Related papers: Neumann spectral problem in a domain with very cor…

We consider singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain. In the boundary we select a set depending on a character small parameter and consisting of a great number of small disjoint parts. On this…

We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with…

We consider the Robin Laplacian in the domains $\Omega$ and $\Omega^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $\Omega$…

In this paper we study the asymptotic behaviour as $\varepsilon\to 0$ of the spectrum of the elliptic operator $\mathcal{A}^\varepsilon=-{1\over b^\varepsilon}\mathrm{div}(a^\varepsilon\nabla)$ posed in a bounded domain…

Neumann domains of Laplacian eigenfunctions form a natural counterpart of nodal domains. The restriction of an eigenfunction to one of its nodal domains is the first Dirichlet eigenfunction of that domain. This simple observation is…

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for…

We consider a 2-dimensional thin domain with order of thickness {\epsilon} which presents oscillations of amplitude also {\epsilon} on both boundaries, top and bottom, but the period of the oscillations are of different order at the top and…

In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the…

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain $\Omega_\varepsilon$ coinciding with two thin rectangles connected through a joint of diameter ${\cal O}(\varepsilon)$. A rigorous procedure…

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-\Delta$ in the perforated domain $\Omega\setminus \bigcup_{ i\in 2\varepsilon\mathbb Z^d }B_{a_\varepsilon}(i),$…

We consider a waveguide-like domain consisting of two thin straight tubular domains connected through a tiny window. The perpendicular size of this waveguide is of order $\varepsilon$. Under the assumption that the window is appropriately…

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain $\Omega$, with magnetic Neumann boundary conditions. If $\Omega$ is simply connected then the spectrum reduces to the spectrum of the usual…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and the space $V^{-1,\alpha}(\partial\Omega)$ of (distributional) normal derivatives on the boundary of $\alpha$-H\"{o}lder…

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the…

We study a planar magnetic Schr\"odinger operator with an Aharonov-Bohm vector potential, under Neumann boundary conditions. Through a gauge transformation, the corresponding eigenvalue problem can be formulated in terms of the Laplacian on…

We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $\Omega\subset\R^{n}$ whose boundary has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we prove that,…

We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that…

In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $\Omega$ with Neumann/Robin boundary condition i.e. $\partial_n u=\alpha u$ when $\alpha$ tends to infinity. This…

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…

In this paper we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type $R^\epsilon = \{(x_1,x_2) \in \R^2 \; | \; x_1 \in (0,1), \, - \, \epsilon \, b(x_1) < x_2 < \epsilon \, G(x_1,…