Minimizers of abstract generalized Orlicz--bounded variation energy

Abstract

A way to measure the lower growth rate of $\phi :\Omega \times [0,\infty )\to [0,\infty )$ is to require $t\mapsto \phi (x,t)t^{-r}$ to be increasing in $(0,\infty )$. If this condition holds with $r=1$, then \[ \inf_{u\in f+W^{1, φ}_0(Ω)}\int_Ωφ(x, |\nabla u|) \, dx \] with boundary values $f\in W^{1,\phi }(\Omega )$ does not necessary 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 convergences when $p\to 1^{+}$ in a suitable $\mathrm{BV}$-type space involving generalized Orlicz growth and obtain the $\Gamma$-convergence of functionals with fixed boundary values and of functionals with fidelity terms. %We complement our results by showing that some previous papers by some of the authors are included in our analysis.