Wulff theorem

Definition 1 (Wulff shape).

Let $\varphi \mathrm{:}{\mathrm{R}}^n\mathrm{\to }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{+}\mathrm{\infty }\mathrm{)}$ be a non-negative, convex, coercive, positively $\mathrm{1}$-homogeneous function. We define the Wulff shape relative to $\varphi$ as the set

$$W_{\varphi }:=\{x\in {ℝ}^n:⟨x,y⟩\le 1\mathit{\text{ for all }}y\mathit{\text{ such that }}\varphi (y)\le 1\}$$

(where $\mathrm{⟨}\mathrm{\cdot }\mathrm{,}\mathrm{\cdot }\mathrm{⟩}$ is the Euclidean inner product in ${\mathrm{R}}^n$.)

Theorem 1 (Wulff).

Let $\varphi \mathrm{:}{\mathrm{R}}^n\mathrm{\to }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{+}\mathrm{\infty }\mathrm{)}$ be a non-negative, convex, coercive, $\mathrm{1}$-homogeneous function. Given a regular open set $D\mathrm{\subset }{\mathrm{R}}^n$ we consider the following anisotropic surface energy:

$$F_{\varphi }(D)={\int }_{\partial D}\varphi ({\nu }_D(x))𝑑\sigma (x)$$

where ${\nu }_D\mathit{}\mathrm{(}x\mathrm{)}$ is the outer unit normal to $\mathrm{\partial }\mathit{}D$, and $\sigma$ is the surface area on $\mathrm{\partial }\mathit{}D$.Then, given any set $D$ with the same volume as $W_{\varphi }$, i.e. $\mathrm{|}D\mathrm{|}\mathrm{=}\mathrm{|}{W}_{\varphi }\mathrm{|}$, one has $F_{\varphi }\mathit{}\mathrm{(}D\mathrm{)}\mathrm{\ge }{F}_{\varphi }\mathit{}\mathrm{(}{W}_{\varphi }\mathrm{)}$.Moreover if $\mathrm{|}D\mathrm{|}\mathrm{=}\mathrm{|}{W}_{\varphi }\mathrm{|}$ and $F_{\varphi }\mathit{}\mathrm{(}D\mathrm{)}\mathrm{=}{F}_{\varphi }\mathit{}\mathrm{(}{W}_{\varphi }\mathrm{)}$ then $D$ is a translation of $W_{\varphi }$ i.e. there exists $v\mathrm{\in }{R}^n$ such that $D\mathrm{=}v\mathrm{+}{W}_{\varphi }$.