Related papers: Stability of Membranes
In general, the system of $2$nd-order partial differential equations made of the Euler-Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of…
We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field $\mathbf{n}$ and its degree of orientation $s$,…
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value…
Smooth solutions of the forced incompressible Euler equations satisfy an energy balance, where the rate-of-change in time of the kinetic energy equals the work done by the force per unit time. Interesting phenomena such as turbulence are…
Motivated by recent work we study rotating ellipsoidal membranes in the framework of the light-cone supermembrane theory. We investigate stability properties of these classical solutions which are important for the quantization of super…
In the stability analysis of an equilibrium, given by a stationary point of a functional F[n] (free energy functional, e.g.), the second derivative of F[n] plays the essential role. If the system in equilibrium is subject to the…
A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$--gradient flow of an energy involving an elastic bending energy and a line…
We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants.…
We consider the periodic problem for two-fluid non-isentropic Euler-Maxwell systems in plasmas. By means of suitable choices of symmetrizers and an induction argument on the order of the time-space derivatives of solutions in energy…
We describe a functional framework suitable to the analysis of the Cahn-Hilliard equation on an evolving surface whose evolution is assumed to be given \textit{a priori}. The model is derived from balance laws for an order parameter with an…
We examine the reaction of a homogeneous spherical fluid vesicle to the force exerted by a rigid circular ring located at its equator in the linear regime. We solve analytically the linearized first integral of the Euler-Lagrange equation…
The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The…
We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character - considering either one of…
Surfaces of revolution in three-dimensional Euclidean space are considered. Several new examples of surfaces of revolution associated with well-known solvable cases of the Schoedinger equation (infinite well, harmonic oscillator, Coulomb…
Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows and general relativity. Recent work by the author and his collaborators attempts to set the foundations for a study of weak solutions defined on…
We deal with boundary value problems for second-order nonlinear elliptic equations in divergence form, which emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of…
Consider a surface, enclosing a fixed volume, described by a free-energy depending only on the local geometry; for example, the Canham-Helfrich energy quadratic in the mean curvature describes a fluid membrane. The stress at any point on…
Using a covariant geometric approach we obtain the effective bending couplings for a 2-dimensional rigid membrane embedded into a $(2+D)$-dimensional Euclidean space. The Hamiltonian for the membrane has three terms: The first one is…
We extend Newton's problem of minimal resistance to the Lorentz-Minkowski space. We derive the functional energy and determine the Euler-Lagrange equation. In contrast to the Euclidean case, this equation is quasilinear elliptic, and thus,…
We introduce a novel set of stability conditions for vacua with broken Lorentz symmetry. The first class of conditions require that the energy be minimized under small geometric deformations, which translates into requiring the positivity…