10 papers
Quantitative Pólya inequalities for annular sectors
Guowei Dai, Yingxin Sun
We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on t…
Pólya's conjecture for Neumann eigenvalues of annuli
Guowei Dai, Yingxin Sun
We prove strict Pólya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity a…
Sharp Neumann eigenvalue means on triangles
Guowei Dai, Yingxin Sun
We prove two conjectures of Laugesen and Siudeja: the equilateral triangle uniquely maximizes the harmonic mean of the first two positive Neumann eigenvalues at fixed perimeter and…
Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics
Guowei Dai, Yingxin Sun, Yong Zhang
We prove Laugesen and Siudeja's conjecture on Neumann eigenvalues of isosceles triangles. After multiplication by the squared diameter, the first positive symmetric eigenvalue incr…
Robin eigenvalue ratios: monotonicity on boxes and counterexamples
Guowei Dai, Yingxin Sun
We disprove Laugesen's conjecture that the ratio of the first two Robin eigenvalues decreases with the positive boundary parameter. On any prescribed compact interval of positive p…
Upper bounds for the ratio of the first two eigenvalues of the Robin Laplacian
Guowei Dai, Yingxin Sun
We sharpen upper bounds for the ratio of the first two Robin eigenvalues by matching the first eigenvalue on a comparison ball with an independently chosen boundary parameter. The…
Complete spectrum of the Robin eigenvalue problem on the ball
Guowei Dai, Yingxin Sun
We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -Δu=μu\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+αu=0 &\text{on}\,\, \partial…
On the Schiffer and Berenstein conjectures with high-frequency for convex domains in the plane
Guowei Dai, Yingxin Sun, Juncheng Wei +1
In this paper, by introducing two-point stationary-phase amplitude defect, we provide a partial positive answer to the Schiffer and Berenstein conjectures in . More p…
Confirmed answer to the Schiffer conjecture and the Berenstein conjecture
Guowei Dai
Let be a bounded domain in with a connected () boundary. We show that, if the following overdetermined elliptic problem \begin{equation}…
Regularity and uniqueness to multi-phase problem with variable exponent
Guowei Dai, Francesca Vetro
In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple differen…