Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints Article Swipe

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Harbir Antil , Rafael Arndt , Carlos N. Rautenberg , Deepanshu Verma ·

YOU? · · 2021 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2106.12680

In this paper, we consider non-diffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a non-standard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solutions to this pre-dual problem under some assumptions. We conclude the paper by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples confirm our theoretical findings.

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Uniqueness Mathematics Sobolev space Square-integrable function Bounded function Constraint (computer-aided design) Space (punctuation) Applied mathematics Function space Integrable system Borel measure Variational inequality Mathematical analysis Boundary (topology) Function (biology) Probability measure Computer science Biology Evolutionary biology Operating system Geometry

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2106.12680
PDF: https://arxiv.org/pdf/2106.12680
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DOI: https://doi.org/10.48550/arxiv.2106.12680

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Title: Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints

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Language: en

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Publication year: 2021

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Publication date: 2021-06-23

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Authors: Harbir Antil, Rafael Arndt, Carlos N. Rautenberg, Deepanshu Verma

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Landing page: https://arxiv.org/abs/2106.12680

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Concepts: Uniqueness, Mathematics, Sobolev space, Square-integrable function, Bounded function, Constraint (computer-aided design), Space (punctuation), Applied mathematics, Function space, Integrable system, Borel measure, Variational inequality, Mathematical analysis, Boundary (topology), Function (biology), Probability measure, Computer science, Biology, Evolutionary biology, Operating system, Geometry

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