Rotationally symmetric hypersurfaces with prescribed mean curvature
type: Item
instance of: scholarly article
title: Rotationally symmetric hypersurfaces with prescribed mean curvature
zbMATH Open document ID: 0853.53006
published in: Pacific Journal of Mathematics
publication date: 1996-08-29T00:00:00Z
review text: The physical problem of characterizing the shape of a rotationally symmetric pendent liquid drop in a gravity field leads to that of finding symmetric solutions to the equation \[ \text{div } {{Du} \over {\sqrt {1+|Du|^2}}}+ \kappa u=0, \qquad \kappa>0, \tag{1} \] for the height \(u(r)\) of the surface near the tip \(u_0= u(0)\) of the drop. \textit{P. Concus} and the reviewer [Invent. Math. 29, 143-148 (1975; Zbl 0319.76007)] showed the local existence of a solution of (1) with an isolated singularity, and obtained an asymptotic expansion valid for any such solution. Later they showed that for large negative \(u_0\) there exists a globally regular solution \(u(r)\) of an associated parametric system, yielding many nearly spherical ``bubbles'' along the negative \(u\)-axis, then becoming a graph \(u(r)\) that oscillates about the \(r\)-axis to infinity. The reviewer proved that a subsequence of these solutions converges, uniformly in compacta, to a solution of (1) with an isolated singularity and the same (divergent) asymptotic expansion. In the present work, the author extends and in some ways improves those results, corresponding to equations of the general form \[ \text{div } {{Du} \over {\sqrt {1+|Du|^2}}}+ f(u)= 0 \tag{2} \] in \(\mathbb{R}^N\), under suitable hypotheses on \(f(u)\). The extensions are of considerable interest and decidedly nontrivial. She obtains a number of explicit estimates, yielding conditions for existence of singular solutions and characterizing the asymptotic and global behavior of such solutions when they exist. Her results are obtained in part via procedures analogous to those of the earlier papers, but in extending them to the present context she makes also effective use of subtle energy relations, connected with identities of Pohozaev type, that did not enter into the earlier contributions. It seems worth remarking that if \(f' (u)\leq 0\), then every isolated singularity of a solution of (2) is removable. Thus, it is clear that the…
reviewed by: Robert Finn
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 76B45
zbMATH DE Number: 910275
zbMATH Keywords: rotationally symmetric liquid drop
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: existence of singular solutions
author: Marie-Françoise Bidaut-Véron
MaRDI profile type: Publication
full work available at URL: pjm.1996.173.29
OpenAlex ID: W2136937877
DOI: 10.2140/PJM.1996.173.29
description: scientific article; zbMATH DE number 910275