In doing so it traps some air and, in trying to minimize area, compresses that air inside; this sets up a pressure differential across the surface that enters into the dynamics of surface formation Soap bubbles are not minimal surfaces; they are what mathematicians call surfaces of constant

mean curvature. These are formed by processes that try to minimize area, subject to the constraint that a fixed volume must be enclosed.

where [H.sub.s] denotes the

mean curvature with respect to the unit normal pointing towards spatial infinity.

Finally we describe the Lagrangian self-similar solutions of the

Mean Curvature Flow which are SO(n)-equivariant.

For readers who have completed their study of linear elliptic differential equations and intend to explore nonlinear ones, Han discusses quasilinear and fully nonlinear equations, focusing on two important nonlinear elliptic differentials closely related to geometry, the

mean curvature equation and the Monge-Ampere equation.

Submanifolds of [E.sup.m] with harmonic

mean curvature vector.

For instance, in [3], Carmo and Dajczer define rotational hyper-surfaces with constant

mean curvature (cmc) in hyperbolic n-space.

The

mean curvature of the bilayer increases gradually (Figure 5) until the stable meander shape form is achieved after approximately 16 [micro]s.

We provide a variational approximation scheme to the anisotropic

mean curvature flow; a family [{[[GAMMA].sub.t]}.sub.t[greater than or equal to]0] of closed hyper-surfaces in [R.sup.N] evolving by the following equation

Smyth: A formula of Simons' type and hypersurfaces with constant

mean curvature, J.

Among the topics are variation on the p-Laplacian, extremal functions in Poincare-Sobolev inequalities for functions of bounded variation, homocline type solutions for a class of differential equations with periodic coefficients, the cooperative case of quasilinear and singular systems, weighted asymmetric problems for an indefinite elliptic operator, multiple non-trivial solutions of the Dirichlet problem for the prescribed

mean curvature equation, and existence of nodal solutions for some nonlinear elliptic problems.

area S = 2[V.sub.2], the integral of

mean curvature M = [pi][V.sub.1], and the Euler characteristic X = [V.sub.0], see e.g.

The trace of -[[??].sub.[xi]] is called the lightlike

mean curvature [H.sub.[xi]] on M associated with [xi].