We establish existence and non-existence results to the BrezisNirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition.

In this note, we establish the sectorial property of the Caputo fractional derivative operator of order α ∈ (1, 2) with a zero Dirichlet boundary condition.

The Casimir effect giving rise to an attractive or repulsive force between the configuration boundaries that confine the massless scalar field is reexamined for one to three-dimensional pistons in this paper. Especially, we consider Casimir pistons with hybrid boundary conditions, where the boundary condition on the piston is Neumann and those on other surfaces are Dirichlet. We show that the C...

Many multiscale methods are based on the idea of extracting macroscopic behavior of solutions by solving an array of microscale models over small domains. A key ingredient in such multiscale methods is the boundary condition and the size of the computational domain over which the microscale problems are solved. This problem is systematically investigated in the present paper in the context of m...

The topological sensitivity analysis consists to study the variation of a shape functional with respect to emerging of small holes in the interior of the domain occupied by the body. In this paper, we consider the case of a hole located at the boundary of the domain. The topological asymptotic expansion is derived for the Laplace equation in three cases: Dirichlet condition, Neumann condition a...

This work considers a reinforcement of a multiconnected domain representing the cross section of a cylindrical bar. Dirichlet boundary condition is imposed on its outer boundary and integral condition on its inner boundary. We assume that the reinforcement is periodic and has small thickness which is of the same order as the period. We describe the limiting behaviour of the reinforced structure...

In this paper we prove the L p − L ´ p estimate for the Schrödinger equation on the half-line and with homogeneous Dirichlet boundary condition at the origin.

We consider Laplace and Stokes operators in domains with rough boundaries and search for an effective boundary condition. The method of homogenization, coupled with the boundary layers, is used to obtain it. In the case of the homogeneous Dirichlet condition at the rough boundary, the effective law is Navier’s slip condition, used in the computations of viscous flows in complex geometries. The ...

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an...

Abstract. The aim of this paper is to study the asymptotic behavior of the solution of a parabolic dynamical boundary -value problem in a periodically perforated domain. The domain is considered to be a fixed bounded open subset Ω⊂R , in which identical and pe riodically distributed perforations (holes) of size ε are made. In the perforated domain we consider a heat equation, with a Dirichlet c...