Inequalities involving \(\Gamma\)-functionals and semi complete lattice homomorphisms
The paper under review deals with the generalization of some inequalities involving \(\Gamma\)-limits and lattice homomorphisms. Some particular cases were already considered by \textit{E. De Giorgi} and \textit{T. Franzoni} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58, 842-850 (1975; Zbl 0339.49005)] in the abstract framework of \(\Gamma\)- convergence theory, and by the reviewer and \textit{G. Dal Maso} [J. Optimization Theory Appl. 38, 385-422 (1982; Zbl 0471.49012)] in the applications of \(\Gamma\)-limits to optimal control problems. The main result of the paper is the following. Let \(\phi\) be a ``joint semi complete lattice homomorphism'' and let \(\alpha_{ij}\), \(\gamma_ i\) be the numbers \(+1\) or -1 \((i=1,...,n\) and \(j=1,...,k)\). If for every \(i\leq n\), \[ \sum^{k}_{j=1}\alpha_{ij}\leq \gamma_ i-k+1, \] then, given \(f_ 1,...,f_ k\) from \(X_ 1\times...\times X_ n\) into a complete lattice L, \[ (*)\quad \phi (\Gamma (F_ 1^{\alpha_{11}},...,F_ n^{\alpha_{n1}})f_ 1,...,\Gamma (F_ 1^{\alpha_{1k}},...,F_ n^{\alpha_{nk}})f_ k)\leq \Gamma (F_ 1^{\gamma_ 1},...,F_ n^{\gamma_ n})\phi (f_ 1,...,f_ k). \] On the other hand, if \(\phi\) is a ``meet semi complete lattice homomorphism'' and for very \(i\leq n\), \[ \sum^{k}_{j=1}\alpha_{ij}\geq \gamma_ i+k-1, \] then the opposite inequality holds in (*).
scientific article; zbMATH DE number 4030335
scientific article