scholarly journals Cloaking for a quasi-linear elliptic partial differential equation

2018 ◽  
Vol 12 (2) ◽  
pp. 461-491 ◽  
Author(s):  
Tuhin Ghosh ◽  
◽  
Karthik Iyer ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Partial Differential Equation ◽  
Partial Differential

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

2013 ◽  
Vol 24 (5) ◽  
pp. 631-656 ◽  
Author(s):  
N. D. BRUBAKER ◽  
A. E. LINDSAY
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Blow Up ◽  
Elliptic Partial Differential Equation ◽  
Classical Solutions ◽  
Necessary Condition ◽  
Partial Differential ◽  
End Point ◽  
Dead End ◽  
The Dead

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.

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