A note on the existence of $H$-bubbles via perturbation methods

Felli, Veronica

doi:10.4171/rmi/419

Cited by 10 publications

(7 citation statements)

References 17 publications

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“…The interested reader can see, e.g., the papers [7,40,41,42,43,45,51,79,130] where these problems are studied essentially by the same methods. However, for the sake of brevity, we will not deal with these topics here but we will focus on elliptic problems.…”

Section: The Abstract Settingmentioning

confidence: 99%

Perturbation Methods and Semilinear Elliptic Problems on Rn

Ambrosetti¹,

Malchiodi²

2006

Progress in Mathematics

262

347

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Section: The Abstract Settingmentioning

confidence: 99%

Perturbation Methods and Semilinear Elliptic Problems on Rn

Ambrosetti¹,

Malchiodi²

2006

Progress in Mathematics

262

347

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“…This is the key step for a finite-dimensional reduction of our problem (see [2,3,Sec. 2.4;[10][11][12]15,23,25,37] for related methods).…”

Section: Introductionmentioning

confidence: 99%

Embedded Disc-Type Surfaces With Large Constant Mean Curvature and Free Boundaries

Fall

2012

Commun. Contemp. Math.

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Given Ω ⊂ R 3 an open bounded set with smooth boundary ∂Ω and H ∈ R, we prove the existence of embedded H-surfaces supported by ∂Ω, that is regular surfaces in R 3 with constant mean curvature H at every point, contained in Ω and with boundary intersecting ∂Ω orthogonally. More precisely, we prove that if Q ∈ ∂Ω is a stable stationary point for the mean curvature of ∂Ω, then there exists a family of embedded 1 ε -surfaces near Q, ε > 0 small, which, once dilated by a factor 1 ε and suitably translated, converges to a hemisphere of radius 1 as ε → 0.

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“…Let us point out that, as a by-product of Theorem 1.2 and with the estimates (3.26) we obtain a new existence result for the H-bubble problem. This result has a perturbative character, in the same direction of other works like [7], [10], [16], [21]. Theorem 3.15 Let K ∈ C 1 (R 3 ) satisfy (K 1 ), (K 2 ), and (K 4 ).…”

Section: Conclusion In Both Cases There Exists Just Onementioning

confidence: 66%

Isovolumetric and Isoperimetric Problems for a Class of Capillarity Functionals

Caldiroli

2015

Arch Rational Mech Anal

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Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R 3 as the sum of the area integral and an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of S 2 and with fixed volume, extremals of capillarity functionals are surfaces whose mean curvature is prescribed up to a constant. For a certain class of anisotropies vanishing at infinity, we prove existence and nonexistence of volumeconstrained, S 2 -type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show existence of extremals for the full isoperimetric inequality.

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A note on the existence of $H$-bubbles via perturbation methods

Abstract: We study the problem of existence of surfaces in R 3 parametrized on the sphere S 2 with prescribed mean curvature H in the perturbative case, i.e. for H = H 0 + εH 1 , where H 0 is a nonzero constant, H 1 is a C 2 function and ε is a small perturbation parameter.

Cited by 10 publications

References 17 publications

Perturbation Methods and Semilinear Elliptic Problems on Rn

Perturbation Methods and Semilinear Elliptic Problems on Rn

Embedded Disc-Type Surfaces With Large Constant Mean Curvature and Free Boundaries

Isovolumetric and Isoperimetric Problems for a Class of Capillarity Functionals

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