Narrow Results

In this paper, we study a Robin boundary value problem driven by the Laplace operator plus an indefinite potential term. The reaction is of the logistic type. Using variational methods together with truncations, comparisons and critical groups, we prove two multiplicity theorems producing smooth solutions with sign information. A special case, which we encounter in the literature, is also considered.

The Casimir forces on two parallel plates in conformally flat de Sitter background due to conformally coupled massless scalar field satisfying mixed boundary conditions on the plates is investigated. In the general case of mixed boundary conditions formulae are derived for the vacuum expectation values of the energy–momentum tensor and vacuum forces acting on boundaries. Different cosmological constants are assumed for the space between and outside of the plates to have general results applicable to the case of domain wall formations in the early universe.

The quantum vacuum effects are investigated for a massive scalar field with general curvature coupling and obeying the Robin boundary conditions given on two concentric spherical shells with radii a and b in the (D+1)-dimensional global monopole background. The expressions are derived for the Wightman function, the vacuum expectation values of the field square, the vacuum energy density, radial and azimuthal stress components in the region between the shells. A regularization procedure is carried out by making use of the generalized Abel–Plana formula for the series over zeros of combinations of the cylinder functions. This formula allows us to extract from the vacuum expectation values the parts due to a single sphere on background of the global monopole gravitational field, and to present the "interference" parts in terms of exponentially convergent integrals, useful, in particular, for numerical evaluations. The vacuum forces acting on the boundaries are presented as a sum of the self-action and interaction terms. The first one contains well-known surface divergences and needs a further regularization. The interaction forces between the spheres are finite for all values a<b and are attractive for a Dirichlet scalar. The asymptotic behavior of the vacuum densities is investigated (i) in the limits a→0 and b→∞, (ii) in the limit a, b→∞ for fixed value b-a, and (iii) for small values of the parameter associated with the solid angle deficit in global monopole geometry. We show that in the case (ii) the results for two parallel Robin plates on the Minkowski bulk are rederived to the leading order.

The Casimir stress on two parallel plates in a de Sitter background corresponding to different metric signatures and cosmological constants is calculated for massless scalar fields satisfying Robin boundary conditions on the plates. Our calculation shows that for the parallel plates with false vacuum between and true vacuum outside, the total Casimir pressure leads to an attraction of the plates at very early universe.

We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain with bounded boundary which is symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain’s geometry. Our approach is close to the Born–Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition

A novel method of order reduction is proposed to the high-dimensional convection-diffusion-reaction equation with Robin boundary condition based on the multiquadric radial basis function-generated finite difference method (MQ RBF-FD). The main motivation is to get not only a second-order accurate solution but also a second-order accurate gradient. Key to the proposed method is introducing the intermediate variables representing the first-order derivatives to reduce the original second-order problem into an equivalent system of first-order partial differential equations. Then a discrete scheme for the latter is constructed, in which MQ RBF-FD method is applied to approximate the first-order derivatives of the original variable at the center point with decoupled method. Moreover, we can obtain an equivalent discrete scheme about the original variable and intermediate variables which can be proven all second-order convergent, that is, the convergence rate of the gradient of solution is also second-order. Finally numerical examples are presented to show the efficiency and accuracy of the proposed method.

A semi-analytical approach is presented to solve three-dimensional dynamic Green’s function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green’s function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green’s functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green’s functions.