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?

2. A more detailed version of the question

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 $A$.

The authors say - right after the definition - that "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 $n-8$".

I would like to investigate the analogue of this problem in the anisotropic case. Let me clarify what I mean: let $f \colon \mathbb R^n \to [0,+\infty)$ 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 and $\mathcal H^{n-1}$ is the 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.

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.