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

Published online by Cambridge University Press:  26 February 2013

N. D. BRUBAKER  and
A. E. LINDSAY
N. D. BRUBAKER
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: brubaker@math.udel.edu
A. E. LINDSAY
Affiliation:
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK email: a.lindsay@hw.ac.uk
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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 curvatureDisappearing solutionsSingular perturbationMEMSSingular non-linearity
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 5 , October 2013 , pp. 631 - 656
Copyright
Copyright © Cambridge University Press 2013 

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