline{f}\ge f_2$ therefore $$ \mathscr{A}(f_2,\Omega)\le \mathscr{A}(\overline{f},\Omega) \iff \mathscr{A}(f_2,K)\le \mathscr{A}(f_1,K). $$ In the same way, defining $\underline{f}=\min\{f_1, f_2\}$, we obviously see that $\mathscr{A}(f_2,K)\ge \mathscr{A}(f_1,K)$, thus $$ \mathscr{A}(f_2,K)= \mathscr{A}(f_1,K). $$ Now, since $f_2=f_1$ on $\partial K$ and $f_1>f_2$ in $K$, it must be $D f_1\neq Df_2$ on a set of positive measure in $K$ therefore by \eqref{5} $$ \mathscr{A}\left(\frac{f_1+f_2}{2},K\right)<\frac{1}{2}\big[\mathscr{A}(f_1,K) + \mathscr{A}(f_2,K)\big] = \mathscr{A}(f_2,K) $$ but this is impossible since $f_2$ is a supersoluion in $\Lip_k(\Omega)$ and thus $$ \mathscr{A}\left(\frac{f_1+f_2}{2},K\right) \ge \mathscr{A}(f_2,K) $$ since $\frac{1}{2}(f_1+f_2)>f_2$. $\blacksquare$

The weak maximum principle in the paper of Bombieri, De Giorgi and Giusti

Let's consider inequalities \eqref{2} and \eqref{3} and the functional derivative of \eqref{1} on the "points" (subsolution and supersolution) $f_1$ and $f_2$: $$ \begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}\varepsilon} \mathscr{A}(f_1+\varepsilon\varphi,D_1)&\le 0 & \iff \int\limits_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{1}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{1}\right|^{2}\right)^{-\frac{1}{2}} \mathrm{d}x \le 0\\ \frac{\mathrm{d}}{\mathrm{d}\varepsilon} \mathscr{A}(f_2+\varepsilon\varphi, D_1)&\ge 0 & \iff \int\limits_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{2}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{2}\right|^{2}\right)^{-\frac{1}{2}} \mathrm{d}x \ge 0 \end{eqnarray}, $$ for all $\varphi \in C^{\infty}(D_1)$ such that $\varphi\ge 0$. Inequalities \eqref{2}and \eqref{3} (whose right side is de facto the weak formulation of the minimal surface operator as shown in problem \eqref{1}) are the functional derivatives of the area functional \eqref{4} and their solutions are respectively subsolutions and supersolutions of the area minimization problem, and thus the weak maximum principle holds for them: this allows the authors deduce the double sided estimate that originated this question.

Notes

  • The weak maximum principle is elementary in the sense that it does not involve any concept outside the realm of basic multivariable real analysis: the tools used for the proofs are simply order relations and the (strict) convexity of the functional \eqref{4}.
  • The weak maximum principle does not requires the existence of sub/superminimizer of the area functional \eqref{4}: in practical cases, when we can effectively construct such minimizers, we can use the pinciple in order to eventually prove relevant existence and uniqueness results for Plateau's problem.
  • The weak maximum principle is "weak" in the sense that the fact that it does not implies that a minimizer is the constant function if it has a minimum/maximum in the interior of $\Omega$, like the strong maximum principle for Laplace's equation does.
  • I was not able to find the original reference for the weak maximum principle: according to Miranda ([2], §1, p. 668), it was used by Von Neumann to prove a maximum principle for the gradients of minimizers of the functional \eqref{4} (see the references therein for the details) and it seems that also Hilbert was aware of the result in a simpler case. It is probably due to the "fog" which seem to surround the original source that Bombieri, De Giorgi, Giusti do not gave precise references.

References

[1] Giusti, Enrico, Minimal surfaces and functions of bounded variation, (English) Monographs in Mathematics, Vol. 80, Boston-Basel-Stuttgart: Birkhäuser, pp. XII+240, ISBN: 0-8176-3153-4, MR0775682, Zbl 0545.49018.

[2] Miranda, Mario, "Maximum principles and minimal surfaces", (English) Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, IV Serie, 25, No. 3-4, 667-681 (1997), MR1655536, Zbl 1015.49028.

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  • $\begingroup$ Thank you for your great, detailed answer to this question and the helpful notes and references!! $\endgroup$
    – Hetong Xu
    Jul 1, 2020 at 8:46
  • $\begingroup$ @HetongXu you are welcome! I am glad to have been of some help. $\endgroup$ Jul 1, 2020 at 8:47
  • $\begingroup$ Maybe I have misunderstood something and I'm wondering how we can deduce the inequality (23) and (24) directly from (25), as you said "whose right side is de facto the weak formulation of the minimal surface operator as shown in problem (25)". Sorry for such a trivial question and thank you for your help! $\endgroup$
    – Hetong Xu
    Jul 2, 2020 at 2:18
  • $\begingroup$ @HetongXu you have correctly understand the statement and do not feel sorry: no motivated question is trivial. Suppose that $f_k\in C^2$, $j=1,2$ and choose $\varphi \in C_c^\infty(D_1)\subset C^\infty(D_1)$: then $$ \int\limits_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{k}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{k}\right|^{2}\right)^{-\frac{1}{2}} \mathrm{d}x=-\int\limits_{D_{1}} \varphi \sum_{i=1}^{8} \frac{\partial }{\partial x_{i}}\bigg(\frac{\partial f_{k}}{\partial x_{i}} \left(1+\left|D f_{k}\right|^{2}\right)^{-\frac{1}{2}} \bigg)\mathrm{d}x$$ $\endgroup$ Jul 2, 2020 at 5:53
  • 1
    $\begingroup$ Thank you for your help and encouragement! $\endgroup$
    – Hetong Xu
    Jul 2, 2020 at 21:20

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