We calculate radiative corrections to the Casimir effect for the massive complex scalar field with the λφ4 self-interaction in d + 1 dimensions. We consider the field submitted to four types of boundary conditions on two parallel planes, namely: (i) quasi-periodic boundary conditions, which interpolates continuously periodic and anti-periodic ones, (ii) Dirichlet conditions on both planes, (iii...

where t ≥ 0 denotes time and x ∈ Ω ⊂ R position typical within a d−dimensional bounded domain Ω with “sufficiently smooth” boundary or Ω = R. The states u = (u1, . . . , um), i = 1, . . . , m describe the concentrations or densities of substances/populations ui = ui(t, x). The functions di(t, x) are called diffusion coefficients or diffusivities. On a bounded domain Ω the eqs. (1) have to be co...

We present and analyze waveform relaxation variants of the Dirichlet-Neumann and NeumannNeumann methods for parabolic problems. These methods are based on a non-overlapping spatial domain decomposition, and each iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, unlike for elliptic problems, each subdoma...

Abstract We study the first eigenvalue of the Laplace equation in a bounded domain in R (d = 2, 3) with mixed Neumann–Dirichlet (Zaremba) boundary conditions. The Neumann condition is imposed on most of the boundary and the Dirichlet boundary consists of a cluster of small windows. When the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problem...

We study the exact observability of systems governed by the Schrödinger equation in a rectangle with homogeneous Dirichlet (respectively Neumann) boundary conditions and with Neumann (respectively Dirichlet) boundary observation. Generalizing results from Ramdani, Takahashi, Tenenbaum and Tucsnak (2005), we prove that these systems are exactly observable in in arbitrarily small time. Moreover, ...

We consider a reaction-diffusion equation with Neumann boundary conditions and show that solutions to this problem may be obtained from a problem with periodic boundary conditions and equivariant under O(2) symmetry. We describe the solutions for Hopf bifurcation and mode interactions involving Hopf bifurcation, namely, steadystate/Hopf and Hopf/Hopf. Neumann boundary conditions constrain the s...

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon-Miranda) discrete maximum principle, and then p...