On structural stability and numerical approximation of diffusion problems of p(x), p(u) and p[u]-laplacian kind

Study of the $p(x)$-laplacian kind problems has known an intense development in years 2000 and 2010; it was motivated, in particular, by modeling of electro- and thermo-rheological non-newtonian fluids (Ruzicka,...), as well as by novel techniques for their mathematical study (Zhikov school, Diening, Finnish school,...). Presently, there is an academic interest into $p(u)$-laplacian “auto-rheological” problems (Antontsev, Chipot,...).The numerical study of these problems is delicate, because of the functional setting that depends (stronger than it usually happens in numerical analysis) on the discretization parameter. The intrication is partly due to the Lavrentiev phenomenon uncovered by Zhikov in 1980ies and still actively studied (Alkhutov, Balci, Surnachev, Diening school,...).We will discuss some of the subtleties of these problems, in particular, the Young measure technique (S. Mueller, Hungerbuehler,...) for passage to the limit in a sequence of exact or approximate solutions of variable-exponent problems will be presented, as a natural alternative to the classical but tricky "Minty trick". We will also discuss the approximation of variable-exponent problems by Finite Volume schemes with gradient discretization "per diamond". The results we obtain are comparable to those of Balci, Ortner, Storn obtained in the Finite Element setting. The contents are taken from joint works with Mostafa Bendahmane (Bordeaux, France) and Stanislas Ouaro (Ouagadougou, Burkina-Faso) dating back to 2010 and from the recent work with ElHoussaine Quenjel (La Rochelle, France).