# On a Level-set Crystalline Curvature Flow of Surfaces

Friday, May 23, 2014 - 10:20am - 11:00am

Keller 3-180

Yoshikazu Giga (University of Tokyo)

It is by now standard that a level-set approach provides a global unique generalized solution (up to fattening) for mean curvature flow equations [G]. Even from the early stage of the theory, it is known that the method is very flexible to apply anisotropic curvature flow equations which correspond to anisotropic interfacial energy.

The anisotropy is very important in materials sciences. However, if the anisotropy is very singular for example a crystalline curvature flow corresponding to crystalline interfacial energy a level set approach was not available except evolution of curves to which a foundation of the theory was established by M.-H. Giga and Y. Giga more than ten years ago.

In this talk we push forward a level-set approach to surface evolution by crystalline curvature. The main difficulty is that crystalline curvature is a nonlocal quantity and it may not be a constant on each flat potion of a surface. (In the case of curve evolution it is always a constant over a segment.) We overcome this difficulty by introducing a suitable notion of viscosity solutions so that a comparison principle holds. We further construct a global-in-time solution as a limit of smoother problem. A delicate analysis is necessary to achieve the goal. A similar but a simpler problem was studied in [MGP1], [MGP2]. We elaborate these approaches for our purpose.

[G] Y. Giga, Surface evolution equations. A level set approach. Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. xii+264 pp.

[MGP1] M.-H. Giga, Y. Giga and N. Požár, Periodic total variation flow of non-divergence type in $R^n$, J. Math. Pures Appl., to appear.

[MGP2] M.-H. Giga, Y. Giga and N. Požár, Anisotropic total variation flow of non-divergence type on a higher dimensional torus. Adv. Math. Sci. Appl. 23 (2013), no. 1, 235–266.

The anisotropy is very important in materials sciences. However, if the anisotropy is very singular for example a crystalline curvature flow corresponding to crystalline interfacial energy a level set approach was not available except evolution of curves to which a foundation of the theory was established by M.-H. Giga and Y. Giga more than ten years ago.

In this talk we push forward a level-set approach to surface evolution by crystalline curvature. The main difficulty is that crystalline curvature is a nonlocal quantity and it may not be a constant on each flat potion of a surface. (In the case of curve evolution it is always a constant over a segment.) We overcome this difficulty by introducing a suitable notion of viscosity solutions so that a comparison principle holds. We further construct a global-in-time solution as a limit of smoother problem. A delicate analysis is necessary to achieve the goal. A similar but a simpler problem was studied in [MGP1], [MGP2]. We elaborate these approaches for our purpose.

[G] Y. Giga, Surface evolution equations. A level set approach. Monographs in Mathematics, 99. Birkhäuser Verlag, Basel, 2006. xii+264 pp.

[MGP1] M.-H. Giga, Y. Giga and N. Požár, Periodic total variation flow of non-divergence type in $R^n$, J. Math. Pures Appl., to appear.

[MGP2] M.-H. Giga, Y. Giga and N. Požár, Anisotropic total variation flow of non-divergence type on a higher dimensional torus. Adv. Math. Sci. Appl. 23 (2013), no. 1, 235–266.

MSC Code:

53C44

Keywords: