Minimizers of abstract generalized Orlicz-bounded variation energy

Abstract

A way to measure the lower growth rate of phi : Omega x [0, infinity) -> [0, infinity) is to require t (sic) phi (x, t)t(-r) to be increasing in (0, infinity). If this condition holds with r = 1, theninf (u is an element of f+W1,phi (Omega)) integral(Omega) phi(x, vertical bar del u vertical bar) dxwith boundary values f is an element of W-1,W-phi (Omega) does not necessarily have a minimizer. However, if phi is replaced by phi(p), then the growth condition holds with r = p > 1 and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence (u(p)) of such minimizers converges when p -> 1(+) in a suitable BV-type space involving generalized Orlicz growth and obtain the Gamma-convergence of functionals with fixed boundary values and of functionals with fidelity terms.