Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth
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| Formato: | Preprint |
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2025
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| _version_ | 1866910934957555712 |
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| author | Boccardo, Lucio Palladino, Maria Antonietta Picerni, Marco |
| author_facet | Boccardo, Lucio Palladino, Maria Antonietta Picerni, Marco |
| contents | We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. More precisely, for a fixed obstacle $ψ\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$, we consider $u\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ satisfying $u\geqψ$ a.e. and $$ \langle A(u),v-u\rangle+\int_ΩH(x,u,\nabla u)(v-u)\geq 0$$ for all $v\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ with $v\geqψ$. Here, $A$ is a Leray-Lions type operator, mapping $W_0^{1,p}(Ω)$ into its dual $W^{-1, p'}(Ω)$, while $H(x, u, D u)$ grows like $|D u|^p$. Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth Boccardo, Lucio Palladino, Maria Antonietta Picerni, Marco Analysis of PDEs 35J85 We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. More precisely, for a fixed obstacle $ψ\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$, we consider $u\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ satisfying $u\geqψ$ a.e. and $$ \langle A(u),v-u\rangle+\int_ΩH(x,u,\nabla u)(v-u)\geq 0$$ for all $v\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ with $v\geqψ$. Here, $A$ is a Leray-Lions type operator, mapping $W_0^{1,p}(Ω)$ into its dual $W^{-1, p'}(Ω)$, while $H(x, u, D u)$ grows like $|D u|^p$. Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems. |
| title | Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth |
| topic | Analysis of PDEs 35J85 |
| url | https://arxiv.org/abs/2505.05899 |