Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

On Steklov-Neumann boundary value problems

We will study a class of Steklov-Neumann boundary value problems for some quasilinear elliptic equations. We obtain result ensuring the existence of solutions when resonance and nonresonance conditions occur. The result was obtained by using variational arguments.

EXTREMAL EIGENVALUE PROBLEMS FOR THE LAPLACIANSteven

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons’ cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin’s theorem [43] is optimal.

Boundary behavior of capillary surfaces possibly with extremal boundary angles

For solutions to the capillarity problem possibly with the boundary contact angle θ being 0 and/or π in a relatively open portion of the boundary which is C2, we will show that if the solution is locally bounded up to this portion of boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Hölder continuous up to the boundary, provided the prescr...

Application of Decoupled Scaled Boundary Finite Element Method to Solve Eigenvalue Helmholtz Problems (Research Note)

A novel element with arbitrary domain shape by using decoupled scaled boundary finite element (DSBFEM) is proposed for eigenvalue analysis of 2D vibrating rods with different boundary conditions. Within the proposed element scheme, the mode shapes of vibrating rods with variable boundary conditions are modelled and results are plotted. All possible conditions for the rods ends are incorporated ...