Blow-up analysis for the prescribed mean curvature equation on \(\mathbb R^2\)
type: Item
instance of: scholarly article
title: Blow-up analysis for the prescribed mean curvature equation on \(\mathbb R^2\)
zbMATH Open document ID: 1180.35264
author: Paolo Caldiroli
published in: Journal of Functional Analysis
publication date: 2009-07-10T00:00:00Z
review text: The \(H\)-system, for a given function \(H:\mathbb R^3\to\mathbb R\), is given by \(\Delta u=2H(u)u_x\wedge u_y\) for \(u:\mathbb R^3\to\mathbb R^3\). If \(u\) is conformal, it describes a surface in \(\mathbb R^3\) which has prescribed mean curvature \(H(u)\) in every point. In the current paper, sequences of approximate solutions to the system (without the requirement of \(u\) being conformal) are considered, that is sequences with uniformly bounded Dirichlet integrals and \(\Delta u^n-2H(u^n)u^n_x\wedge u^n_y\to 0\) in the dual of some weighted Sobolev space natural for the problem. For such a sequence, the following is proven: Either \(\nabla u^n\to 0\) strongly in \(L^2\), or the sequence can be decomposed (using a different sequence of conformal transformations of the plane for each bubble) in the limit to give a finite set of \(\widetilde{H}_i\)-bubbles, i.e.\ solutions of the \(\widetilde{H}_i\)-system (for suitable \(\widetilde{H}_i\) derived from \(H\)) on \(\mathbb R^2\). The set of bubbles is complete in the sense that their Dirichlet energies add up to the limit energy of the sequence.
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 35J93
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 5577358
zbMATH Keywords: \(H\)-systems
zbMATH Keywords: prescribed mean curvature equation
zbMATH Keywords: blowup
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.12.003
OpenAlex ID: W2008933623
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DOI: 10.1016/J.JFA.2008.12.003
description: scientific article; zbMATH DE number 5577358