Existence of \(H\)-bubbles in a perturbative setting
type: Item
instance of: scholarly article
title: Existence of \(H\)-bubbles in a perturbative setting
zbMATH Open document ID: 1066.53018
DOI: 10.4171/RMI/402
author: Paolo Caldiroli
author: Roberta Musina
published in: Revista Matemática Iberoamericana
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39723
review text: Let \(H: \mathbb R^3 \to \mathbb R\) be a \(C^1\) function and consider the problem of finding smooth non-constant maps \(\omega :\mathbb R^2 \to \mathbb R^3\) satisfying \[ \Delta \omega = 2 H(\omega) \omega_x \wedge \omega_y \quad\text{in } \mathbb R^2,\quad \text{and}\quad \int_{\mathbb R^2} | \nabla \omega | ^2 < + \infty.\tag{1} \] It is known that such a solution must be conformal and, at each regular point \(p=\omega (z)\), \(H(p)\) is the mean curvature of the surface parameterized by \(\omega\) at the point \(p\). Moreover, if \(\sigma : S^2 \to \mathbb R^2\) denotes the stereographic projection, the map \(\omega \circ \sigma : S^2 \to \mathbb R^3\) defines an \(S^2\)-type parametric surface in \(\mathbb R^3\) having prescribed mean curvature, simply called an \(H\)-bubble. The problem has also a variational description in which \(H\)-bubbles can be found as critical points of an energy functional. When the prescribed mean curvature is a non-zero constant function, \(H(u)\equiv H_0\), \textit{H. Brezis} and \textit{J.-M. Coron} [Arch. Ration. Mech. Anal. 89, 21--56 (1985; Zbl 0584.49024)] have proved that the only non-zero solutions to (1) are spheres of radius \(| H_0| ^{-1}\) placed anywhere in \(\mathbb R^3\). The present paper deals with the case \[ H(u)=H_0(u)+\varepsilon H_1(u)=:H_{\varepsilon}(u),\tag{2} \] where \(H_0 \in C^1(\mathbb R^3)\) satisfies some particular conditions, \(| \varepsilon| \) is small, and \(H_1 : \mathbb R^3 \to \mathbb R\) is any \(C^1\) function. The conditions for \(H_0\) are motivated by a previous result of the authors [Commun. Contemp. Math. 4, No. 2, 177--209 (2002; Zbl 1009.53008)] regarding the existence of solutions to (1) when \(H\) is nonconstant. For such an \(H_0\), the authors have proved the existence of \(H_0\) bubbles which are minimal in the sense of having minimal energy in the variational description mentioned above. The main result of the paper states that for any \(H_{\varepsilon}\) as in (2) there exist…
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 2110200
zbMATH Keywords: parametric surfaces
zbMATH Keywords: prescribed mean curvature
MaRDI profile type: Publication
cites work: Multiple solutions to the Plateau problem for nonconstant mean curvature
cites work: Convergence of solutions of H-systems or how to blow bubbles
cites work: EXISTENCE OF MINIMAL H-BUBBLES
OpenAlex ID: W2035109999
description: scientific article; zbMATH DE number 2110200