We discuss the following question: for a function of two or more variables which is convex in the directions of coordinate axes, what can its trace look like? In the two-dimensional case, we provide some necessary and sufficient conditions, as well as some examples illustrating that our approach does not seem to be appropriate for finding a characterization in full generality. For a concave function , however, a characterization in the two-dimensional case is established.
Accepté le :
DOI : 10.1051/cocv/2016066
Mots clés : Separately convex function, trace problem
@article{COCV_2017__23_4_1617_0,
author = {Kurka, Ond\v{r}ej and Pokorn\'y, Du\v{s}an},
title = {Notes on the trace problem for separately convex functions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1617--1648},
publisher = {EDP-Sciences},
volume = {23},
number = {4},
year = {2017},
doi = {10.1051/cocv/2016066},
zbl = {1390.26023},
mrnumber = {3716935},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2016066/}
}
TY - JOUR AU - Kurka, Ondřej AU - Pokorný, Dušan TI - Notes on the trace problem for separately convex functions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1617 EP - 1648 VL - 23 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016066/ DO - 10.1051/cocv/2016066 LA - en ID - COCV_2017__23_4_1617_0 ER -
%0 Journal Article %A Kurka, Ondřej %A Pokorný, Dušan %T Notes on the trace problem for separately convex functions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1617-1648 %V 23 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016066/ %R 10.1051/cocv/2016066 %G en %F COCV_2017__23_4_1617_0
Kurka, Ondřej; Pokorný, Dušan. Notes on the trace problem for separately convex functions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1617-1648. doi : 10.1051/cocv/2016066. http://archive.numdam.org/articles/10.1051/cocv/2016066/
and , Bi-convexity and bi-martingales. Israel J. Math. 54 (1986) 159–180. | DOI | MR | Zbl
, and , A new approach to counterexamples to estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Ration. Mech. Anal. 175 (2005) 287–300. | DOI | MR | Zbl
, , and , Rank-one convex functions on symmetric matrices and laminates on rank-three lines. Calc. Var. Partial Differ. Equ. 24 (2005) 479–493. | DOI | MR | Zbl
and , Semiconvex functions: representations as suprema of smooth functions and extensions. J. Convex Anal. 16 (2009) 239–260. | MR | Zbl
and , Smallness of singular sets of semiconvex functions in separable Banach spaces. J. Convex Anal. 20 (2013) 573–598. | MR | Zbl
, and , Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Method Oper. Res. 66 (2007) 373–407. | DOI | MR | Zbl
B. Kirchheim, S. Müller and V. Šverák, Studying nonlinear pde by geometry in matrix space. Geometric analysis and nonlinear partial differential equations. Springer, Berlin (2003). | MR | Zbl
, and , Compensated compactness, separately convex functions and interpolatory estimates between Riesz transforms and Haar projections. Commun. Partial Differ. Equ. 36 (2011) 547–601. | DOI | MR | Zbl
, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 1999 (1999), 1087–1095. | DOI | MR | Zbl
and , On removable sets for convex functions. J. Math. Anal. Appl. 415 (2014) 803–815. | DOI | MR | Zbl
L. Tartar, Some remarks on separately convex functions. Microstructure and phase transition. Vol. 54 of IMA Vol. Math. Appl. Springer, New York (1993). | MR | Zbl
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