Related papers: Stability Estimates for Commutativity Properties o…

Let $(M,g)$ and $(M',g')$ be non-orientable Riemannian surfaces with fixed boundary $\Gamma$ and fixed Euler characterictic $m$, and $\Lambda$ and $\Lambda'$ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of…

We present first results on the Dirichlet-to-Neumann operator associated with the $1$-Laplace operator in $L^1$. In particular, we show that this operator can be realized as a sub-differential operator in $L^1\times L^{\infty}$ of a…

Let (M,g) be a compact Riemmanian manifold with non-empty boundary. Consider the second order hyperbolic initial-boundary value problem (\delta_t^2 + P(x,D))u = 0 in (0,T) x M, u(0,x) = \delta_t u(0,x) = 0 for x in M, u(t,x) = f(t,x) on…

In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is…

In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(\Omega) =…

In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains in general Riemannian manifolds and establish a criterion for stability. We classify the stable extremal domains in the 2-sphere and…

In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\Lambda$ is shown to be self-adjoint…

As is known, the Dirichlet-to-Neumann operator $\Lambda$ of a Riemannian surface $(M,g)$ determines the surface up to conformal equivalence class $[(M,g)]$. Such classes constitute the Teichm\"uller space with the distance ${\rm d}_T$. We…

For $M\subset \mathbb{R}^{d\geq 3}$ a smooth, connected, compact $d$-dimensional submanifold with boundary, equipped with the standard metric, the Laplacian on $\partial M$ is known to commute with the corresponding Dirichlet-to-Neumann map…

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a…

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an…

In this paper we are interested in establishing stability estimates in the inverse problem of determining on a compact Riemannian manifold the electric potential or the conformal factor in a Schr\"odinger equation with Dirichlet data from…

We study the relationship between the symbol of the Dirichlet-to-Neumann operator associated with a connection Laplacian, and the geometry on and near the boundary. As a consequence, we show that the geometric data on the boundary, and when…

We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_\lambda$ be its nodal set. Assume that zero is a regular value of…

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(\lambda)= \kappa_0\lambda^d…

In this paper we consider the inverse problem of determining on a compact Riemannian manifold the metric tensor in the wave equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

We consider a Laplace eigenfunction $\varphi_\lambda$ on a smooth closed Riemannian manifold, that is, satisfying $-\Delta \varphi_\lambda = \lambda \varphi_\lambda$. We introduce several observations about the geometry of its vanishing…

The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to…

The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace…