Asymptotic expansion at a corner for the capillary problem: The singular case
type: Item
instance of: scholarly article
title: Asymptotic expansion at a corner for the capillary problem: The singular case
zbMATH Open document ID: 0796.76020
author: Erich Miersemann
published in: Pacific Journal of Mathematics
publication date: 1994-10-11T00:00:00Z
review text: Consider the capillary equation \[ \text{div} Tu=\kappa u,\qquad Tu={\nabla u\over\sqrt{1+|\nabla u|^ 2}},\qquad\kappa>0, \] in a domain \(\Omega\) containing a protruding corner of opening angle \(2\alpha\), and with contact angle \(\gamma\) that is constant on \(\partial\Omega\) in a neighborhood of the vertex point \(V\), \(0\leq\gamma<\pi/2\). In earlier work, Concus and Finn demonstrated a dichotomy of behavior, showing that every solution is bounded at \(V\) if \(\alpha+\gamma\geq\pi/2\), while in the complementary (singular) case there holds \(| u-h_{-1}(\vartheta)r^{-1}|<\text{const}\), with \(h_{-1}(\vartheta)=\Bigl(\cos\vartheta-\sqrt{k^ 2-\sin^ 2\vartheta}\Bigr)(k\kappa )^{-1}\), \(k=\sin\alpha/\cos\gamma\) in polar coordinates centered at \(V\), so that all solutions are unbounded. In the present paper, the author establishes for the singular case the definitive result that for any given \(m\geq 0\) there are constant \(r_ 0\), \(A\) and functions \(h_{4l-1}(\vartheta)\), \(l=0,\ldots,m\), all independent of the particular solution considered, such that \[ \mid u(x)- \sum^ m_{l=0}h_{4l-1}(\vartheta)r^{4l-1}\mid\leq Ar^{4m+3} \] in the sector \(\Omega_{r_ 0}\). A particular consequence is that the expression \(h_{-1}(\vartheta)r^{-1}\) originally introduced by Concus and Finn is asymptotic to the solution up to \(O(r^ 3)\). The proof of the expansion is something of a tour de force, using the same comparison principle that was exploited for the original result. The functions \(h_{4l-1}(\vartheta)\) appear as solutions of linear two point boundary value problems on \((-\alpha,\alpha)\). The case \(\gamma=0\) leads to singular equations and requires special consideration.
Mathematics Subject Classification ID: 76B45
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 508854
zbMATH Keywords: dichotomy
zbMATH Keywords: comparison principle
zbMATH Keywords: linear two point boundary value problems
reviewed by: Robert Finn
MaRDI profile type: Publication
full work available at URL: pjm.1993.157.95
OpenAlex ID: W1985547953
DOI: 10.2140/PJM.1993.157.95
description: scientific article; zbMATH DE number 508854