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

• Published in: European journal of applied mathematics 2013-10, Vol.24 (5), p.631-656
• Main Authors: BRUBAKER, N. D., LINDSAY, A. E.
• Format: Article
• Language: English
• Subjects: Applied mathematics; Asymptotic properties; Bifurcations; Curvature; Linear equations; Mathematical analysis; Mathematical models; Nonlinear equations; Nonlinearity; Parametrization; Partial differential equations
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792513000077

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.