We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functio...

We consider the hp–version interior penalty discontinuous Galerkin finite element method (hp–DGFEM) for semilinear parabolic equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the error analysis of the hp–DGFEM on shape–regular spatial meshes. We derive error bounds under various hypotheses on the regularity of the solution, for both the symmetric and non–symmet...

In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ∂X. In the case of a two-dimensional waveguide, two particular en...

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the...

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell’s equations. These results are analogous to those obtained by the author for the scalar wave equation [23] and for the wave equation on systems ...

Domain decomposition methods are widely used to solve the parabolic partial differential equations with Dirichlet, Neumann, or mixed boundary conditions. Modified implicit prediction (MIP) algorithm is unconditionally stable domain decomposition method. In this paper, the SOR iterative technique is applied to the MIP algorithm and the optimum over-relaxation parameter is provided.

Let (SDΩ) be the Stokes operator defined in a bounded domain Ω of R with Dirichlet boundary conditions. We first prove that, generically with respect to the domain Ω, all the eigenvalues of (SDΩ) are simple. That answers positively a question raised by J. H. Ortega and E. Zuazua in [18, Section 6]. Our second result states that, generically with respect to the domain Ω, the spectrum of (SDΩ) ve...

Quantum graphs having one cycle are considered. It is shown that if the cycle contains at least three vertices, then the potential on the graph can be uniquely reconstructed from the corresponding Titchmarsh-Weyl function (Dirichlet-to-Neumann map) associated with graph’s boundary, provided certain non-resonant conditions are satisfied. PACS numbers: 03.65.Nk, 73.63.-b, 85.35.-p.

We consider a general formulation for shape optimization problems involving the eigenvalues of the Laplace operator. Both the cases of Dirichlet and Neumann conditions on the free boundary are studied. We survey the most recent results concerning the existence of optimal domains, and list some conjectures and open problems. Some open problems are supported by efficient numerical computations.

We consider an operator ∆ + A(x) · D + q(x) with the Navier boundary conditions on a bounded domain in R, n ≥ 3. We show that a first order perturbation A(x) ·D+ q can be determined uniquely by measuring the Dirichlet–to–Neumann map on possibly very small subsets of the boundary of the domain. Notice that the corresponding result does not hold in general for a first order perturbation of the La...