We regularize non-convex anisotropic surface energy of a two-dimensional surface, given as a graph over the two-dimensional unit disk, by the Willmore functional and investigate existence of the corresponding global minimizers. Restricting to the rotationally symmetric case, we obtain a one-dimensional variational problem which permits to derive substantial qualitative information on the minimizers. We show that minimizers tend to a “cone”-like solution as the regularization parameter tends to zero. Areas where the solutions are either convex or concave are identified. It turns out that the structure of the chosen anisotropy hardly affects the qualitative shape of the minimizers.

Accepted:

DOI: 10.1051/cocv/2016024

Keywords: Non-convex anisotropy, regularization, Willmore functional, rotationally symmetric solutions

^{1}; Reiter, Philipp

^{1}

@article{COCV_2017__23_3_1047_0, author = {Pozzi, Paola and Reiter, Philipp}, title = {On non-convex anisotropic surface energy regularized via the {Willmore} functional: {The} two-dimensional graph setting}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1047--1071}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016024}, mrnumber = {3660459}, zbl = {1371.35073}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016024/} }

TY - JOUR AU - Pozzi, Paola AU - Reiter, Philipp TI - On non-convex anisotropic surface energy regularized via the Willmore functional: The two-dimensional graph setting JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1047 EP - 1071 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016024/ DO - 10.1051/cocv/2016024 LA - en ID - COCV_2017__23_3_1047_0 ER -

%0 Journal Article %A Pozzi, Paola %A Reiter, Philipp %T On non-convex anisotropic surface energy regularized via the Willmore functional: The two-dimensional graph setting %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1047-1071 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016024/ %R 10.1051/cocv/2016024 %G en %F COCV_2017__23_3_1047_0

Pozzi, Paola; Reiter, Philipp. On non-convex anisotropic surface energy regularized via the Willmore functional: The two-dimensional graph setting. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1047-1071. doi : 10.1051/cocv/2016024. http://archive.numdam.org/articles/10.1051/cocv/2016024/

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