Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the selfintersecting minimal surface that fills the trefoil knot. UDC 514.763.85, 517.972.6 Introduction. In this paper we consider the Euler–Lagrange minimal surface equation

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in [2]. Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence. We prove a Sobolev regularity result for their minimizers despite ...

Abstract. We study the composite membrane problem in all dimensions. We prove that the minimizing solutions exhibit a weak uniqueness property which under certain conditions can be turned into a full uniqueness result. Next we study the partial regularity of the solutions to the Euler– Lagrange equation associated to the composite problem and also the regularity of the free boundary for solutio...

A quaternionic partial differential equation is shown to be a generalisation of the traditional Riccati equation and its relationship with the Schrödinger equation is established. Various approaches to the problem of finding particular solutions to this equation are explored, and the generalisations of two theorems of Euler on the Riccati equation, which correspond to this partial differential ...

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

. Consider a mechanical system consisting of N particles in R subject to some forces. Let xi ∈ R denote the position vector of the ith particle. Then all possible positions of the system are described by N -tuples (x1, . . . , xN ) ∈ (R) . The space (R) is an example of a configuration space. The time evolution of the system is described by a curve (x1(t), . . . , xN (t)) in (R) and is governed...

Let M be a Kähler surface and Σ be a closed symplectic surface which is smoothly immersed in M . Let α be the Kähler angle of Σ in M . We first deduce the Euler-Lagrange equation of the functional L = ∫ Σ 1 cosα dμ in the class of symplectic surfaces. It is cos αH = (J(J∇ cosα)), where H is the mean curvature vector of Σ in M , J is the complex structure compatible with the Kähler form ω in M ,...

We prove higher integrability properties of solutions to the problem of minimizing ∫ Ω L(x, u(x),∇u(x))dx, where ξ → L(x, u, ξ) is a convex function satisfying some additional conditions. As an application, we prove the validity of the Euler–Lagrange equation for a class of functionals with growth faster than exponential.

The auto-parallel equation over spaces with affine connections and metrics [(Ln, g)-spaces] is considered as a result of the application of the method of Lagrangians with covariant derivatives (MLCD) on a given Lagrangian density.

Under some regularity assumptions on the boundary datum u (assumptions automatically satisfied in the classical case when the growth of the integrand is bounded by a+ b‖ξ‖), we prove the validity of the Euler Lagrange equation for the functional