On semiconvexity properties of rotationally invariant functions in two dimensions

Miroslav Šilhavý

Displaying similar documents to “On semiconvexity properties of rotationally invariant functions in two dimensions”

Rank 1 convex hulls of isotropic functions in dimension 2 by 2

Miroslav Šilhavý (2001)

Mathematica Bohemica

Similarity:

Let $f$ be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set $M^{2 \times 2}$ of all $2$ by $2$ matrices. Based on conditions for the rank 1 convexity of $f$ in terms of signed invariants of $𝔸$ (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given. ...

A note on equality of functional envelopes

Martin Kružík (2003)

Mathematica Bohemica

Similarity:

We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in $ℝ^{m \times n}$ , $min (m, n) \leq 2$ , then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

A complete characterization of invariant jointly rank- convex quadratic forms and applications to composite materials

Vincenzo Nesi, Enrico Rogora (2007)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank- convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank- convex forms arises. In the present paper, we define the concept of extremal -forms and characterize them in the rotationally invariant...

Critique of "Two-dimensional examples of rank-one convex functions that are not quasiconvex" by M. K. Benaouda and J. J. Telega

Patrizio Neff (2005)

Annales Polonici Mathematici

Similarity:

It is noted that the examples provided in the paper "Two-dimensional examples of rank-one convex functions that are not quasiconvex" by M. K. Benaouda and J. J. Telega, Ann. Polon. Math. 73 (2000), 291-295, contain unrecoverable errors.

Generalized characterization of the convex envelope of a function

Fethi Kadhi (2002)

RAIRO - Operations Research - Recherche Opérationnelle

Similarity:

We investigate the minima of functionals of the form $\int_{[a, b]} g (\dot{u} (s)) d s$ where $g$ is strictly convex. The admissible functions $u : [a, b] \to ℝ$ are not necessarily convex and satisfy $u \leq f$ on $[a, b]$ , $u (a) = f (a)$ , $u (b) = f (b)$ , $f$ is a fixed function on $[a, b]$ . We show that the minimum is attained by $\bar{f}$ , the convex envelope of $f$ .