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This paper is concerned with the existence and qualitative property of solutions for a Hénon-like equationwhere Ω = {x ∈ ℝN : 1 < |x| < 3} with N ≥ 4, 2* = 2N/(N − 2), τ > 0 and ε > 0 is a small parameter. For any given k ∈ ℤ+, we construct positive solutions concentrating simultaneously at 2k different points for ε sufficiently small, among which k points are near the interior boundary {x ∈ ℝN : |x| = 1} and the other k points are near the outward boundary {x ∈ ℝN : |x| = 3}. Moreover, the 2k points tend to the boundary of Ω as ε goes to 0.

The aim of this paper is to study a boundary time-optimal control problem for the heat equation in a two-dimensional ball. The main ingredient is the extension of a result concerning Müntz polynomials due to Borwein and Erdélyi that allows us to prove an observability inequality for the dynamical system's truncation to a finite number of modes. This result, combined with a well-known Lebeau–Robbiano argument used to show the null-controllability of parabolic type equations, enables us to deduce the existence, uniqueness and bang-bang properties for the boundary time-optimal control.

This paper is devoted to studying the asymptotic behaviour of a sequence of elliptic systems posed in a sequence of rough domains Ωn. The solutions un are assumed to satisfy un(x) ϵ Vn(x), where Vn(x) is a vectorial space depending on $\smash{x\in\bar\varOmega_n}$. This enables one to consider several types of boundary conditions posed in variable sets of the boundary. For some choices of the vectorial spaces Vn(x), our study provides, in particular, some classical results for the homogenization of Dirichlet elliptic problems in varying domains.

Dirichlet–Neumann operators arise in many applications in the sciences, and this has inspired a number of studies on their analytical properties. In this paper we further investigate the analyticity properties of Dirichlet–Neumann operators as functions of the boundary shape. In particular, we study the size of the disc of convergence of their Taylor-series representation. For this we use a complexification technique which requires a novel reformulation of the problem, coupled with methods for systems of elliptic partial differential equations. Numerical results to illustrate our theoretical conclusions are presented.

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

SynopsisIn this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the gratingwith respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables (x, y, δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

SynopsisIn this paper it is shown that the analysis of Titchmarsh's book [32] for regular Sturm-Liouville problems on a finite closed interval carries over readily to regular problems involving the eigenvalue parameter in the boundary condition at one end-point. The manner in which this type of problem is associated with a self-adjoint operator in Hilbert space has recently been pointed out by Walter in [36], and his operator-theoretic formulation is adopted here. The use of the eigenfunction expansion is illustrated by applying it to solve a heat-conduction problem for a solid in contact with a fluid.

SynopsisThe paper considers a linear non-homogeneous boundary value problem for a class of neutral type functional differential equations. A necessary and sufficient condition for the existence of a unique solution of that problem is obtained.

SynopsisThis paper is concerned with the existence of solutions of a two point boundary value problem for neutral functional differential equations. We consider the problemwhere M and N are n × n matrices. This is examined by using the “shooting method”. Also, an example is given to illustrate how our result can be applied to yield the existence of solutions of a periodic boundary value problem.

SynopsisThe paper deals with boundary value problems for second-order vector differential equations x″ = f (t, x, x′). Given a region Ω in (t, x)-space we ask whether there exists a solution x(t) of the problem satisfying (t, x(t)) ∊Ω. We arrive at a rather general type of conditions which are sufficient in order that Ω has the desired property. One of these conditions is geometric in nature and depends upon the boundary data only. The second condition can be expressed in terms of inequalities and depends upon the values of f on ∂Ω. These inequalities turn out to be the common background of a variety of conditions which can be found in the literature on boundary value problems and which in the case of a scalar equation reduce to the well-known properties of upper and lower solutions.

SynopsisThe present paper is concerned with the problem of regularity of weak solutions of boundary value problems. We shall present a new method to prove differentiability on the boundary. This method was developed in our thesis [12] within the theory of abstract Sobolev spaces, introduced by Stummel [14]. Here, we shall describe it by applying it to elliptic boundary value problems. It will be seen that the advantage of this method consists in the fact that it is based on functional analysis only and therefore may be used for other types of differential equations as well.