A variational principle for gradient flows in metric spaces

We present a novel variational approach to gradient-flow evolution in metric spaces. In particular, we advance a functional defined on entire trajectories, whose minimizers converge to curves of maximal slope for geodesically convex energies. The crucial step of the argument is the reformulation of the variational approach in terms of a dynamic programming principle, and the use of the corresponding Hamilton–Jacobi equation. The result is applicable to a large class of nonlinear evolution PDEs including nonlinear drift-diffusion, Fokker–Planck, and heat flows on metric-measure spaces.     

Constant vorticity Riabouchinsky flows from a variational principle

Summary Flows with constant vorticity regions bounded by vortex sheets are obtained by minimizing a functional which is the difference of energy in the external (irrotational) flow and the internal flow. In the zero vorticity case this reduces to the functional used by Garabedian, Lewy, and Schiffer for Riabouchinsky's problem. The discretization is done using Schwarz-Christoffel transformations for approximating polygons and FFT's to compute required Dirichlet integrals.     

A variational principle for domino tilings

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

     

A variational principle for fractal dimensions

A new definition of the dimension of probability measures is introduced. It is related with the fractal dimension of sets by a variational principle. This principle is applied in the theory of iterated function systems.     

A variational principle for vector equilibrium problems

A variational principle is described and analysed for the solutions of vector equilibrium problems.     

A variational principle for the Navier-Stokes equation

Motivated from Arnold's variational characterization of the Euler equation in terms of geodesic families of diffeomorphisms, a variational principle for the motion of incompressible viscous fluids is presented. A volume preserving diffusion process with drift velocity field subject to the Navier-Stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations.     

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.     

A variational principle for doubly nonlinear evolution

The weighted energy-dissipation principle stands as a novel variational tool for the study of dissipative evolution and has already been applied to rate-independent systems and gradient flows. We provide here an example of its application to a specific yet critical doubly nonlinear equation featuring a super-quadratic dissipation.     

A pre-order principle and set-valued Ekeland variational principle

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi-ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results.     

A relative local variational principle for topological pressure

We define the relative local topological pressure for any given factor map and open cover,and prove the relative local variational principle of this pressure.More precisely,for a given factor map π:(X,T)→(Y,S) between two topological dynamical systems,an open cover U of X,a continuous,real-valued function f on X and an S-invariant measure ν on Y,we show that the corresponding relative local pressure P(T,f,U,y) satisfies sup μ∈M(X,T){ hμ(T,U|Y)+∫X f(x)dμ(x) :πμ=ν}=∫Y P(T,f,U,y)dν(y),where M(X,T) denotes the family of all T-invariant measures on X.Moreover,the supremum can be attained by a T-invariant measure.     

A parametric variational principle and residuality

We prove a parametric version of a smooth convex variational principle with constraints using a Baire category approach. We examine in depth the necessity of the assumptions of our variational principle by providing counterexamples.     

A convexity theorem for noncommutative gradient flows

In this survey we shall prove a convexity theorem for gradient actions of reductive Lie groups on Riemannian symmetric spaces. After studying general properties of gradient maps, this proof is established by (1) an explicit calculation on the hyperbolic plane followed by a transfer of the results to general reductive Lie groups, (2) a reduction to a problem on abelian spaces using Kostant's Convexity Theorem, (3) an application of Fenchel's Convexity Theorem. In the final section the theorem is applied to gradient actions on other homogeneous spaces and we show, that Hilgert's Convexity Theorem for moment maps can be derived from the results.     

A variational principle for the nonstationary linear navier-stokes equations

A time-nonlocal boundary value problem for the linear evolution Navier-Stokes equations is considered, with time-averaged data being prescribed instead of initial ones. The solving of the problem is reduced to the minimizing of a quadratic functional.     

A Gibbs variational principle in space-time for infinite-dimensional diffusions

 We show that the set of stationary weak solutions for a class of infinite dimensional stochastic differential equations coincides with the set of shift invariant, space-time Gibbs fields for a certain potential. The key step consists in proving the Gibbs variational principle for space-time Gibbs fields. Received: 20 May 1999 / Revised version: 14 May 2001 / Published online: 11 December 2001     

Vector variational principle

We prove an Ekeland’s type vector variational principle for monotonically semicontinuous mappings with perturbations given by a convex bounded subset of directions multiplied by the distance function. This generalizes the existing results where directions of perturbations are singletons.     

A non-smooth version of Minty variational principle

The aim of this article is to study the relationship between generalized Minty vector variational inequalities and non-smooth vector optimization problems. Under pseudoconvexity or pseudomonotonicity, we establish the relationship between an efficient solution of a non-smooth vector optimization problem and a generalized Minty vector variational inequality. This offers a non-smooth version of existing Minty variational principle.     

A variational principle and best proximity points

We generalize Ekeland's Variational Principle for cyclic maps. We present applications of this version of the variational principle for proving of existence and uniqueness of best proximity points for different classes of cyclic maps.     

A variational principle for problems with a hint of convexity

A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.     

A variational principle for eigenvalues of pencils of Hermitian matrices

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number is an eigenvalue ofH ifdet(A–B)=0 (we include = ifdetB=0). For nonsingularH (i.e., for which some is not an eigenvalue), we show precisely which eigenvalues can be characterized as _k ⁺ =sup{inf{*A:*B=1,S},SS _k},S _k being the set of subspaces of C ^N of codimensionk–1.Dedicated to the memory of our friend and colleague Branko NajmanResearch supported by NSERC of Canada and the I.W.Killam FoundationProfessor Najman died suddenly while this work was at its final stage. His research was supported by the Ministry of Science of CroatiaResearch supported by NSERC of Canada