We approximate a hypersurface Sigma with prescribed anisotropic mean curvature with solutions u(epsilon), of suitable nonlinear elliptic equations depending on a small parameter epsilon > O. We work in relative geometry, by endowing R-N with a Finsler norm phi describing the anisotropy. The main result states that Sigma and (u(epsilon) = 0) are close of order epsilon(2)/log epsilon/(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 Sigma. 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 Sigma; (iii) the expansion of the phi-mean curvature of Sigma in a suitable tubular neighbourhood.
