Minimizers of abstract generalized Orlicz--bounded variation energy

A way to measure the lower growth rate of $\varphi:\Omega\times [0,\infty) \to [0,\infty)$istorequire$t \mapsto \varphi(x,t)t^{-r}$ to be increasing in $(0,\mathrm{\infty })$. If this condition holds with $r=1$, then $$\underset{u\in f+W_0^{1,\phi }(\mathrm{\Omega })}{\inf }{\int }_{\mathrm{\Omega }}\phi (x,\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })dx$$ with boundary values $f\in W^{1,\phi }(\mathrm{\Omega })$ 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 $\mathrm{B}\mathrm{V}$-type space involving generalized Orlicz growth and obtain the $\mathrm{\Gamma }$-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.