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1. Introduction.

By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.

Question. What are the known regularity results for anisotropic minimal surfaces?

1.2 Preliminaries

For instance, reading a 1972 paper by Bombieri and Giusti I found the following definition:

Definition. A set $A \subset \mathbb R^n$ has an oriented boundary of least area if $\chi_A \in BV(\mathbb R^n)$ and for every $g \in BV(\mathbb R^n)$ with compact support $K$ we have $$\tag{1} P(A) := \vert D \chi_A\vert(K) \le \vert D(\chi_A + g)\vert(K) $$ in the sense of measures, being $\chi_A$ the characteristic function of the set $A$ and $P(A)$ the Euclidean perimeter of $A$.

Right after this definition, the authors say:

It is known that if $A$ has oriented boundary of least area then [..] the boundary is an analytic hypersurface, except possibly for a closed set whose Hausdorff dimension does not exceed $n-8$.

2. My question

I would like to know what are the known results for the analogue of this problem in the anisotropic case. Let me clarify what I mean: let $f \colon \mathbb R^n \to \mathbb R$ be some good function (say non-negative, convex and positively 1-homogeneous, as usual in Calculus of Variations). We define the anisotropic perimeter of a set $A$ (of finite perimeter) by $$ P_f(A) := \int_{\partial^e A} f(\nu_A(x))\, d\mathcal H^{n-1}(x) $$ where $\nu_A$ is the measure theoretic outer unit normal, $\partial^e A$ is the essential boundary of $A$ and $\mathcal H^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.

What is the analogue of (1) in this case? And are there known regularity results for the solutions to this minimum problem? Can you point out some reference to the literature investigating this problem? Thanks.

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Very little is known in terms of regularity theory compared with the level it is known for area-minimizers. However there are recent breakthroughs in other senses i.e. existence and rectifiability.

See

De Philippis, De Rosa, Ghiraldin. "Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies." Communications on Pure and Applied Mathematics 71.6 (2018): 1123-1148.

De Lellis, De Rosa, Ghiraldin. "A direct approach to the anisotropic Plateau problem." Advances in Calculus of Variations (2016)

...And the other related papers by this circle of Italians.

A key issue is the lack of a monotonicity formula for varifolds stationary with respect to an anisotropic integrand. This is discussed in an old and somewhat hard to find paper of Allard, but Allard proved a result that suggests that perhaps it is not useful to look for monotonicity formulae for other functionals.

EDIT: I should clarify a bit: There are some older regularity results when the integrand is assumed to satisfy certain nice ellipticity hypotheses. e.g. one of the main papers is:

Schoen, Simon, Almgren. "Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals." Acta Mathematica 139.1 (1977): 217-265.

The characterization of ellipticity/convexity conditions for these functionals is a lore unto itself, but the situation in which most progress was made by papers like the one above is this: If your minimizer is a Lipschitz graph then you have a Lipschitz function that minimizes some functional. Look at the Euler--Lagrange equation of that functional. Is it a nice uniformly elliptic 2nd order elliptic equation? If yes, then some of the same techniques as for area minimizers will open up. But some of the recent theorems like I mention above have weaker hypotheses on the integrands. So, more general notions of anisotropic convex integrands end up corresponding to the theory of elliptic equations with $$ a_{ij}\xi^i\xi^j \geq 0 $$ type conditions, rather than $$ a_{ij}\xi^i\xi^j > \lambda |\xi|^2 \qquad \lambda > 0\ \text{fixed} $$ type conditions. This is a tricky area of PDE, let alone GMT.

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  • $\begingroup$ Thanks a lot for your answer! I found rather surprising that very little is known on this topic, I did expect a huge literature on this. I will look through the papers you mentioned. A part from regularity, which seems out-of-reach, do you know whether some maximum principles are available in this setting? In the aforementioned paper by Bombieri & co. it is proved e.g. (using regularity) that if $U,V$ are open sets s.t. their boundaries of least area and $U,V$ coincide out of a compact then they are indeed equal $U=V$. Any ideas for a similar principle in anisotropic case? Thanks again. $\endgroup$ – Romeo Jan 8 at 19:52
  • $\begingroup$ Maximum principles is an excellent question, but as I allude to, a lot of these questions can't be answered in general for degenerate elliptic PDE and these minimizers are going to be harder. $\endgroup$ – T_M Jan 9 at 4:52

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