2023
DOI: 10.5802/jep.250
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On the polygonal Faber-Krahn inequality

Beniamin Bogosel,
Dorin Bucur

Abstract: It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with n sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each n ⩾ 5 the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon … Show more

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Cited by 7 publications
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“…They provide increased accuracy for numerical approximations, as evidenced for instance in [20,21,27,28], where both strong and weak forms of shape derivatives have been used and methodically compared for various numerical applications. Moreover, the weak form can be employed to study optimality conditions for shape optimization problems involving domains with low smoothness such as polygons [29,30]. In particular, shape optimization problems on polygons, polyhedrons and in general domains with corners and edges often appear in applications.…”
Section: Introductionmentioning
confidence: 99%
“…They provide increased accuracy for numerical approximations, as evidenced for instance in [20,21,27,28], where both strong and weak forms of shape derivatives have been used and methodically compared for various numerical applications. Moreover, the weak form can be employed to study optimality conditions for shape optimization problems involving domains with low smoothness such as polygons [29,30]. In particular, shape optimization problems on polygons, polyhedrons and in general domains with corners and edges often appear in applications.…”
Section: Introductionmentioning
confidence: 99%