Nontrivial periodic solutions to the one-phase free boundary problem
Abstract
We construct new classical global solutions of the one-phase Bernoulli free boundary problem in ℝ3 whose positive phases are periodic perturbations of the product of the Hauswirth--Hélein--Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in ℝn+1, n ≥ 3, bifurcating from product solutions U(x,t)=u(x), when the base u:ℝn→ [0,∞) has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions n∈{3,4}, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in ℝ4 and ℝ5 bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.
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BibTeX
@misc{basulto2026nontrivial,
title = {{Nontrivial periodic solutions to the one-phase free boundary problem}},
author = {José Basulto and Nikola Kamburov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05257},
eprint = {2610.05257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05257},
}
RIS
TY - UNPB
TI - Nontrivial periodic solutions to the one-phase free boundary problem
AU - José Basulto
AU - Nikola Kamburov
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05257
UR - https://arxiv.org/abs/2610.05257
AB - We construct new classical global solutions of the one-phase Bernoulli free boundary problem in ℝ^3 whose positive phases are periodic perturbations of the product of the Hauswirth--Hélein--Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in ℝ^n+1, n ≥ 3, bifurcating from product solutions U(x,t)=u(x), when the base u:ℝ^n→ [0,∞) has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions n∈{3,4}, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in ℝ^4 and ℝ^5 bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.
ER -