Archive for 22nd June 2010

Variation Formulae for Perimeter and Volume Densities

22nd June 2010, 07:45 am

For Rn+1 with volume density f and perimeter density g, for a normal variation u of a surface with classical mean curvature H, the first variation of volume and perimeter are given by:

\delta ^1V=-\int uf,

\delta ^1P=-\int [(g/f)nH - (1/f)(\partial g/\partial n)] uf.

For a volume-preserving variation, the second variation of perimeter is given by:
\delta ^2P=\int g|\nabla u|^2-g|\sigma|^2u^2-f\frac{\partial (g/f)}{\partial n}u^2nH+u^2\frac{\partial ^2g}{\partial n^2}-\frac 1fu^2\frac{\partial f}{\partial n}\frac{\partial g}{\partial n},

where \sigma is the second fundamental form, so that \sigma^2 is the sum of the squares of the principal curvatures. Continue reading ‘Variation Formulae for Perimeter and Volume Densities’ »

Category: Math  |  1 Comment