I will explain the existence of two capillary embedded geodesics on Riemannian 2-disks with a strictly convex boundary with a certain condition via multi-parameter min-max construction.  I will then explain the existence of two free boundary embedded geodesics on Riemannian 2-disks with a strictly convex boundary by free boundary curve shortening flow on surfaces, which is a free boundary analog of Grayson’s theorem in 1989.  Moreover, I will explain the Morse Index bound of such geodesics. Finally, I will introduce recent developments to obtain p-widths of surfaces, nonlinear analog of the spectrum of Laplacian, by Allen-Cahn type PDE approaches.