Optimizing the Principal Coefficient of Elliptic Equations using $L^p$-regularity, $p < \infty$

Mathematics > Optimization and Control [Submitted on 16 Jun 2026] Title:Optimizing the Principal Coefficient of Elliptic Equations using $L^p$-regularity, $p < \infty$ View PDF HTML (experimental)Abstract:We study coefficient identification problems for elliptic partial differential equations with total variation regularization and control constraints. Existing related literature relies on continuity and differentiability properties of the control-to-state operator with respect to the $L^\infty$-norm. While this is sufficient for deriving optimality conditions, it is not well-suited for numerical algorithms, as it neglects the spatial extent of perturbations and leads to a qualitative discrepancy compared to $L^q$-norms with $q < \infty$. In this work, we address this gap by exploiting $W^{1,s}$-regularity results to establish differentiability properties of the control-to-state operator with respect to $L^q$-norms for finite $q$. Based on this framework, we derive first- and second-order differentiability results for the reduced objective functional and establish first-order optimality conditions involving a restricted subdifferential characterization of the total variation seminorm and corresponding regularity of the associated multipliers. Building on this, we analyze a nonsmooth trust-region method based on an $L^r$-trust region for $r > 0.5 q$ and prove its convergence to first-order stationary points. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.