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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The limit as $ p\rightarrow\infty$ in free boundary problems with fractional $ p$-Laplacians


Authors: João Vítor da Silva and Julio D. Rossi
Journal: Trans. Amer. Math. Soc. 371 (2019), 2739-2769
MSC (2010): Primary 35J60, 35B65
DOI: https://doi.org/10.1090/tran/7559
Published electronically: October 23, 2018
MathSciNet review: 3896096
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Abstract: We study the $ p$-fractional optimal design problem under volume constraint taking special care of the case when $ p$ is large, obtaining in the limit a free boundary problem modeled by the Hölder infinity Laplacian operator. A necessary and sufficient condition is imposed in order to obtain the uniqueness of solutions to the limiting problem, and, under this condition, we find precisely the optimal configuration for the limit problem. We also prove the sharp regularity (locally $ C^{0, s}$) for any limiting solution. Finally, we establish some geometric properties for solutions and their free boundaries.


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João Vítor da Silva
Affiliation: FCEyN, Department of Mathematics, Universidad de Buenos Aires, Ciudad Universitaria-Pabellón I-(C1428EGA), Buenos Aires, Argentina
Email: jdasilva@dm.uba.ar

Julio D. Rossi
Affiliation: FCEyN, Department of Mathematics, Universidad de Buenos Aires, Ciudad Universitaria-Pabellón I-(C1428EGA), Buenos Aires, Argentina
Email: jrossi@dm.uba.ar

DOI: https://doi.org/10.1090/tran/7559
Keywords: Optimal design problems, fractional diffusion, sharp regularity, H\"older Infinity Laplacian
Received by editor(s): December 6, 2016
Received by editor(s) in revised form: January 7, 2018
Published electronically: October 23, 2018
Additional Notes: This work has been partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina) under grant PIP GI No 11220150100036CO
Article copyright: © Copyright 2018 American Mathematical Society