Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19156 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908339949010944 |
|---|---|
| author | Azevedo, Davide Santos, Lisa |
| author_facet | Azevedo, Davide Santos, Lisa |
| contents | We prove existence and uniqueness of solution of an evolutionary vector-valued variational inequality defined in the convex set of vector valued functions $\boldsymbol v$ subject to the constraint $|\boldsymbol v|\le1$. We show that we can write the variational inequality as a system of equations on the unknowns $(λ,\boldsymbol u)$, where $λ$ is a (unique) Lagrange multiplier belonging to $L^p$ and $\boldsymbol u$ solves the variational inequality.
Given data $(\boldsymbol f_n,\boldsymbol u_{n0})$ converging to $(\boldsymbol f,\boldsymbol u_0)$ in $\boldsymbol L^\infty(Q_T)\times H^1_0(Ω)$, we prove the convergence of the solutions $(λ_n,\boldsymbol u_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\rightarrow \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An evolutionary vector-valued variational inequality and Lagrange multiplier Azevedo, Davide Santos, Lisa Analysis of PDEs We prove existence and uniqueness of solution of an evolutionary vector-valued variational inequality defined in the convex set of vector valued functions $\boldsymbol v$ subject to the constraint $|\boldsymbol v|\le1$. We show that we can write the variational inequality as a system of equations on the unknowns $(λ,\boldsymbol u)$, where $λ$ is a (unique) Lagrange multiplier belonging to $L^p$ and $\boldsymbol u$ solves the variational inequality. Given data $(\boldsymbol f_n,\boldsymbol u_{n0})$ converging to $(\boldsymbol f,\boldsymbol u_0)$ in $\boldsymbol L^\infty(Q_T)\times H^1_0(Ω)$, we prove the convergence of the solutions $(λ_n,\boldsymbol u_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\rightarrow \infty$. |
| title | An evolutionary vector-valued variational inequality and Lagrange multiplier |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.19156 |