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A characterization for elliptic problems on fractal sets


Authors: Giovanni Molica Bisci and Vicenţiu D. Rădulescu
Journal: Proc. Amer. Math. Soc. 143 (2015), 2959-2968
MSC (2010): Primary 35J20; Secondary 28A80, 35J25, 35J60, 47J30, 49J52
DOI: https://doi.org/10.1090/S0002-9939-2015-12475-6
Published electronically: February 13, 2015
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Abstract: In this paper we prove a characterization theorem on the existence of one non-zero strong solution for elliptic equations defined on the Sierpiński gasket. More generally, the validity of our result can be checked studying elliptic equations defined on self-similar fractal domains whose spectral dimension $ \nu \in (0,2)$. Our theorem can be viewed as an elliptic version on fractal domains of a recent contribution obtained in a recent work of Ricceri for a two-point boundary value problem.


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Giovanni Molica Bisci
Affiliation: Patrimonio, Architettura e Urbanistica, Department, University of Reggio Calabria, 89124 - Reggio Calabria, Italy
Email: gmolica@unirc.it

Vicenţiu D. Rădulescu
Affiliation: Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Email: vicentiu.radulescu@math.cnrs.fr

DOI: https://doi.org/10.1090/S0002-9939-2015-12475-6
Keywords: Sierpi\'nski gasket, non-linear elliptic equation, Dirichlet form, weak Laplacian
Received by editor(s): December 27, 2013
Received by editor(s) in revised form: February 3, 2014, and February 9, 2014
Published electronically: February 13, 2015
Communicated by: Catherine Sulem
Article copyright: © Copyright 2015 American Mathematical Society