Minimizers of abstract generalized Orlicz‐bounded variation energy

• Published in: Mathematical methods in the applied sciences 2024-10, Vol.47 (15), p.11795-11809
• Main Authors: Eleuteri, Michela, Harjulehto, Petteri, Hästö, Peter
• Format: Article
• Language: English
• Subjects: generalized bounded variation; generalized Orlicz space; minimizer; Musielak–Orlicz space; Γ$$ \Gamma $$‐convergence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9042

A way to measure the lower growth rate of φ:Ω×[0,∞)→[0,∞)$$ \varphi :\Omega \times \left[0,\infty \right)\to \left[0,\infty \right) $$ is to require t↦φ(x,t)t−r$$ t\mapsto \varphi \left(x,t\right){t}^{-r} $$ to be increasing in (0,∞)$$ \left(0,\infty \right) $$. If this condition holds with r=1$$ r=1 $$, then infu∈f+W01,φ(Ω)∫Ωφ(x,|∇u|)dx$$ \underset{u\in f+{W}_0^{1,\varphi}\left(\Omega \right)}{\operatorname{inf}}{\int}_{\Omega}\varphi \left(x,|\nabla u|\right)\kern0.1em dx $$ with boundary values f∈W1,φ(Ω)$$ f\in {W}^{1,\varphi}\left(\Omega \right) $$ does not necessarily have a minimizer. However, if φ$$ \varphi $$ is replaced by φp$$ {\varphi}^p $$, then the growth condition holds with r=p>1$$ r=p>1 $$ and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence (up)$$ \left({u}_p\right) $$ of such minimizers converges when p→1+$$ p\to {1}^{+} $$ in a suitable BV$$ \mathrm{BV} $$‐type space involving generalized Orlicz growth and obtain the Γ$$ \Gamma $$‐convergence of functionals with fixed boundary values and of functionals with fidelity terms.