In this paper we consider, in a bounded and smooth domain, the wave equation with delay term boundary condition. We also consider delayed velocity mixed Dirichlet–Neumann both cases, under suitable assumptions, prove exponential stability of solution. These results are obtained by introducing energies using some observability inequalities. If one above assumptions is not satisfied, instability given constructing sequences delays for which energy solutions does tend to zero.

We investigate time harmonic Maxwell equations in heterogeneous media, where the permeability μ and permittivity ε are piecewise constant. The associated boundary value problem can be interpreted as a transmission problem. In very natural way interfaces have edges corners. give detailed description of edge corner singularities electromagnetic fields.

We consider the wave equation in a bounded region with smooth boundary distributed delay on or into domain. In both cases, under suitable assumptions, we prove exponential stability of solution. These results are obtained by introducing energies and proving some observability inequalities. For an internal delay, further show instability results.

Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in boundary conditions. The function admitted be an <em>a priori</em> given upper bound on its derivative, which less than $1$. Sufficient explicit conditions are derived that guarantee exponential stability. Moreover decay rate can explicitly computed if data given.

Abstract We study transmission problems for elliptic operators of order 2 m with general boundary and interface conditions, introducing new covering conditions. This allows to prove solvability, regularity asymptotics solutions in weighted Sobolev spaces. give some numerical examples the location singular exponents.

In this paper we consider the wave equation on 1-d networks with a delay term in boundary and/or transmission conditions. We first show well posedness of problem and decay an appropriate energy. give necessary sufficient condition that guarantees to zero further conditions lead exponential or polynomial stability solution. Some examples are also given.

We consider the wave equation with a time-varying delay term inthe boundary condition in bounded and smooth domain $\Omega\subset\RR^n.$ Under suitableassumptions, we prove exponential stability of solution.These results are obtained by introducing suitable energies Lyapunov functionals. Such analysis is also extended to nonlinear versionof model.

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains corners and edges. First, the anisotropic singular behaviour of solution described. Then method anisotropic, graded meshes piecewise linear shape functions investigated such problems; schemes exhibit optimal convergence rates decreasing mesh size. For proof, new local interpolation error estimates from anisotropically weighted spaces are derived. Finally, numerical...

We consider abstract second order evolution equations with unbounded feedback delay. Existence results are obtained under some realistic assumptions. Sufficient and explicit conditions derived that guarantee the exponential or polynomial stability. Some new examples enter into our framework presented.

Abstract We analyse the convergence of finite element discretizations time-harmonic wave propagation problems. propose a general methodology to derive stability conditions and error estimates that are explicit with respect wavenumber $k$. This is formally based on an expansion solution in powers $k$, which permits split into regular, but oscillating part, another component rough, behaves nicely when increases. The method developed its full generality illustrated by three particular cases:...

In the two first parts of this work [RAIRO Modél. Math. Anal. Numér., 24 (1990), pp. 27é52], 343–367] formulas giving coefficients arising in singular expansion solutions elliptic boundary value problems on nonsmooth domains are investigated. Now, for case homogeneous strongly operators with constant polygonal domains, solution such by finite element method is considered. order to approximate or coefficients, different methods used based expressions that were obtained parts; dual function...

We consider abstract second order evolution equations with unbounded feedback time-varying delay. Existence results are obtained under some realistic assumptions. prove the exponential decay conditions by introducing an Lyapunov functional. Our framework is applied to wave, beam, and plate boundary delays.