Related papers: A mixed problem for the Laplace operator in a doma…

The Dirichlet problem on a bounded planar domain is more readily understood and solved for the Laplace operator than it is for a Schrodinger operator. When the potential function is small, we might hope to approximate the solution to the…

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A…

This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the…

We analyze the existence and multiplicity of positive solutions to a nonlocal elliptic problem involving the spectral fractional Laplace operator endowed with homogeneous mixed Dirichlet-Neumann boundary conditions and weighted critical…

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…

We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image $\phi(\partial\Omega)$ of a reference set…

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is…

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem…

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.

In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…

In this paper we study a bounded domain with a small hole removed. Our main result concerns the spectrum of the Laplace operator with the Robin conditions imposed at the hole boundary. Moreover we prove that under some suitable assumptions…

We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirich- let boundary condition and transmission condition, subject to the small geometric perturbation and the high…

We study the critical problem {equation} {{array}{ll} \Delta ^{2}u=u^{\frac{N+4}{N-4}} & {in}\Omega\setminus \bar{B(\xi_0,\varepsilon)},\medskip u>0&{in}\Omega\setminus \bar{B(\xi_0,\varepsilon)},\medskip u=\Delta u=0 & {on}\partial (\Omega…

We show that the Neumann problem for Laplace's equation in a convex domain $\Omega$ with boundary data in $L^p(\partial\Omega)$ is uniquely solvable for $1<p<\infty$. As a consequence, we obtain the Helmholtz decomposition of vector fields…

In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $\epsilon>0$ proportional to the radius of the holes and a map $\phi$,…

We consider Laplace's equation in a periodically perforated domain with Robin boundary conditions on the holes, where the Robin coefficient is scaled proportionally to the inverse total surface area of the performations. We identify a…

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous…