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Regularity for optimal compliance problems with length penalization

Abstract : We investigate the regularity and topological structure of a set minimizing the p-compliance functional with a length penalization. The key feature of our work is that we study the regularity of minimizers for some free boundary type problem with a high codimensional free boundary set. We prove that in any spatial dimension N that is greater than or equal to 2 and for every real number p that is strictly greater than N-1, if S is a minimizer of the p-compliance functional with a length penalization, then S cannot contain closed loops (i.e., homeomorphic images of the unit circle S^{1}), S is C^{1,alpha} regular at almost every point (with respect to the one-dimensional Hausdorff measure) that is in a given open bounded set O, and S cannot contain quadruple points in O, namely, there is no ball centered on S and contained in O such that S in this ball is a union of four distinct C^{1} arcs, each of which meets exactly one of the other three at an angle of 180 degrees, and each of the other two at an angle of 90 degrees. We also prove that in dimension 2 and for every p that is strictly greater than 1, if S is a minimizer of the p-compliance functional with a length penalization containing at least two points, then S is Ahlfors regular up to the boundary for a Lipschitz domain. Finally, we provide a proof of the importance of the connectedness assumption in the statement of the optimal p-compliance problem with length penalization and in the statement of the constrained form of this problem for the existence of solutions under the sharp assumptions. The results of this dissertation generalize some of the results obtained in [CLLS], but also contain better results in dimension 2 and for the special case p = 2 as well. Also, in some sense and in dimension 2, the result of this dissertation can be considered as establishing a link between the result in [CLLS] and the result in [Sle].
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Submitted on : Friday, July 1, 2022 - 12:22:41 PM
Last modification on : Friday, August 5, 2022 - 12:02:00 PM

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Bohdan Bulanyi. Regularity for optimal compliance problems with length penalization. Optimization and Control [math.OC]. Université Paris Cité, 2021. English. ⟨NNT : 2021UNIP7112⟩. ⟨tel-03711331⟩

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