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

5 Citations (Scopus)

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

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