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Structure of approximate solutions of variational problems with extended-valued convex integrands

Alexander J. Zaslavski — 2009

ESAIM: Control, Optimisation and Calculus of Variations

In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand f : R n × R n R 1 { } , where R n is the n -dimensional euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

On a variational problem arising in crystallography

Alexander J. Zaslavski — 2007

ESAIM: Control, Optimisation and Calculus of Variations

We study a variational problem which was introduced by Hannon, Marcus and Mizel [ (2003) 145–149] to describe step-terraces on surfaces of so-called “unorthodox” crystals. We show that there is no nondegenerate intervals on which the absolute value of a minimizer is π / 2 identically.

Structure of approximate solutions of variational problems with extended-valued convex integrands

Alexander J. Zaslavski — 2008

ESAIM: Control, Optimisation and Calculus of Variations

In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand : × { } , where is the -dimensional Euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

Anisotropic functions : a genericity result with crystallographic implications

Victor J. MizelAlexander J. Zaslavski — 2004

ESAIM: Control, Optimisation and Calculus of Variations

In the 1950’s and 1960’s surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was anisotropic. By examining step/terrace (vicinal) surface defects it was discovered through lengthy and tedious experiments...

Two generic results in fixed point theory

Simeon ReichAlexander J. Zaslavski — 2007

Banach Center Publications

We give two examples of the generic approach to fixed point theory. The first example is concerned with the asymptotic behavior of infinite products of nonexpansive mappings in Banach spaces and the second with the existence and stability of fixed points of continuous mappings in finite-dimensional Euclidean spaces.

Anisotropic functions: a genericity result with crystallographic implications

Victor J. MizelAlexander J. Zaslavski — 2010

ESAIM: Control, Optimisation and Calculus of Variations

In the 1950's and 1960's surface physicists/metallurgists such as Herring and Mullins applied ingenious thermodynamic arguments to explain a number of experimentally observed surface phenomena in crystals. These insights permitted the successful engineering of a large number of alloys, where the major mathematical novelty was that the surface response to external stress was . By examining step/terrace () surface defects it was discovered through lengthy and tedious experiments that the stored...

Best approximations and porous sets

Simeon ReichAlexander J. Zaslavski — 2003

Commentationes Mathematicae Universitatis Carolinae

Let D be a nonempty compact subset of a Banach space X and denote by S ( X ) the family of all nonempty bounded closed convex subsets of X . We endow S ( X ) with the Hausdorff metric and show that there exists a set S ( X ) such that its complement S ( X ) is σ -porous and such that for each A and each x ˜ D , the set of solutions of the best approximation problem x ˜ - z min , z A , is nonempty and compact, and each minimizing sequence has a convergent subsequence.

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