Existence of Minimizers for a Variational Problem in Two-Dimensional Nonlinear Magnetoelasticity

1998 ◽  
Vol 144 (2) ◽  
pp. 107-120 ◽  
Author(s):  
Antonio DeSimone ◽  
Georg Dolzmann
Keyword(s):  
Variational Problem ◽  
Two Dimensional ◽  
Existence Of Minimizers ◽  
Nonlinear Magnetoelasticity

scholarly journals Relaxation of some multi-well problems

2001 ◽  
Vol 131 (2) ◽  
pp. 279-320 ◽  
Author(s):  
Kaushik Bhattacharya ◽  
Georg Dolzmann
Keyword(s):  
Thin Films ◽  
Phase Transitions ◽  
Young Measure ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Convex Hulls ◽  
Quasiconvex Functions ◽  
Two Dimensional ◽  
Existence Of Minimizers

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫Ω W (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U1 ∪ … ∪SO(2)Uk or K = O(2)U1 ∪ … ∪ O(2)Uk for U1, … , Uk ∈ M2×2 with det Ui = δ in three dimensions when the matrices Ui are essentially two-dimensional and also for K = SO(3)Û1 ∪ … ∪ SO(3)Ûk for U1, … , Uk ∈ M3×3 with , which arises in the study of thin films. Here, Ûi denotes the (3×2) matrix formed with the first two columns of Ui. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.

scholarly journals Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions

2014 ◽  
Vol 24 (14) ◽  
pp. 2929-2942 ◽  
Author(s):  
Sergio Conti ◽  
Georg Dolzmann
Keyword(s):  
Phase Transformation ◽  
Energy Density ◽  
Variational Problem ◽  
Tetragonal Phase ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Potential Wells ◽  
Quasiconvex Envelope ◽  
Positive Determinant

We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant.

scholarly journals Steady vortex flows obtained from a constrained variational problem

2002 ◽  
Vol 30 (5) ◽  
pp. 283-300 ◽  
Author(s):  
B. Emamizadeh ◽  
M. H. Mehrabi
Keyword(s):  
Kinetic Energy ◽  
Variational Problem ◽  
Linear Constraint ◽  
Vortex Flows ◽  
Two Dimensional

We prove the existence of steady two-dimensional ideal vortex flows occupying the first quadrant and containing a bounded vortex; this is done by solving a constrained variational problem. Kinetic energy is maximized subject to the vorticity, being a rearrangement of a prescribed function and subject to a linear constraint.

scholarly journals A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

2015 ◽  
Vol 25 (14) ◽  
pp. 2749-2793 ◽  
Author(s):  
M. Hassan Farshbaf-Shaker ◽  
Christian Heinemann
Keyword(s):  
Phase Field ◽  
A Priori Estimates ◽  
Control Variable ◽  
External Boundary ◽  
Two Dimensional ◽  
Cost Functional ◽  
Viscoelastic Media ◽  
Existence Of Minimizers ◽  
Damage Processes ◽  
Cost Functionals

In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

scholarly journals SYMMETRY BREAKING RESULTS FOR PROBLEMS WITH EXPONENTIAL GROWTH IN THE UNIT DISK

2006 ◽  
Vol 08 (06) ◽  
pp. 823-839 ◽  
Author(s):  
SIMONE SECCHI ◽  
ENRICO SERRA
Keyword(s):  
Symmetry Breaking ◽  
Variational Problem ◽  
Unit Disk ◽  
Exponential Growth ◽  
Asymptotic Properties ◽  
Two Dimensional ◽  
Dimensional Variational Problem

We investigate some asymptotic properties of extrema uα to the two-dimensional variational problem [Formula: see text] as α → +∞. Here B is the unit disk of ℝ2 and 0 < γ ≤ 4π is a given parameter. We prove that in a certain range of γ′s, the maximizers are not radial for α large.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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