Duality in reconstruction systems

We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.     

Analysis on an ODE arisen from studying the shape of a red blood cell

We study the existence problem of a special solution to the Helfrich functional such that it corresponds to a surface of the shape of a red blood cell. The Helfrich functional is also a perturbation of the Willmore functional involving some parameters with physical meanings. With the expected symmetry of the surface, it reduces to an analysis on an ODE with certain shape requirements. We discover a sufficient condition on the parameters which ensures the existence of such a special solution to the ODE.     

A free boundary problem for capillary surfaces

We prove existence and regularity of capillary surfaces between two parallel hyperplanes in ℝⁿ. By geometrical arguments we also derive some properties of the minimizers of the functional related to the problem.     

Existence and uniqueness for variational problem from progressive lens design

We study a functional modelling the progressive lens design, which is a combination of Willmore functional and total Gauss curvature. First, we prove the existence for the minimizers of this class of functionals among the class of revolution surfaces rotated by the curves y = f(x) about the x-axis. Then, choosing such a minimiser as background surfaces to approximate the functional by a quadratic functional, we prove the existence and uniqueness of the solution to the Euler-Lagrange equation for the quadratic functionals. Our results not only provide a strictly mathematical proof for numerical methods, but also give a more reasonable and more extensive choice for the background surfaces.     

A variational view at the time-dependent minimal surface equation

We present a global variational approach to the L ²-gradient flow of the area functional of cartesian surfaces through the study of the so-called weighted energy-dissipation (WED) functional. In particular, we prove a relaxation result which allows us to show that minimizers of the WED converge in a quantitatively prescribed way to gradient-flow trajectories of the relaxed area functional. The result is then extended to general parabolic quasilinear equations arising as gradient flows of convex functionals with linear growth.     

Limiting properties of the second order Ginzburg–Landau minimizers

The author studies the weak convergence for the gradient of the minimizers for a second order energy functional when the parameter tends to 0. And this paper is also concerned with the location of the zeros and the blow-up points of the gradient of the minimizers of this functional. Finally, the strong convergence of the gradient of the radial minimizers is obtained.     

Uniqueness Theorem of the Regularizable Redial Ginzburg-Landau Type Minimizers

The author proves the uniqueness of the regularizable radial minimizers of a Ginzburg-Landau type functional in the case n - 1 < p < n,and the location of the zeros of the regularizable radial minimizers of this functional is discussed.     

Asymptotic analysis for the radial minimizer of a second-order energy functional

The convergence for the radial minimizers of a second-order energy functional, when the parameter tends to 0 is studied. And the location of the zeros of the radial minimizers of this functional is presented. Based on this result, the uniqueness of the radial minimizer is discussed.     

Local minimizers in micromagnetics and related problems

Let be a smooth bounded domain and consider the energy functional Here is a small parameter and the admissible function m lies in the Sobolev space of vector-valued functions and satisfies the pointwise constraint for a.e. . The induced magnetic field is related to m via Maxwell's equations and the function is assumed to be a sufficiently smooth, non-negative energy density with a multi-well structure. Finally is a constant vector. The energy functional arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown [9]. In this paper we aim to construct local energy minimizers for this functional. Our approach is based on studying the corresponding Euler-Lagrange equation and proving a local existence result for this equation around a fixed constant solution. Our main device for doing so is a suitable version of the implicit function theorem. We then show that these solutions are local minimizers of in appropriate topologies by use of certain sufficiency theorems for local minimizers. Our analysis is applicable to a much broader class of functionals than the ones introduced above and on the way to proving our main results we reflect on some related problems. Received: 20 November 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001     

Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints: a double isosceles configuration and an isosceles trapezoid configuration, while order constraints are introduced on the boundary configurations. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.     

一类Ginzburg-Landau型泛函的径向极小元的渐近性

The behavior of radial minimizers for a Ginzburg-Landau type functional is considered. The weak convergence of minimizers in W1,n is improved to the strong convergence in W1,n. Some estimates of the rate of the convergence for the module of minimizers are presented.     

Convergence of Equilibria of Thin Elastic Plates – The Von Kármán Case

We study the behaviour of thin elastic bodies of fixed cross-section and of height h, with h → 0. We show that critical points of the energy functional of nonlinear three-dimensional elasticity converge to critical points of the von Kármán functional, provided that their energy per unit height is bounded by Ch ⁴ (and that the stored energy density function satisfies a technical growth condition). This extends recent convergence results for absolute minimizers.     

Minimum of a functional in a metric space and fixed points

The existence of minimizers is examined for a function defined on a metric space. Theorems are proved that assert the existence of minimizers, and examples of the functions for which these theorems are valid are given. Then, these theorems are applied to proving theorems on fixed points of univalent and multivalued mappings of metric spaces. Finally, coincident points of two mappings are examined.     

On the ground states of the Ostrovskyi equation and their stability

The Ostrovskyi (Ostrovskyi-Vakhnenko/short pulse) equations are ubiquitous models in mathematical physics. They describe water waves under the action of a Coriolis force as well as the amplitude of a “short” pulse in an optical fiber. In this paper, we rigorously construct ground traveling waves for these models as minimizers of the Hamiltonian functional for any fixed L² norm. The existence argument proceeds via the method of compensated compactness, but it requires surprisingly detailed Fourier analysis arguments to rule out the nonvanishing of the limits of the minimizing sequences. We show that all of these waves are weakly nondegenerate and spectrally stable.     

On Equilibrium Shape of Charged Flat Drops

The equilibrium shapes of two‐dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed. © 2018 Wiley Periodicals, Inc.     

Laguerre minimal surfaces,isotropic geometry and linear elasticity

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.     

Size of Planar Domains and Existence of Minimizers of the Ginzburg–Landau Energy with Semistiff Boundary Conditions

The Ginzburg–Landau energy with semistiff boundary conditions is an intermediate model between the full Ginzburg–Landau equations, which leads to the appearance of both a condensate wave function and a magnetic potential, and the simplified Ginzburg–Landau model, coupling the condensate wave function to a Dirichlet boundary condition. In the semistiff model, there is no magnetic potential. The boundary data are not fixed, but circulation is prescribed on the boundary. Mathematically, this leads to prescribing the degrees on the components of the boundary. The corresponding problem is variational, but noncompact: in general, energy minimizers do not exist. Existence of minimizers is governed by the topology and the size of the underlying domain. We propose here various notions of domain size related to existence of minimizers, and discuss existence of minimizers or critical points, as well as their uniqueness and asymptotic behavior. We also present the state of the art in the study of this model, accounting for results obtained during the last decade by Berlyand, Dos Santos, Farina, Golovaty, Rybalko, Sandier, and the author.     

An inverse problem for Hamiltonian systems

In a linear Hamiltonian system for which the Dirichlet principle is valid, solutions to boundary value problems can be identified as the unique minimizers of the quadratic functional associated with the system. The inverse problem, in which coefficient functions in the differential equations are identified as unique minimizers of a related functional, is discussed, together with conditions under which recovery can occur.     

On Problems in the Calculus of Variations in Increasingly Elongated Domains

This paper deals with minimization problems in the calculus of variations set in a sequence of domains, the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. The authors study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions.