In this paper, we study the shape optimization problem for first eigenvalue of p-Laplace operator with mixed Neumann-Dirichlet boundary conditions on multiply-connected domains in hyperbolic space. Precisely, establish that among all a given volume and prescribed (n−1)-th quermassintegral convex Dirichlet (inner boundary), concentric annular region produces largest eigenvalue. We also derive Na...

We consider three problems for the Helmholtz equation in interior and exterior domains in R, (d = 2, 3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for ex...

We give a new derivation of Robin boundary conditions and interface jump conditions for the diffusion equation in one dimension. To derive a Robin boundary condition, we consider the diffusion equation with a boundary condition that randomly switches between a Dirichlet and a Neumann condition. We prove that, in the limit of infinitely fast switching rate with the proportion of time spent in th...

The existence of trapped modes near obstacles in two-dimensional waveguides is well established when the centre-line of the guide is a line of symmetry for the geometry. In this paper we examine cases where no such line of symmetry exists. The boundary condition on the obstacle is of Neumann type and both Neumann and Dirichlet conditions on the guide walls are treated. A variety of techniques (...

In this paper we take up the question of analyticity properties of Dirichlet–Neumann operators with respect to boundary deformations. In two separate results, we show that if the deformation is sufficiently small and lies either in the class of C1+α (any α > 0) or Lipschitz functions, then the Dirichlet–Neumann operator is analytic with respect to this deformation. The proofs of both results ut...

If Ω is any compact Lipschitz domain, possibly in a Riemannian manifold, with boundary Γ = ∂Ω, the Dirichlet-to-Neumann operator Dλ is defined on L2(Γ) for any real λ. We prove a close relationship between the eigenvalues of Dλ and those of the Robin Laplacian ∆μ, i.e. the Laplacian with Robin boundary conditions ∂νu = μu. This is used to give another proof of the Friedlander inequalities betwe...

This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of n intervals (n ≥ 1). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as fa...

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians −∆Θ,Ω in L (Ω; dx) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω ⊂ R, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonl...

the isogeometric analysis is increasingly used in various engineering problems. it is based on non-uniform rational b-splines (nurbs) basis function applied for the solution field approximation and the geometry description. one of the major concerns with this method is finding an efficient approach to impose essential boundary conditions, especially for inhomogeneous boundaries. the main contri...