Given a regular function H: ℝ^3 → ℝ, we look for H-bubbles, that is, regular surfaces in R^3 parametrized on the sphere S^2, with mean curvature H at every point. Here we study the case of H(u) = H_0 + εH_1(u) =: H_ε(u), where H_0 is a nonzero constant, ε is the smallness parameter, and H_1 is any C^2-function. We prove that if p̄ ∈ ℝ^3 is a "good" stationary point for a suitable Melnikov-type function Γ, then for |ε| small there exists an H_ε-bubble ω_ε that converges to a sphere of radius 1/ |H_0| centered at at p̄, as ε → 0.

H-bubbles in a perturbative setting: The finite-dimensional reduction method

Musina, Roberta
2004

Abstract

Given a regular function H: ℝ^3 → ℝ, we look for H-bubbles, that is, regular surfaces in R^3 parametrized on the sphere S^2, with mean curvature H at every point. Here we study the case of H(u) = H_0 + εH_1(u) =: H_ε(u), where H_0 is a nonzero constant, ε is the smallness parameter, and H_1 is any C^2-function. We prove that if p̄ ∈ ℝ^3 is a "good" stationary point for a suitable Melnikov-type function Γ, then for |ε| small there exists an H_ε-bubble ω_ε that converges to a sphere of radius 1/ |H_0| centered at at p̄, as ε → 0.
DUKE MATHEMATICAL JOURNAL
122
3
457
484
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1082665285
Caldiroli, P; Musina, Roberta
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/32218
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