Related papers: Norm-Resolvent Convergence in Perforated Domains

We investigate the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } \Omega, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial \Omega, \leqno{(P_\lambda)} $$ where $\Omega$ is a bounded smooth…

In this paper, we consider the Brezis-Nirenberg problem $$ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, \Omega,\quad u=0,\quad\mbox{on}\,\, \partial\Omega, $$ where $\lambda\in\mathbb{R}$, $\Omega\subset\mathbb R^N$ is a…

We consider a boundary value problem for a general second order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries…

We study the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem for the Sturm-Liouville operator with general boundary conditions and the weight function perturbed by the so-called $\delta'$-like sequence…

We consider Schr\"odinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to…

This work addresses the resolvent convergence of generalized MIT bag operators to Dirac operators with zigzag type boundary conditions. We prove that the convergence holds in strong but not in norm resolvent sense. Moreover, we show that…

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq…

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 consider the Hele-Shaw problem in a randomly perforated domain with zero Neumann boundary conditions. A homogenization limit is obtained as the characteristic scale of the domain goes to zero. Specifically, we prove that the solutions as…

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin…

We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square…

Let $\Delta_{\Omega_\varepsilon}$ be the Dirichlet Laplacian in the domain $\Omega_\varepsilon:=\Omega\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $\Omega\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny…

Assume that $Au=f,\quad (1)$ is a solvable linear equation in a Hilbert space $H$, $A$ is a linear, closed, densely defined, unbounded operator in $H$, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the…

The two scale convergence of the solution to a Robin's type-like problem of a stationary diffusion problem in a periodically perforated domain is investigated. It is shown that the Robin's problem converges to a problem associated to a new…

We consider a boundary value problem for a general second order linear equation in a perforated domain. The perforation is made by small cavities, a minimal distance between the cavities is also small. We impose minimal natural geometric…

Norm-resolvent convergence with order-sharp error estimate is established for Neumann Laplacians on thin domains in $\mathbb{R}^d,$ $d\ge2,$ converging to metric graphs in the limit of vanishing thickness parameter in the resonant case. The…

Let $\Omega$ be a bounded domain in $\mathbb R^2$ with smooth boundary $\partial\Omega$, and let $\omega_h$ be the set of points in $\Omega$ whose distance from the boundary is smaller than $h$. We prove that the eigenvalues of the…

We prove H\"older continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This…

We demonstrate how to approximate one-dimensional Schr\"odinger operators with $\delta$-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small…

We analyze the behavior of the eigenvalues and eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions when the domain is perturbed. We focus on exterior perturbations of the domain, that is, the limit domain is…