Differences Between Robin and Neumann Eigenvalues

Let $$\Omega {\subset } {\mathbb {R}}^2$$ Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary $$\partial \Omega $$ ∂ Ω . For $$\sigma >0$$ σ > 0 , we consider the Robin boundary value problem $$\begin{aligned} -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \text{ on } \partial \Omega \end{aligned}$$ - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where $$ \frac{\partial f}{\partial n} $$ ∂ f ∂ n is the derivative in the direction of the outward pointing normal to $$\partial \Omega $$ ∂ Ω . Let $$0<\lambda ^\sigma _0\le \lambda ^\sigma _1\le \ldots $$ 0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps $$\begin{aligned} d_n(\sigma ):=\lambda _n^\sigma -\lambda _n^0 . \end{aligned}$$ d n ( σ ) : = λ n σ - λ n 0 . For a wide class of planar domains we show that there is a limiting mean value, equal to $$2{\text {length}}(\partial \Omega )/{\text {area}}(\Omega )\cdot \sigma $$ 2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of $$d_n(\sigma )\le C(\Omega ) n^{1/3}\sigma $$ d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

On functions with the boundary Morera property in domains with piecewise-smooth boundary

The problem of holomorphic extension of functions defined on the boundary of a domain into this domain is actual in multidimensional complex analysis. It has a long history, starting with the proceedings of Poincaré and Hartogs. This paper considers continuous functions defined on the boundary of a bounded domain $ D $ in $ \mathbb C ^ n $, $ n> 1 $, with piecewise-smooth boundary, and having the generalized boundary Morera property along the family of complex lines that intersect the boundary of a domain. Morera property is that the integral of a given function is equal to zero over the intersection of the boundary of the domain with the complex line. It is shown that such functions extend holomorphically to the domain $ D $. For functions of one complex variable, the Morera property obviously does not imply a holomorphic extension. Therefore, this problem should be considered only in the multidimensional case $ (n> 1) $. The main method for studying such functions is the method of multidimensional integral representations, in particular, the Bochner-Martinelli integral representation.

A heat trace anomaly on polygons

Let Ω0be a polygon in$\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωεis a family of surfaces with${\mathcal C}$∞boundary which converges to Ω0smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient oft0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domainZwhich models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.

Effective limit distribution of the Frobenius numbers

The Frobenius number$F(\boldsymbol{a})$of a lattice point$\boldsymbol{a}$in$\mathbb{R}^{d}$with positive coprime coordinates, is the largest integer which cannotbe expressed as a non-negative integer linear combination of the coordinates of$\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math.181(2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when$\boldsymbol{a}$is taken to be random in an enlarging domain in$\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.