TY - JOUR
T1 - The simultaneous asymmetric perturbation method for overdetermined free boundary problems
AU - Cavallina, Lorenzo
N1 - Funding Information:
This research was partially supported by the JSPS Grant-in-Aid for Research Activity Startup Grant Number JP20K22298 .
Publisher Copyright:
© 2021 Elsevier Ltd
PY - 2022/2
Y1 - 2022/2
N2 - 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.
AB - 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.
KW - Free boundary problem
KW - Implicit function theorem
KW - Overdetermined problem
KW - Shape derivatives
KW - Two-phase
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U2 - 10.1016/j.na.2021.112685
DO - 10.1016/j.na.2021.112685
M3 - Article
AN - SCOPUS:85119200730
VL - 215
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
SN - 0362-546X
M1 - 112685
ER -