Elliptic approximations of prescribed mean curvature surfaces in Finsler geometry

Article type: Research Article

Authors: Bellettini, Giovanni | Fragalà, Ilaria

Affiliations: Istituto di Matematica Applicata “U. Dini”, Via Bonanno Pisano, 25/b, 56126 Pisa, Italy E‐mail: [email protected] | Dipartimento di Matematica “L. Tonelli”, Via Buonarroti, 2, 56127 Pisa, Italy E‐mail: [email protected]

Abstract: 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.

Keywords: Finsler spaces, surfaces with prescribed mean curvature, second variation, asymptotic behaviour of solutions of PDE’s, nonlinear elliptic equations

Journal: Asymptotic Analysis, vol. 22, no. 2, pp. 87-111, 2000