We approximate a hypersurface
$\varSigma$
with prescribed anisotropic mean curvature with solutions
$u _{\varepsilon}$
of suitable nonlinear elliptic equations depending on a small parameter
$\varepsilon>0$
. We work in relative geometry, by endowing
$\mathbb{R}^N$
with a Finsler norm
$\phi$
describing the anisotropy. The main result states that
$\varSigma$
and
$\{u _{\varepsilon}=0\}$
are close of order
$\varepsilon^2\vert\log\varepsilon\vert^2$
, and this estimate is optimal. This is obtained for two different elliptic equations by sub‐ and supersolutions technique, under smoothness and nondegeneracy assumptions on
$\varSigma$
. Basic steps are: (i) an explicit computation of the second variation of the
$\phi$
‐Minkowski content along geodesics; (ii) the definition of a Laplace–Beltrami operator on
$\varSigma$
; (iii) the expansion of the
$\phi$
‐mean curvature of
$\varSigma$
in a suitable tubular neighbourhood.
