The C1 and C2 quasi-Plateau problems

In this paper, we study the Ck quasi-Plateau problem: to find the parametric surface of minimal area defined on a rectangular parametric domain among all the surfaces which are Ck continuous on the border (k=1,2). An approach is proposed based on the Coons surface and the MRA (Multi-Resolution Analysis) formed by the B-spline of orders 3 and 4. The Coons surface is constructed firstly to satisfy the prescribed border conditions. Then replacing the area functional with the Dirichlet functional and using MRA, the problem reduces into solving a system of linear equations. The linear equations have simple and sparse structure. Finally, the method is concluded into six algorithms according to the different boundary conditions. Examples are provided to illustrate that the proposed method is effective and flexible.