Accepted Paper

Inserted: 31 jul 2026
Last Updated: 31 jul 2026

Journal: Mathematical Methods in the Applied Sciences
Volume: 47
Number: (15)
Pages: 11795–11809
Year: 2024
Doi: 10.1002/mma.9042

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 \[ \inf_{u\in f+W^{1, \phi}_0(\Omega)}\int_\Omega \phi(x,
\nabla u
) \, dx \] 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 $\BV$-type space involving generalized Orlicz growth and obtain the $\Gamma$-convergence of functionals with fixed boundary values and of functionals with fidelity terms.

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