A variational principle for gradient flows

We verify – after appropriate modifications – an old conjecture of Brezis-Ekeland ([3], [4]) concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals. Our approach is based on a concept of self-duality inherent in many parabolic evolution equations, and motivated by Bolza-type problems in the classical calculus of variations. The modified principle allows to identify the extremal value –which was the missing ingredient in [3]– and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media, fast diffusion and more general dissipative evolution equations.Both authors were partially supported by a grant from the Natural Science and Engineering Research Council of Canada.This paper is part of this authors Masters thesis under the supervision of the first named author.Revised version: 31 March 2004     

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.     

A variational approach to the determination of domain for Euler flows with constant vorticity

An alternative variational method for a steady, two-dimensional inviscid flow problem to determine the domain given the constant wall velocity and constant vorticity in the whole domain is presented. The uniqueness result is obtained for convex domains.     

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.     

On the variational principle

The variational principle states that if a differentiable functional F attains its minimum at some point $\text{u}\text{?}$, then $\text{F′(}\text{u}\text{?}\text{) = 0}$; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every ? > 0, there exists some point u_?, where $\text{∥F′(u}{}_{\mathrm{?}}\text{)∥}{}^1\text{? ?}$, i.e., its derivative can be made arbitrarily small. Applications are given to Plateau's problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infinite-dimensional manifolds, and to control theory.     

Parametric Ekeland's variational principle

A parametrized version of Ekeland's variational principle is proved, showing that under suitable conditions, the minimum point of the perturbed function can be chosen to depend continuously on a parameter. Applications of this result are given.     

Plane Hydrodynamic flows with steady vorticity: a reduction theorem

Summary Besides steady plane flows and unsteady plane flows of constant and steady vorticity, there are only two simple types of plane hydrodynamic flows with steady vorticity. These two types of unsteady flows have steady streamlines that are parallel straight lines or concentric circles.

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Zusammenfassung Neben stationären und nichtstationären ebenen Strömungen konstanter und stationärer Wirbelstärke, gibt es nur zwei einfache Typen von ebenen hydrodynamischen Strömungen mit stationärer Wirbelstärke. Diese zwei Typen von nichtstätionaren Strömungen haben stationäre Stromlinien, welche aus parallelen Geraden oder aus konzentrischen Kreisen bestehen.

     

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.     

The strong Ekeland variational principle

In this paper, we consider the strong Ekeland variational principle due to Georgiev [P.G. Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988) 1–21]. We discuss it for functions defined on Banach spaces and on compact metric spaces. We also prove the τ-distance version of it.     

Feynman's variational principle for a polaron in a magnetic field

Theoretical and Mathematical Physics -     

Electromagnetic finite elements based on a four-potential variational principle

We derive electromagnetic finite elements based on a variational principle that uses the electromagnetic four-potential as primary variable. This choice is used to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of electromagnetic/mechanical systems that involve superconductors. The main advantages of the four-potential as a basis for finite element formulation are: the number of degrees of freedom per node remains modest as the problem dimensionality increases, jump discontinuities on interfaces are naturally accomodated, and statics as well as dynamics may be treated without any a priori approximations. The new elements are tested on an axisymmetric problem under steady-state forcing conditions. The results are in excellent agreement with analytical solutions.     

Parametric Borwein-Preiss variational principle and applications

A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.

The tool for proving this parametric smooth variational principle is a useful lemma about continuous -minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

     

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 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 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.