On non-convex anisotropic surface energy regularized via the Willmore functional: The two-dimensional graph setting

Pozzi, Paola; Reiter, Philipp

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 1047-1071.

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.

Classification : 35J35, 35B65, 35B07
Mots clés : Non-convex anisotropy, regularization, Willmore functional, rotationally symmetric solutions

S. Angenent and M.E. Gurtin, Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface. Arch. Rational Mech. Anal. 108 (1989) 323–391. | DOI | MR | Zbl

P. Aviles and Y. Giga, A mathematical problem related to the physical theory of liquid crystal configurations. In Miniconference on geometry and partial differential equations, 2 (Canberra (1986)). Vol. 12 of Proc. Centre Math. Anal. Austral. Nat. Univ. Austral. Nat. Univ., Canberra (1987) 1–16. | MR

G. Bellettini, Anisotropic and crystalline mean curvature flow. In A sampler of Riemann-Finsler geometry. Vol. 50 of Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge (2004) 49–82. | MR | Zbl

G. Buttazzo, M. Giaquinta and S. Hildebrandt. One-dimensional variational problems. Vol. 15 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998). | MR | Zbl

D.G. De Figueiredo, E.M. Dos Santos and O.H. Miyagaki, Sobolev spaces of symmetric functions and applications. J. Funct. Anal. 261 (2011) 3735–3770. | DOI | MR | Zbl

K. Deckelnick, G. Dziuk and C. M. Elliott. Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232. | DOI | MR | Zbl

A. Di Carlo, M.E. Gurtin and P. Podio-Guidugli, A regularized equation for anisotropic motion-by-curvature. SIAM J. Appl. Math. 52 (1992) 1111–1119. | DOI | MR | Zbl

L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Revised third printing. Springer-Verlag, Berlin, 2nd edition (1998). | MR

A. Lorent, A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity. ESAIM: COCV 18 (2012) 383–400. | Numdam | MR | Zbl

P. Pozzi and Ph. Reiter, Approximation of non-convex anisotropic energies via Willmore energy. CD-ROM Proc. of the 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), September 10-14, 2012, Vienna, Austria, edited by J. Eberhardsteiner, H.J. Böhm and F.G. Rammerstorfer. Vienna University of Technology, Austria (2012).

P. Pozzi and Ph. Reiter, Willmore-type regularization of mean curvature flow in the presence of a non-convex anisotropy. The graph setting: analysis of the stationary case and numerics for the evolution problem. Adv. Differ. Eq. 18 (2013) 265–308. | MR | Zbl

A. Sard, The measure of the critical values of differentiable maps. Bull. Amer. Math. Soc. 48 (1942) 883–890. | DOI | MR | Zbl