About Regularity of Collections of Sets

The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular.     

Extremality and the global Markov property II: The global markov property for non-FKG maximal Gibbs measures

We give a condition on a Gibbs measure for an attractive Markov specification, which assures extremality and the global Markov property. As an example of application we consider the class of attractive Markov specifications defined on a compact configuration space over a two-dimensional lattice by the interaction Hamiltonians (assumed to have a finite set of periodic ground configurations) satisfying Peierl's condition. We prove that each extremal Gibbs measure for such a specification, at sufficiently low temperature, has the global Markov property.On leave of absence from the Institute of Theoretical Physics, University of Wrocaw, Poland.     

On the extremality of quasiconformal mappings and quasiconformal deformations

Given a family of quasiconformal deformations such that has a uniform bound , the solution of the Löwner-type differential equation

is an -quasiconformal mapping. An open question is to determine, for each fixed , whether the extremality of is equivalent to that of . The note gives this a negative approach in both directions.

     

Nonsmooth Calculus, Minimality, and Monotonicity of Convexificators

Noncompact convexificators, which provide upper convex and lower concave approximations for a continuous function, are defined. Various calculus rules, including extremality and mean-value properties, are presented. Regularity conditions are given for convexificators to be minimal. A characterization of quasiconvexity of a continuous function is obtained in terms of the quasimonotonicity of convexificators.     

Some characterizations of unique extremality

In this paper, it is shown that some necessary characteristic conditions for unique extremality obtained by Zhu and Chen are also sufficient and some sufficient ones by them actually imply that the uniquely extremal Beltrami differentials have a constant modulus. In addition, some local properties of uniquely extremal Beltrami differentials are given.     

Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

Let be the set of holomorphic functions on the unit disc with and Dirichlet integral not exceeding one, and let be the set of complex-valued harmonic functions on the unit disc with and Dirichlet integral not exceeding one. For a (semi)continuous function , define the nonlinear functional on or by . We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.

     

On criteria for extremality of Teichmüller mappings

Let be a Teichmüller self-mapping of the unit disk corresponding to a holomorphic quadratic differential . If satisfies the growth condition (as ), for any given 0$">, then is extremal, and for any given , there exists a subsequence of such that

is a Hamilton sequence. In addition, it is shown that there exists with bounded Bers norm such that the corresponding Teichmüller mapping is not extremal, which gives a negative answer to a conjecture by Huang in 1995.

     

Proper exhausters and coexhausters in nonsmooth analysis

There exist many tools to analyze nonsmooth functions. For convex and max-type functions, the notion of subdifferential is used, for quasidifferentiable functions – that of quasidifferential. By means of these tools, one is able to solve, e.g. the following problems: to get an approximation of the increment of a functional, to formulate conditions for an extremum, to find steepest descent and ascent directions and to construct numerical methods. For arbitrary directionally differentiable functions, these problems are solved by employing the notions of upper and lower exhausters and coexhausters, which are generalizations of such notions of nonsmooth analysis as sub- and superdifferentials, quasidifferentials and codifferentials. Exhausters allow one to construct homogeneous approximations of the increment of a functional while coexhausters provide nonhomogeneous approximations. It became possible to formulate conditions for an extremum in terms of exhausters and coexhausters. It turns out that conditions for a minimum are expressed by an upper exhauster, and conditions for a maximum are formulated via a lower one. This is why an upper exhauster is called a proper one for the minimization problem (and adjoint for the maximization problem) while a lower exhauster is called a proper one for the maximization problem (and adjoint for the minimization problem). The conditions obtained provide a simple geometric interpretation and allow one to find steepest descent and ascent directions. In this article, optimization problems are treated by means of proper exhausters and coexhausters.     

Truncations of extremal quasiconformal mappings and their applications

In this paper we give a necessary and sufficient condition to decide whether the Teichmüller equivalency class [α] of a truncation α induced by a uniquely extremal Beltrami differential is a Strebel point in T. We also obtain a necessary and sufficient condition of the unique extremality of α. Using the properties of truncations we provide a method to construct Hamilton sequences. We also get a sufficient condition for the extremality of f(z,t) to be equivalent to that of F(w,t). The corresponding results in the infinitesimal case are obtained, too.     

Nontrivial Critical Networks. Singularities of Lagrangians and a Criterion for Criticality

We single out the class of so-called quasiregular Lagrangians, which have singularities on the zero section of the cotangent bundle to the manifold on which extremal networks are considered. A criterion for a network to be extremal is proved for such Lagrangians: the Euler--Lagrange equations must be satisfied on each edge, and some matching conditions must be valid at the vertices.