Peter A. Djondjorov

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An interesting class of axially symmetric surfaces, which generalizes Delaunay's unduloids and provides solutions of the shape equation is described in explicit parametric form. This class provide the first analytical examples of surfaces with periodic curvatures studied by K. Kenmotsu and leads to some unexpected relationships among Jacobian elliptic(More)
The membrane shape equation derived by Helfrich and Ou-Yang describes the equilibrium shapes of biomembranes, built by bilayers of am-phiphilic molecules, in terms of the mean and Gaussian curvatures of their middle-surfaces. Here, we present a new class of translationally-invariant solutions to this equation in terms of the elliptic functions which(More)
—In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived. A general statement of the associated group-classification problem is given. A simple intrinsic(More)
The six-parameter group of three dimensional Euclidean motions is recognized as the largest group of point transformations admitted by the membrane shape equation in Mongé representation. This equation describes the equilibrium shapes of biomembranes being the Euler-Lagrange equation associated with the Helfrich curvature energy functional under the(More)
This study is concerned with the determination of the mechanical behaviour of closed fluid lipid bilayer membranes (vesicles) under a uniform hydrostatic pressure, pressed against and adhering onto a flat homogeneous rigid substrate. Assuming that the initial and deformed shapes of the vesicle are axisymmetric, a variational statement of the problem is(More)
The consideration of some non-standard parametric Lagrangian leads to a fictitious dynamical system which turns out to be equivalent to the Euler problem for finding out all possible shapes of the lamina. Integrating the respective differential equations one arrives at novel explicit parameteri-zations of the Euler's elastica curves. The geometry of the(More)
Recent results concerning the application of Lie transformation group methods to structural mechanics are presented. Focus is placed on the point Lie symmetries and conservation laws inherent to the Bernoulli–Euler and Timoshenko beam theories as well as to the Marguerre-von Kármán equations describing the large deflection of thin elastic shallow shells(More)
The Sturm spirals which can be introduced as those plane curves whose curvature radius is equal to the distance from the origin are embedded into one-parameter family of curves. Explicit parametrization of the ordinary Sturmian spirals along with that of a wider family of curves are found and depicted graphically. 1. Introduction. The fundamental existence(More)
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