A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

A variational characterization of contact metric structures

Annals of Global Analysis and Geometry - On a manifold with a given nowhere vanishing vector field, we examine the squared $$L^2$$ -norm of the integrability tensor of the orthogonal complement of...     

A variational mcshane integral characterization of the Radon-Nikodym property

We present a new characterization of Banach spaces possessing the Radon-Nikodym property in terms of additive interval functions whose McShane variational measures are absolutely continuous with respect to the Lebesgue measure.     

A variational characterization of canonical angles between subspaces

Canonical angles between subspaces of a unitary space are characterized by a min-max property which involves inner products.     

A variational characterization of rate‐independent evolution

Rate‐independent evolution driven by non‐convex potentials is by nature non‐smooth and some weak solvability notions have been recently advanced. This note is intended to contribute to this discussion by proposing a variational characterization of rate‐independent evolution based on a variational principle and a maximal dissipation criterion. The resulting novel solution notion is assessed in an elementary yet critical scalar case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new variational characterization of Jacobi fields along geodesies

A classical result of Riemannian geometry states that Jacobi fields along geodesics of a Riemannian manifold (Q, g) can be obtained as geodesies of the so-called «complete lift» of the metric g itself to the tangent bundle TQ. We show that this classical result is in fact a very simple consequence of a completely general theorem of Calculus of Variations.     

Area-minimizing subsurfaces of Scherk's singly periodic surface and the catenoid

An unbounded portion of the catenoid and a class of subsurfaces of Scherk's singly periodic surface defined over nonconvex domains are shown to be area-minimizing. Mathematics Subject Classification (1991) Primary 49Q05 Secondary 53A10, 30C45     

A variational characterization of a hyperelastic rod with hard self-contact

We consider an elastic rod, modeled as a curve in space with an impenetrable surrounding tube of radius ρ, subject to a general class of boundary conditions. The impossibility of self-intersection is then imposed as a family of scalar constraints on the physical separation of nonlocal pairs of points on the rod. Thus, the usual variational formulation of energy minimization is considered in a context of nonconvex, nonsmooth optimization. We show the existence of minimizers within suitably defined homotopy classes associated with both the centerline and the frame along the rod. The principle results are then concerned with derivation of first-order necessary conditions for optimality and some consequences of these for the contact forces and for regularity.     

On the variational characterization and convergence of bivariate splines

It is shown that bivariate interpolatory splines defined on a rectangleR can be characterized as being unique solutions to certain variational problems. This variational property is used to prove the uniform convergence of bivariate polynomial splines interpolating moderately smooth functions at data which includes interpolation to values on a rectangular grid. These results are then extended to bivariate splines defined on anL-shaped region.This research was supported by a University of Kansas General Research Grant.     

A variational characterization for $\sigma_{n/2}$

We present here a conformal variational characterization in dimension n = 2k of the equation , where A is the Schouten tensor. Using the fully nonlinear parabolic flow introduced in [3], we apply this characterization to the global minimization of the functional.Received: 31 March 2003, Accepted: 10 July 2003, Published online: 25 February 2004Research supported in part by an NSF Postdoctoral Fellowship.     

A variational theory of the Hessian equation

By studying a negative gradient flow of certain Hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous Hessian equations. To achieve this we derive uniform, first‐ and second‐order a priori estimates for the elliptic and parabolic Hessian equations. Our results generalize well‐known results for semilinear elliptic equations and the Monge‐Ampère equation. © 2001 John Wiley & Sons, Inc.     

A variational proof of the Gauss-Bonnet formula

The classical Gauss-Bonnet formula has the form I(g_ij)=2, where I(g_ij) represents a sum of three terms each of which depends on the metric tensor g_ij. It is shown that the first variation I of I(g_ij) with respect to the metric g_ij vanishes and that for the Euclidean metric _ij we have I(_ij)=2. From this the formula I(g_ij)=2 follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(g_ij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor g_ij and its derivatives up to the second order is obtained with the aid of results of Rund [1].     

A variational approach of the rank function

In the same spirit as the one of the paper (Hiriart-Urruty and Malick in J. Optim. Theory Appl. 153(3):551–577, 2012) on positive semidefinite matrices, we survey several useful properties of the rank function (of a matrix) and add some new ones. Since the so-called rank minimization problems are the subject of intense studies, we adopt the viewpoint of variational analysis, that is the one considering all the properties useful for optimizing, approximating or regularizing the rank function.     

Higher genus minimal surfaces by growing handles out of a catenoid

Starting with the catenoid we derive global Weierstraß representations for minimal surfaces obtained by pushing and pulling handles and by adding ends. The Weierstraß data are obtained from Riemann surfaces given by algebraic functions. Modifying the equations for these curves gives the new surfaces. New and well known examples are treated.