Notes on the trace problem for separately convex functions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1617-1648.

We discuss the following question: for a function f of two or more variables which is convex in the directions of coordinate axes, what can its trace g(x)=f(x,x,...,x) 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 g, however, a characterization in the two-dimensional case is established.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016066
Classification : 26B25
Mots clés : Separately convex function, trace problem
Kurka, Ondřej 1 ; Pokorný, Dušan 1

1 Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic.
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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://www.numdam.org/articles/10.1051/cocv/2016066/

R.J. Aumann and S. Hart, Bi-convexity and bi-martingales. Israel J. Math. 54 (1986) 159–180. | DOI | MR | Zbl

S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L 1 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

S. Conti, D. Faraco, F. Maggi and S. Müller, Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines. Calc. Var. Partial Differ. Equ. 24 (2005) 479–493. | DOI | MR | Zbl

J. Duda and L. Zajíček, Semiconvex functions: representations as suprema of smooth functions and extensions. J. Convex Anal. 16 (2009) 239–260. | MR | Zbl

J. Duda and L. Zajíček, Smallness of singular sets of semiconvex functions in separable Banach spaces. J. Convex Anal. 20 (2013) 573–598. | MR | Zbl

J. Gorski, F. Pfeuffer and K. Klamroth, 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

J. Lee, P.F.X. Müller and S. Müller, 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

S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 1999 (1999), 1087–1095. | DOI | MR | Zbl

D. Pokorný and M. Rmoutil, 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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