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The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not.

Premises

I am analysing the following Plateau problem.
Let $G\subsetneq\Bbb R^n$ be a general open set, its topological boundary and $\Bbb B^n(x, R)$ a ball of radius $R>0$ centred in $x\in\partial G$: moreover let the boundary $n-2$-dimensional contour (the Dirichlet data four our Plateau problem) be given by the intersection $\partial G\cap\partial \Bbb B^n(x, R)$.
It is a standard result, found for example in the monograph [1] by Enrico Giusti (§2, p. 17, due to Ennio De Giorgi), that there exist (at least a) minimal (hyper)surface $M$ spanning this contour: possibly $\partial G$ may act as an obstacle, forcing $M$ not to be exactly the spanning minimal surface (see for example [1] §2, remark 1.22, p. 19) and moreover uniqueness does not hold in general.

The question

Assumed that there isn't uniqueness, I am wondering if a phenomenon ow well posedness similar to the one seen for the first time by Calogero Vinti [2] for (nonlinear) hyperbolic equations may occur (see also here for a relevant Q&A). Precisely, Let's suppose that there exist multiple minimal (hyper)surfaces $M_i$ spanning the same contour $\partial G\cap\partial \Bbb B^n(x, R)$ and suppose that I can approximate this "contour", for example

  • by mollyfinig the characteristic function $\chi_G$ and considering a sequence of intersections $\partial G_k\cap\partial \Bbb B^n(x, R)$ ($\partial G_k$ being an appropriate level set of the mollified characteristic function), or
  • by stretching the ball $\Bbb B^n\big(x, R+{1\over k}\big)$ and considering the sequence of intersections $\partial G\cap\partial \Bbb B^n\big(x, R+{1\over k}\big)$.

Is it reasonable to expect the existence of sequences of approximating minimal surfaces $\{M_{i}^j\}_{j\in\Bbb N}$ converging uniformly to each "branch" $M_i$ respect to a proper norm $\|\cdot\|$, i.e. such that $\|M_i^j - M_i\|\to 0 $ uniformly in $x$?

I think this should be possible thanks to the existence of a maximum principle for the minimal surface equation, but not being an expert, I decided to ask here for an expert advice.

Edit

  • After a clarification requested by Daniel Asimov, I added this picture (taken from [1], §2, p. 19, picture) I've done in order to clarify (in the next future...) an earlier Math.SE answer I gave a few months ago. In this picture context,

    • $\Omega\equiv G$ is the the starting open set, also acting as an obstacle,
    • $L\equiv \Bbb B^n(x, R)$ is the Caccioppoli set which, jointly with $\partial\Omega$, defines the contour
    • the dashed set $E$ is the minimising Caccioppoli set, the part of whose boundary shown in red is the minimal surface $M$.

    In the picture we see that $M$ (i.e. $\partial E\setminus \partial L$) is a minimal surface in its rectilinear parts, while it coincides with $\partial\Omega$ otherwise. enter image description here

A loosely related note

Recently (26th of March 2024) Enrico Giusti passed away: I believe that the best thing I can do in order to remember him is recalling Karl Weierstraß's words for Sofya Kovalevskaya (as also Francesco Tricomi did in remembering Vito Volterra):

Die Menschen sterben, die Gedanken bleiben.

References

[1] Enrico Giusti, 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] Calogero Vinti "Su una specie di dipendenza continua delle soluzioni dal dato iniziale, per l’equazione $p=f(q)$, in una classe ove manca l’unicità [On a kind of continuous dependence of solutions from the initial data, for the $p=f(q)$ equation, in a class where uniqueness lacks]" (in Italian), Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 3, Volume 19 (1965) no. 2, p. 251-263, MR185249, Zbl 0133.04602.

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  • $\begingroup$ The phrase "forcing 𝑀 not to be exactly the spanning minimal surface" leaves me wondering exactly what the result of De Giorgi states. What is M, then? $\endgroup$ Apr 11 at 16:48
  • $\begingroup$ @DanielAsimov $M$ is partly a minimal surface and partly (and possibly) the boundary of $G$ which acts as an obstacle. I will ad an explanatory picture taken from Giusti's monograph. $\endgroup$ Apr 12 at 8:25

1 Answer 1

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I am far from being an expert, but if I understand your question correctly, the answer is "no", at least for the Plateau problem in the realm of Caccioppoli sets.

Vinti's cited result [2] proves (the Theorem at bottom page 254) that for every approximating sequence of boundary datums, you can find a sequence of solutions (with this data at the boundary) converging uniformly. The desired statement for minimal surfaces would be the following.

Say $\Omega = B_1 \subset \mathbb{R}^3$ and you have a $1$-dimensional boundary datum $\gamma \subset \partial B_1$. Then you claim that for every $\gamma_k \to \gamma$ then, there exists a sequence of minimizers $M_k$ with $M_k\cap \partial B_1=\gamma_k$ with least area (not necessarily unique) that converges uniformly.

Take the following example. In $B_1$ there exists exactly one catenoid $C\subset B_1$ whose boundary is two parallel circles in $\partial B_1$ and such that its area is equal to the area of the two circles. That is, the Plateau minimizer for the boundary datum $C\cap \partial B_1$ is not unique. Moreover, if you move the two circles a bit closer, then the minimizer is unique, and it's the catenoid connecting them, and if you move them a bit further, the minimizer is unique and is the union of the two parallel disks. Now take an approximating sequence $\Gamma_k \subset \partial B_1$ of boundary data that oscillates between these two cases; that is, you always swap between two circles slightly closer and slightly further than the critical configuration. Then, for every $k$, there is a unique Plateau minimizer with boundary $\Gamma_k$, and it oscillates between a catenoid and two disks, so it cannot converge uniformly.

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  • $\begingroup$ Thank you very much Michele: I suspected that the non uniqueness could be a problem when, for example, $\gamma_n$ changes its connectivity (i.e. when a loop on $\partial B_1$ collapses to a single point or joins with another loop): now I see that the problem is much worser as in the example you described, when $\gamma_n$ has the same connectivity of $\gamma$ but a branch minimal surface disappears suddenly. $\endgroup$ Apr 12 at 14:19

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