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.

Escobar constants of planar domains

We initiate the study of the higher-order Escobar constants $$I_k(M)$$ I k ( M ) , $$k\ge 3$$ k ≥ 3 , on bounded planar domains M. The Escobar constants $$I_k$$ I k of the unit disk and a family of polygons are provided.

PLANAR IMMERSIONS WITH PRESCRIBED CURL AND JACOBIAN DETERMINANT ARE UNIQUE

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is $$8\pi k$$ 8 π k , which is the best upper bound for the $$k^{\text {th}}$$ k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $$k=1$$ k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.