6 papers
Maximizing the fundamental Laplace--Neumann eigenvalue on quadrilaterals
Ryoki Endo, Braxton Osting
We prove that among convex planar quadrilaterals of equal area, the square uniquely maximizes the first nonzero Laplace--Neumann eigenvalue. This is the quadrilateral case of the N…
Sharp Dirichlet eigenvalue inequalities on triangles
Ryoki Endo, Xuefeng Liu, Phanuel Mariano
We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a…
Stable Computation of Laplacian Eigenfunctions Corresponding to Clustered Eigenvalues
Ryoki Endo, Xuefeng Liu
The accurate computation of eigenfunctions corresponding to tightly clustered Laplacian eigenvalues remains an extremely difficult problem. In this paper, using the shape differenc…
The second Dirichlet eigenvalue is simple on every non-equilateral triangle
Ryoki Endo, Xuefeng Liu
The Dirichlet eigenvalues of the Laplacian on a triangle that collapses into a line segment diverge to infinity. In this paper, to track the behavior of the eigenvalues during the…
The Second Dirichlet Eigenvalue is Simple on Every Non-equilateral Triangle, Part II: Nearly Equilateral Triangles
Ryoki Endo, Xuefeng Liu
This paper solves the open problem of the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a complete solution to Conjecture 6.47 posed by R…
Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation
Ryoki Endo, Xuefeng Liu
A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed g…