The simultaneous asymmetric perturbation method for overdetermined free boundary problems
In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work w...
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| Veröffentlicht in: | Nonlinear analysis Jg. 215; S. 112685 |
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| Sprache: | Englisch |
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01.02.2022
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| Abstract | In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin’s overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed. |
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| AbstractList | In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin’s overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed. |
| ArticleNumber | 112685 |
| Author | Cavallina, Lorenzo |
| Author_xml | – sequence: 1 givenname: Lorenzo orcidid: 0000-0003-4157-3078 surname: Cavallina fullname: Cavallina, Lorenzo email: cavallina.lorenzo.e6@tohoku.ac.jp organization: Mathematical Institute, Tohoku University, Aoba Sendai, 980-8578, Japan |
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| Cites_doi | 10.1016/j.anihpc.2020.05.001 10.1051/cocv/2019048 10.4171/178 10.1007/s00205-021-01620-z 10.2307/2372973 10.1007/s00028-002-8093-y 10.1016/0362-546X(95)00192-X 10.1007/BF00250468 |
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| References | L. Cavallina, T. Yachimura, Symmetry breaking solutions for a two-phase overdetermined problem of Serrin-type, in: Current Trends in Analysis, its Applications and Computation, Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019 (Editors: Paula Cerejeiras, Michael Reissig, Irene Sabadini, Joachim Toft) Birkhäuser , 2022, In press Foote (b10) 1984; 92 Kinderlehrer, Nirenberg (b16) 1977; 4 Cavallina, Yachimura (b7) 2020 Cavallina (b6) 2020 Novruzi, Pierre (b17) 2002; 2 Serrin (b19) 1971; 43 Henrot, Shahgholian (b14) 1997; 28 Delfour, Zolésio (b9) 2001 . Cavallina (b5) 2018 Henrot, Pierre (b13) 2018 Ambrosetti, Prodi (b3) 1983 Sacksteder (b18) 1960; 82 Henrot, Onodera (b12) 2021 Acker (b1) 1989; 393 Alt, Caffarelli (b2) 1981; 325 Beurling (b4) 1957 Kamburov, Sciaraffia (b15) 2021; 38 Gilbarg, Trudinger (b11) 1983 Henrot (10.1016/j.na.2021.112685_b13) 2018 10.1016/j.na.2021.112685_b8 Cavallina (10.1016/j.na.2021.112685_b5) 2018 Henrot (10.1016/j.na.2021.112685_b14) 1997; 28 Acker (10.1016/j.na.2021.112685_b1) 1989; 393 Cavallina (10.1016/j.na.2021.112685_b6) 2020 Delfour (10.1016/j.na.2021.112685_b9) 2001 Alt (10.1016/j.na.2021.112685_b2) 1981; 325 Cavallina (10.1016/j.na.2021.112685_b7) 2020 Ambrosetti (10.1016/j.na.2021.112685_b3) 1983 Kamburov (10.1016/j.na.2021.112685_b15) 2021; 38 Novruzi (10.1016/j.na.2021.112685_b17) 2002; 2 Serrin (10.1016/j.na.2021.112685_b19) 1971; 43 Sacksteder (10.1016/j.na.2021.112685_b18) 1960; 82 Foote (10.1016/j.na.2021.112685_b10) 1984; 92 Beurling (10.1016/j.na.2021.112685_b4) 1957 Gilbarg (10.1016/j.na.2021.112685_b11) 1983 Henrot (10.1016/j.na.2021.112685_b12) 2021 Kinderlehrer (10.1016/j.na.2021.112685_b16) 1977; 4 |
| References_xml | – volume: 92 start-page: 153 year: 1984 end-page: 155 ident: b10 article-title: Regularity of the distance function publication-title: Proc. Amer. Math. Soc. – volume: 82 start-page: 609 year: 1960 end-page: 630 ident: b18 article-title: On hypersurfaces with no negative sectional curvatures publication-title: Amer. J. Math. – volume: 4 start-page: 373 year: 1977 end-page: 391 ident: b16 article-title: Regularity in free boundary problems publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – volume: 28 start-page: 815 year: 1997 end-page: 823 ident: b14 article-title: Convexity of free boundaries with Bernoulli type boundary condition publication-title: Nonlinear Anal. 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| SubjectTerms | Free boundary problem Implicit function theorem Overdetermined problem Shape derivatives Two-phase |
| Title | The simultaneous asymmetric perturbation method for overdetermined free boundary problems |
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