PreprintCC BY 4.0arXiv
Existence and uniqueness of the dual solution for obstacle problems in the space BV
Abstract
We consider the variational problem of minimizing the multiple integral ∫Ωf(∇ u) dx with a convex density f being of linear growth among functions u: Ω→ ℝ satisfying the Dirichlet boundary condition u = u0 on ∂Ω and for which u ≥ Ψ holds throughout the domain Ω. We prove that the dual problem has a unique solution. This result is extended to the case of interior obstacle problems, where the validity of u ≥ Ψ is only required on a subset E of Ω. Moreover, we discuss the vectorial setting, i.e. the case of functions u: Ω→ ℝN, N > 1, combined with the additional condition u(Ω) ⊂ A for a closed convex set A ⊂ ℝN, again showing the existence and the uniqueness of the associated dual solution.
Related
- Existence and uniqueness of solutions to Hardy-Henon boundary-value problems
- Directional differentiability for the solution map of the bilateral parabolic obstacle problem
- The Bourgain, Brezis, Mironescu localization for deterministic obstacle problems
- Global Weighted Gradient Estimates for Double Obstacle Problems with Orlicz Growth
- Triple Junctions in Multi-Phase Obstacle type Problems
BibTeX
@misc{bildhauer2026existence,
title = {{Existence and uniqueness of the dual solution for obstacle problems in the space BV}},
author = {Michael Bildhauer and Martin Fuchs},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41853},
eprint = {2610.41853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41853},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Existence and uniqueness of the dual solution for obstacle problems in the space BV
AU - Michael Bildhauer
AU - Martin Fuchs
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41853
UR - https://arxiv.org/abs/2610.41853
AB - We consider the variational problem of minimizing the multiple integral ∫_Ωf(∇ u) dx with a convex density f being of linear growth among functions u: Ω→ ℝ satisfying the Dirichlet boundary condition u = u_0 on ∂Ω and for which u ≥ Ψ holds throughout the domain Ω. We prove that the dual problem has a unique solution. This result is extended to the case of interior obstacle problems, where the validity of u ≥ Ψ is only required on a subset E of Ω. Moreover, we discuss the vectorial setting, i.e. the case of functions u: Ω→ ℝ^N, N > 1, combined with the additional condition u(Ω) ⊂ A for a closed convex set A ⊂ ℝ^N, again showing the existence and the uniqueness of the associated dual solution.
ER -