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Proceedings of the American Mathematical Society

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Serrin-type theorems for triangles


Authors: Ilaria Fragalà and Bozhidar Velichkov
Journal: Proc. Amer. Math. Soc. 147 (2019), 1615-1626
MSC (2010): Primary 35N25, 35J57, 49Q10.
DOI: https://doi.org/10.1090/proc/14352
Published electronically: January 8, 2019
MathSciNet review: 3910426
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Abstract: We investigate interior and exterior overdetermined boundary value problems on triangles, which corresponds to studying stationary triangles for variational functionals under volume or perimeter constraint. We prove that in certain cases the only triangle supporting solution is the equilateral one. In some other cases, we obtain that all triangles support solutions, thus extending (through a simpler proof) what was recently shown by Hans Christianson [Proc. Amer. Math. Soc. 145 (2017), no. 12, 5247–5255].


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Additional Information

Ilaria Fragalà
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy
MR Author ID: 629098
Email: ilaria.fragala@polimi.it

Bozhidar Velichkov
Affiliation: Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38401 Saint-Martin-d’Hères, France
MR Author ID: 1000813
Email: bozhidar.velichkov@univ-grenoble-alpes.fr

Keywords: Overdetermined problems, triangles, torsional rigidity, principal frequency, $p$-capacity.
Received by editor(s): May 14, 2018
Received by editor(s) in revised form: August 9, 2018
Published electronically: January 8, 2019
Additional Notes: The authors have been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The second author has been partially supported by ANR through the projects GeoSpec (LabEx PERSYVAL-Lab, ANR-11- LABX-0025-01) and CoMeDiC (ANR-15-CE40-0006).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2019 American Mathematical Society