The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

Nicholas Brubaker, Alan Lindsay

Research output: Contribution to journalArticle

Abstract

The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

Original languageEnglish
Pages (from-to)631-656
Number of pages26
JournalEuropean Journal of Applied Mathematics
Volume24
Issue number5
Early online date26 Feb 2013
DOIs
Publication statusPublished - Oct 2013

Keywords

  • Prescribed mean curvature
  • Disappearing solutions
  • Singular perturbation
  • MEMS
  • Singular non-linearity
  • EXPONENTIAL NONLINEARITY
  • TIME MAPS
  • MODELS

Cite this

@article{1e14da0891664e7fa3c8f06989f567fb,
title = "The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity",
abstract = "The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.",
keywords = "Prescribed mean curvature, Disappearing solutions, Singular perturbation, MEMS, Singular non-linearity, EXPONENTIAL NONLINEARITY, TIME MAPS, MODELS",
author = "Nicholas Brubaker and Alan Lindsay",
year = "2013",
month = "10",
doi = "10.1017/S0956792513000077",
language = "English",
volume = "24",
pages = "631--656",
journal = "European Journal of Applied Mathematics",
issn = "0956-7925",
publisher = "Cambridge University Press",
number = "5",

}

The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity. / Brubaker, Nicholas; Lindsay, Alan.

In: European Journal of Applied Mathematics, Vol. 24, No. 5, 10.2013, p. 631-656.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The onset of multi-valued solutions of a prescribed mean curvature equation with singular non-linearity

AU - Brubaker, Nicholas

AU - Lindsay, Alan

PY - 2013/10

Y1 - 2013/10

N2 - The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

AB - The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation with singular non-linearity is analysed. The partial differential equation is a recently derived variant of a canonical model used in the modelling of micro-electromechanical systems. It is observed that the bifurcation curve of solutions terminates at single dead-end point, beyond which no classical solutions exist. A necessary condition for the existence of solutions is developed, revealing that this dead-end point corresponds to a blow-up in the solution's gradient at a point internal to the domain. By employing a novel asymptotic analysis in terms of two small parameters, an accurate characterization of this dead-end point is obtained. An arc length parameterization of the solution curve can be employed to continue solutions beyond the dead-end point; however, all extra solutions are found to be multi-valued. This analysis therefore suggests that the dead-end is a bifurcation point associated with the onset of multi-valued solutions for the system.

KW - Prescribed mean curvature

KW - Disappearing solutions

KW - Singular perturbation

KW - MEMS

KW - Singular non-linearity

KW - EXPONENTIAL NONLINEARITY

KW - TIME MAPS

KW - MODELS

U2 - 10.1017/S0956792513000077

DO - 10.1017/S0956792513000077

M3 - Article

VL - 24

SP - 631

EP - 656

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 5

ER -