Bellido, Jose C

Evgrafov, Anton

English

    Aalborg UniversitetNonlocal Basis Pursuit and the Cost of Antisymmetric FluxesEvgrafov, Anton; Bellido, Jose CPublication date:2021Link to publication from Aalborg UniversityCitation for published version (APA):Evgrafov, A., & Bellido, J. C. (2021). Nonlocal Basis Pursuit and the Cost of Antisymmetric Fluxes. Abstract fromALOP Workshop on Nonlocal Models, Trier, Germany.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: August 24, 2021Nonlocal Basis Pursuit and the Cost of Antisymmetric FluxesA. Evgrafov∗1 and J.C. Bellido21 Department of Mathematical Sciences, Aalborg University, DK–9210 Aalborg, Denmark; anev@math.aau.dk2 Department of Mathematics, University of Castilla-La Mancha, 13071–Ciudad Real, Spain;JoseCarlos.Bellido@uclm.esWe consider a nonlocal version of the classical optimal design problem of distributing a limited amountof conductive material in a given design domain [1, 2]. In the local case, such problems belong to theclass of optimal control problems in the coefficients of PDEs.Of particular interest for these problems is the study of the limiting case, which appears when the amountof available material is driven to zero. Such a limiting process is of both theoretical and practical interestand continues to be a subject of active study [3, 4]. For example in the context of linear elasticity theresulting limiting problem corresponds to the celebrated case of Michell trusses [3, 4]. The limitingproblem can be formulated in terms of fluxes only and is convex. Unfortunately, it admits solutions inthe space of Radon measures, and as such is not easy to solve numerically.With this in mind we focus on identifying the vanishing material limit for the corresponding nonlocaloptimal design problem. The resulting limiting nonlocal optimization problem is convex, can be ex-pressed in terms of nonlocal antisymmetric two-point fluxes, and admits solutions in Lebesgue spaceswith mixed exponents. When the nonlocal interaction horizon is driven to zero, the “vanishing mate-rial limit” nonlocal problems provide a one-sided estimate for the corresponding local measure-valuedoptimal design problem.The surprising fact is that in order to transform the one-sided estimate into a true limiting process it issufficient to disregard the antisymmetry requirement on the two-point fluxes. This result relies on dualityand requires generalizing some of the nonlocal estimates established in [5] to the case of mixed Lebesgueexponents.References[1] Allaire, Grégoire. Shape optimization by the homogenization method (Vol. 146). Springer Science & Business Media,(2012).[2] Andrés, Fuensanta, and Julio Muñoz. Nonlocal optimal design: a new perspective about the approximation of solutionsin optimal design. Journal of Mathematical Analysis and Applications 429, no. 1 (2015): 288-310.[3] Olbermann, Heiner. Michell trusses in two dimensions as a Γ-limit of optimal design problems in linear elasticity. Cal-culus of Variations and Partial Differential Equations 56, no. 6 (2017): 1-40.[4] Babadjian, Jean-Francois, Iurlano, Flaviana, and Rindler, Filip. Shape optimization of light structures and the vanishingmass conjecture. arXiv preprint arXiv:2102.09911 (2021).[5] Bourgain, Jean and Brezis, Haı̈m and Mironescu, Petru. Another look at Sobolev spaces. In: Optimal Control and PartialDifferential Equations: A volume in honor of A. Bensoussan’s 60th birthday, eds: Menaldi, J.L. and Rofman, E. andSulem, A., 2001.1

Nonlocal Basis Pursuit and the Cost of Antisymmetric Fluxes

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Nonlocal Basis Pursuit and the Cost of Antisymmetric Fluxes

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