Directional differentiability for the solution map of the bilateral parabolic obstacle problem
Abstract
We prove that the solution map of the classical bilateral parabolic obstacle problem is directionally differentiable when interpreted as a function between suitable Lebesgue and Sobolev spaces. To the best of our knowledge, this paper is the first to establish this kind of differentiability for a parabolic evolution variational inequality with two-sided fully distributed pointwise inequality constraints. Our analysis relies on known differentiability results for unilateral parabolic obstacle problems and a new theorem on the directional differentiability of solution operators of non-expansive parameterized fixed-point equations. The latter may also be of interest for other applications. Our results can be used, for example, to derive optimality conditions for optimal control problems governed by bilateral parabolic obstacle problems.
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BibTeX
@misc{blomeyer2026directional,
title = {{Directional differentiability for the solution map of the bilateral parabolic obstacle problem}},
author = {Simon Blomeyer and Constantin Christof},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12979},
eprint = {2609.12979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12979},
}
RIS
TY - UNPB
TI - Directional differentiability for the solution map of the bilateral parabolic obstacle problem
AU - Simon Blomeyer
AU - Constantin Christof
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12979
UR - https://arxiv.org/abs/2609.12979
AB - We prove that the solution map of the classical bilateral parabolic obstacle problem is directionally differentiable when interpreted as a function between suitable Lebesgue and Sobolev spaces. To the best of our knowledge, this paper is the first to establish this kind of differentiability for a parabolic evolution variational inequality with two-sided fully distributed pointwise inequality constraints. Our analysis relies on known differentiability results for unilateral parabolic obstacle problems and a new theorem on the directional differentiability of solution operators of non-expansive parameterized fixed-point equations. The latter may also be of interest for other applications. Our results can be used, for example, to derive optimality conditions for optimal control problems governed by bilateral parabolic obstacle problems.
ER -