A novel mathematical model which converts the solar collection problem into a nonlinear boundary problem for the elliptic type Monge-Ampére equations based on differential geometry, refraction law, reflection law and the conservation of energy is proposed. A freeform solar concentrator is designed based on this mathematical model. The freeform concentrator system is simulated by ray-tracing software. The results indicate that considering the Fresnel loss and the absorption of optical materials, when the sunlight possesses 0.27° divergence half-angle, the geometric concentration ratio and the focusing efficiency of the 1000 mm freeform solar concentrator reach 500 and 77.5%, respectively. The freeform designed by this method has higher degree of freedom, which makes it possible to design solar concentrators with different degrees of concentrating requirements.