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The boundary value problem zÃ¢Â€Â³=((ÃÂÃ¢Â‹Â…gÃ¢Â‹Â…zÃ¢ÂˆÂ’p)/ÃŽÂ³)[1+(zÃ¢Â€Â²)2]3/2Ã¢ÂˆÂ’(1/r)Ã¢Â‹Â…[1+(zÃ¢Â€Â²)2]Ã¢Â‹Â…zÃ¢Â€Â²,Ã¢Â€Â‰rÃ¢ÂˆÂˆ[r1, r0], zÃ¢Â€Â²(r1)=Ã¢ÂˆÂ’tanÃ¢ÂÂ¡(ÃÂ€/2Ã¢ÂˆÂ’ÃŽÂ±g), zÃ¢Â€Â²(r0)=Ã¢ÂˆÂ’tanÃ¢ÂÂ¡ÃŽÂ±c, z(r0)=0, and  z(r) is strictly decreasing on [r1,r0], is considered. Here, 0<r1<r0,Ã¢Â€Â‰Ã¢Â€Â‰ÃÂ,Ã¢Â€Â‰Ã¢Â€Â‰g,Ã¢ÂÂ‰Ã¢Â€Â‰ÃŽÂ³,Ã¢Â€Â‰Ã¢Â€Â‰p,Ã¢Â€Â‰Ã¢Â€Â‰ÃŽÂ±c,Ã¢Â€Â‰Ã¢Â€Â‰ÃŽÂ±g are constants having the following properties: ÃÂ,Ã¢Â€Â‰Ã¢Â€Â‰g,Ã¢Â€Â‰Ã¢Â€Â‰ÃŽÂ³ are strictly positive and  0<ÃÂ€/2Ã¢ÂˆÂ’ÃŽÂ±g<ÃŽÂ±c<ÃÂ€/2. Necessary or sufficient conditions are given in terms of p for the existence of concave solutions of the above nonlinear boundary value problem (NLBVP). Numerical illustration is given. This kind of results is useful in the experiment planning and technology design of single crystal rod growth from the melt by edge-defined film-fed growth (EFG) method. With this aim, this study was undertaken

Nonlinear Boundary Value Problem for Concave Capillary Surfaces Occurring in Single Crystal Rod Growth from the Melt