Perturbing singular solutions of the Gelfand problem

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

Dirichlet problems for Poisson equation and a second order linear equation are studied in the unit ball by using an integral representation formula with respect to the Laplacian in the complex Clifford algebra \(\mathbb {C}_m\) for m ≥ 3. Iterating the Green type kernel function, representation of the solution of the bi-Poisson equation with homogeneous Dirichlet condition is presented.

The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L 1 -norm control cost with a weight γ > 0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L 2 -norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L 1 -norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L 2 -sense.

Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation

We consider a class of elliptic-hyperbolic degenerate equations g(u) - Delta b(u) + div phi (u) = f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)(t) - Delta b(u) + div phi (u) = f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function phi satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for phi continuous.

Analytic solutions for unsteady flows over a superhydrophobic surface

The hydrodynamic impact problem is investigated within the framework of potential-flow theory. The vertical load acting on the rigid body is derived based on either momentum or energy conservation, and using the concept of added mass together with a homogeneous Dirichlet condition for the potential on the free surface as usually done to model an impact problem. It is demonstrated that the use of this simplified dynamic free-surface condition, instead of the fully nonlinear one, has a direct influence on the computation of the loads. In particular, the equivalence of momentum and energy analysis is in general not recovered. The situation is then highlighted by performing an asymptotic analysis of the two-dimensional blunt-body asymmetric impact problem. The asymptotic solution is given explicitly and validated through comparisons with experimental results. The energy distribution is then studied. It is shown that the contradiction between momentum and energy analysis can be removed, provided that the flux of energy through the jets is taken into account in the energy balance. If the simplified free-surface condition is indeed valid in the far-field, nonlinear terms must be retained near the body, in the spray-root domains. To leading order, the energy distribution during the gravity-free inertia stage does not depend on the blunt-body shape. The general analysis based on momentum or energy conservation suggests that this result also applies for arbitrary body shape as soon as a homogeneous Dirichlet condition can be applied as a dynamical free-surface boundary condition. In this case, and for a constant vertical impact velocity, half the work performed by the body would seem to be transferred to the fluid as kinetic energy within the spray.

The mathematical design of a realistic three-position quantum switch controlled by the classical electric field is suggested in form of a circular quantum well — a unit disc on a plane with four straight channels attached to it. This device implements a triple splitting of an input waveguide. The magnitude of the constant electric field directed parallel to the disc may be defined such that rotation of this field in the plane of the device permits manipulation of the electron current through the triple splitting. The problem of calculating of the current through the switch is reduced to the construction of scattered waves for the Schrödinger operator on the corresponding composite domain with the homogeneous Dirichlet conditions on the boundary. The Dirichlet boundary conditions are found to correspond most closely to real experimental conditions on the boundary of a deep quantum well. Explicit expression for transmission coefficient from one channel to another is obtained.Technically the analysis of the corresponding infinitely-dimensional spectral problem is reduced to the analysis of a relevant finite-dimensional analytic matrix function. We estimate the errors that arise from replacement of infinite-dimensional operator by the finite matrix. Our main result is the calculation of the working point of the switch in the multi-dimensional space of the numerical parameters of the switch which permits the resonance manipulation of the current.KeywordsDirichlet Boundary ConditionScattered WaveBasic DomainConstant Electric FieldHomogeneous Dirichlet Boundary ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let us denote by (λk ) the eigenvalues of the operator −∆ with homogeneous Dirichlet condition. First studied in [1] when β < λ1 < α < λ2, this problem has been widely investigated in the case that some eigenvalue λk belongs to the interval ]β, α[ (see e.g. [14], [13], [10] and references therein). The methods used are, often, a combination of topological and variational techniques. In this paper, we will be concerned with a jumping problem associated to a quasilinear elliptic equation of the form − n ∑

Analysis of the loss of boundary conditions for the diffusive Hamilton–Jacobi equation

We consider boundary value problems for operators Δ and Δ2 in periodically perforated domains with homogeneous Dirichlet conditions on the boundaries of the holes. The period of perforation and the “size” of the hole with respect to the period of perforation are regarded as two small parameters. We study asymptotic behavior of solutions, eigenvalues, and eigenfunctions for boundary value problems, under various assumptions on the relation between the two parameters. Bibliography: 13 titles.

A parabolic obstacle-type problem without sign restriction on the solution is considered. An exact representation of the global solutions (i.e., solutions in the entire half- space {(x,t) ∈ R n+1 |x 1 > 0}) is found. Finally, the local properties of the free boundary near a fixed boundary is studied. Under the homogeneous Dirichlet condition on the given boundary, the smoothness of the free one is proved.

An irregular grid for the numerical solution of linear elliptic partial differential equations

We consider the boundary value problem for the Laplace operator in a two-dimensional domain which is partially perforated along the boundary. The homogeneous Neumann condition is imposed on the outer boundary, whereas the homogeneous Dirichlet condition is stated on the boundary of small cavities. We construct two-term asymptotics with respect to a small parameter for eigenvalues of this boundary value problem converging to a multiple eigenvalue of the homogenized (limit) problem. We obtain conditions under which a multiple eigenvalue of the homogenized problem splits into simple ones. We also obtain the limit of the corresponding eigenfunctions.

On the Fučik spectrum of the wave operator and an asymptotically linear problem