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Sinan Ariturk

Publications and source records attributed to Sinan Ariturk.

11 recordsLinked to original sources
arXiv2017

Optimal stretching for lattice points under convex curves

Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the area under the curve, and again count lattice points. Which choice of stretch factor will maximize the lattice point count? We show the optimal stretch factor approaches $1$ as the area approaches infinity. In particular, when $0 0$, the one enclosing the most first-quadrant lattice points approaches a $p$-circle ($s=1$) as $r \to \infty$. The case $p=2$ was established by Antunes and Freitas, with generalization to $1<p<\infty$ by Laugesen and Liu. The case $p=1$ remains open, where the question is: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity? Our results for $p<1$ lend support to the conjecture that in all dimensions, the rectangular box of given volume that minimizes the $n$-th eigenvalue of the Dirichlet Laplacian will approach a cube as $n \to \infty$. This conjecture remains open in dimensions four and higher.

math.SP↗
arXiv2016

Simons' cone and equivariant maximization of the first $p$-Laplace eigenvalue

We consider an optimization problem for the first Dirichlet eigenvalue of the $p$-Laplacian on a hypersurface in $\mathbb{R}^{2n}$, with $n \ge 2$. If $p \ge 2n-1$, then among hypersurfaces in $\mathbb{R}^{2n}$ which are $O(n) \times O(n)$-invariant and have one fixed boundary component, there is a surface which maximizes the first Dirichlet eigenvalue of the $p$-Laplacian. This surface is either Simons' cone or a $C^1$ hypersurface, depending on $p$ and $n$. If $n$ is fixed and $p$ is large, then the maximizing surface is not Simons' cone. If $p=2$ and $n \le 5$, then Simons' cone does not maximize the first eigenvalue.

math.AP↗
arXiv2016

Eigenvalue estimates on quantum graphs

On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees, these estimates are sharp. We also establish sharp upper bounds for the spectral gap of the complete graph $K_4$. The proofs are based on estimates for eigenvalues on graphs with Dirichlet conditions imposed at the pendant vertices.

math.SP↗
arXiv2016

Maximal spectral surfaces of revolution converge to a catenoid

We consider a maximization problem for eigenvalues of the Laplace-Beltrami operator on surfaces of revolution in $\mathbb{R}^3$ with two prescribed boundary components. For every $j$, we show that there is a surface $Σ_j$ which maximizes the $j$-th Dirichlet eigenvalue. The maximizing surface has a meridian which is a rectifiable curve. If there is a catenoid which is the unique area minimizing surface with the prescribed boundary, then the eigenvalue maximizing surfaces of revolution converge to this catenoid.

math.SP↗
arXiv2016

Pucci eigenvalues on geodesic balls

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng's bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng's upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an $O(n)$-invariant hypersurface immersed in $\mathbb{R}^{n+1}$ with one smooth boundary component are smaller than the eigenvalues of an $n$-dimensional Euclidean ball with the same boundary.

math.AP↗
arXiv2015

An annulus and a half-helicoid maximize Laplace eigenvalues

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinder successively increases the eigenvalues. A similar argument shows that the Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a half-helicoid than on any other screw surface in $\mathbb{R}^2 \times \mathbb{S}^1$ with the same boundary.

math.AP↗
arXiv2014

Eigenvalue maximization for surfaces of revolution with prescribed boundary

Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the circles are sufficiently close together, then this surface is unique.

math.AP↗
arXiv2011

Concentration of eigenfunctions near a concave boundary

This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in different ways. We also prove one of these inequalities, which bounds the L^p norms of the restrictions of eigenfunctions to broken geodesics.

math.AP↗