We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infinite strips. As an application, we derive certain stability of the spectrum for the Laplacian in Dirichlet-Neumann strips along a class of curves of sign-changing curvature, improving in this way an initial result of Dittrich and Kriz.
We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that the projections on each transverse mode generically give rise to decoupling conditions between the edges of the graph while exceptionally a coupling can occur. The non decoupling conditions are related to the existence of resonances at the thresholds of the continuum spectrum.
We consider a magnetic Schroedinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for…
We investigate spectral properties of a quantum particle confined to an infinite straight planar strip by imposing Robin boundary conditions with variable coupling. Assuming that the coupling function tends to a constant at infinity, we localize the essential spectrum and derive a sufficient condition which guarantees the existence of bound states. Further properties of the associated eigenvalues and eigenfunctions are studied numerically by the mode-matching technique.
We consider the Laplacian in a tubular neighbourhood of a hyperplane subjected to non-self-adjoint $\mathcal{PT}$-symmetric Robin boundary conditions. Its spectrum is found to be purely essential and real for constant boundary conditions. The influence of the perturbation in the boundary conditions on the threshold of the essential spectrum is studied using the Birman-Schwinger principle. Our aim is to derive a sufficient condition for existence, uniqueness and reality of discrete eigenvalues. We show that discrete spectrum exists when the perturbation acts in the mean against the unperturbed …
We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive …
We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distr…
We consider the Laplacian in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to a Hardy inequality for the Laplacian. As a byproduct of our method, we obtain a simple proof of a theorem of Dittrich and Kriz.
We consider the Laplace operator in a thin three dimensional tube with a Robin type condition on its boundary and study, asymptotically, the spectrum of such operator as the diameter of the tube's cross section becomes infinitesimal. In contrast with the Dirichlet condition case, we evidence different behaviors depending on a symmetry criterium for the fundamental mode in the cross section. If that symmetry condition fails, then we prove the localization of lower energy levels in the vicinity of the minimum point of a suitable function on the tube's axis depending on the curvature and the rota…
The purpose of this paper is the mathematical analysis of the weak solution of the mixed Dirichlet-Robin boundary value problem for the nonlinear Darcy-Forchheimer- Brinkman system in a bounded, two-dimensional Lipschitz domain, and the application of the corresponding results to the study of the lid-driven flow problem of an incompressible viscous fluid located in a square cavity filled with a porous medium. First we obtain a well-posedness result for the linear Brinkman system with Dirichlet-Neumann boundary conditions, employing a variational approach for the corresponding boundary integral…
We consider a boundary value problem for a general second order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries of the cavities are subject either to a Dirichlet or a nonlinear Robin condition. On the perforation, certain rather weak conditions are imposed to ensure that under the homogenization we obtain a similar problem in a non-perforated domain with an additional potential in the equation usually called a strange term. Our main results state the convergence of th…
Quantum waveguide with the shape of planar infinite straight strip and combined Dirichlet and Neumann boundary conditions on the opposite half-lines of the boundary is considered. The absence of the point as well as of the singular continuous spectrum is proved.
A simple transformation converts a solution of a partial differential equation with a Dirichlet boundary condition to a function satisfying a Robin (generalized Neumann) condition. In the simplest cases this observation enables the exact construction of the Green functions for the wave, heat, and Schrodinger problems with a Robin boundary condition. The resulting physical picture is that the field can exchange energy with the boundary, and a delayed reflection from the boundary results. In more general situations the method allows at least approximate and local construction of the appropriate …
This works deals with one dimensional infinite perturbation - namely line defects - in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann (DtN) approach, we show that this problem is equivalent to one set on a small neighborhood…
Consider the Dirichlet Laplacian operator $-Δ^D$ in a periodic waveguide $Ω$. On the condition that $Ω$ is sufficiently thin, we show that its spectrum $σ(-Δ^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $σ(-Δ^D)$ and locate it.
We describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric graph.
The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N-1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly general hypotheses, including the special case where the obstacles consist of line segments placed parallel to the waveguide walls. This work should be viewed as an extension of "Periodic structures on wav…
In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of $δ$-type boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.
We consider the heat equation in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to an improvement of the decay rate of the heat semigroup of the order of $t^{-1/2}$. The proof employs similarity variables that lead to a non-autonomous parabolic equation in a thin strip contracting to the real line, that can be analyzed on weighted Sobolev spaces in which the operators under consideration have discrete spectra. A careful analysis of its asymptotic behaviour shows that an added Dirichlet b…
We consider the Dirichlet Laplacian with uniform magnetic field on a curved strip in two dimensions. We give a sufficient condition ensuring the existence of the discrete spectrum in the strong magnetic field limit.
We consider a periodic strip in the plane and the associated quantum waveguide with Dirichlet boundary conditions. We analyse finite segments of the waveguide consisting of $L$ periodicity cells, equipped with periodic boundary conditions at the ``new'' boundaries. Our main result is that the distance between the first and second eigenvalue of such a finite segment behaves like $L^{-2}$.
In the junction $Ω$ of several semi-infinite cylindrical waveguides we consider the Dirichlet Laplacian whose continuous spectrum is the ray $[λ_\dagger, +\infty)$ with a positive cut-off value $λ_\dagger$. We give two different criteria for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation $-Δu=λ_\dagger u$ in $Ω$. The first criterion is quite simple and is convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreove…
The heat trace asymptotics are discussed for operators of Laplace type with Dirichlet, Robin, spectral, D/N, and transmittal boundary conditions. The heat content asymptotics are discussed for operators with time dependent coefficients and Dirichlet or Robin boundary conditions.
We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critic…
Spatial cruciform quantum waveguides (the Dirichlet problem for Laplace operator) are constructed such that the total multiplicity of the discrete spectrum exceeds any preassigned number.
This paper is concerned with the study of theexistence/non-existence of the discrete spectrum of the Laplaceoperator on a domain of $\mathbb R ^3$ which consists in atwisted tube. This operator is defined by means of mixed boundaryconditions. Here we impose Neumann Boundary conditions on abounded open subset of the boundary of the domain (the Neumannwindow) and Dirichlet boundary conditions elsewhere.
The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin…
We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_\circ \ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_\circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichle…
We consider the Dirichlet Laplacian in a waveguide of uniform width and infinite length which is ideally divided into three parts: a "vertex region", compactly supported and with non zero curvature, and two "edge regions" which are semi-infinite straight strips. We make the waveguide collapse onto a graph by squeezing the edge regions to half-lines and the vertex region to a point. In a setting in which the ratio between the width of the waveguide and the longitudinal extension of the vertex region goes to zero, we prove the convergence of the operator to a selfadjoint realization of the Lapla…
For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expre…
We consider a planar waveguide with combined Dirichlet and Neumann conditions imposed in a "twisted" way. We study the discrete spectrum and describe it dependence on the configuration of the boundary conditions. In particular, we show that in certain cases the model can have discrete eigenvalues emerging from the threshold of the essential spectrum. We give a criterium for their existence and construct them as convergent holomorphic series.
We consider regular and singular perturbations of the Dirichlet and Neumann boundary value problems for the Helmholtz equation in $n$-dimensional cylinders. Existence of eigenvalues and their asymptotics are studied.
We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to…
One dimensional Stefan problems for a semi-infinite material with temperature dependent thermal coefficients are considered. Existence and uniqueness of solution are obtained imposing a Dirichlet or a Robin type condition at fixed face $x=0$. Moreover, it is proved that the solution of the problem with the Robin type condition converges to the solution of the problem with the Dirichlet condition at the fixed face. Computational examples are provided.
We establish various Hardy-type inequalities for the Dirichlet Laplacian in perturbed periodically twisted tubes of non-circular cross-sections. We also state conjectures about the existence of such inequalities in more general regimes, which we support by heuristic and numerical arguments.
We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results …
This paper studies the third boundary problem of the Laplace equation with azimuthal symmetry.Many solutions of the boundary value problems in spherical coordinates are available in the form of infinite series of Legendre polynomials but the evaluation of the summing series shows many computational difficulties. Integral transform is a challenge as it involves an inverse Legendre transform. Here, the closed-form solution of the Laplace equation with the Robin boundary conditions on a sphere is solved by the Legendre transform. This analytical solution is expressed with the Appell hypergeometri…
The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.
We introduce a planar waveguide of constant width with non-Hermitian PT-symmetric Robin boundary conditions. We study the spectrum of this system in the regime when the boundary coupling function is a compactly supported perturbation of a homogeneous coupling. We prove that the essential spectrum is positive and independent of such perturbation, and that the residual spectrum is empty. Assuming that the perturbation is small in the supremum norm, we show that it gives rise to real weakly-coupled eigenvalues converging to the threshold of the essential spectrum. We derive sufficient conditions …
We construct multi-soliton solutions of the n-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the approach of dressing the integrable BCs: soliton solutions are generated in preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes d…
In this paper, a one-phase Stefan-type problem for a semi-infinite material which has as its main feature a variable latent heat that depends on the power of the position and the velocity of the moving boundary is studied. Exact solutions of similarity type are obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face. Required relationships between data are presented in order that these problems become equivalent to the problem where a Dirichlet condition at the fixed face is considered. Moreover, in the case where a Robin condition is prescribed, the limi…
We introduce an analog model for the conformally coupled scalar field on the BTZ black hole. The model is based on the propagation of acoustic waves in a Laval nozzle. Since the BTZ black hole is not a globally hyperbolic spacetime, the dynamics of the scalar field is not well defined until extra boundary conditions are prescribed at its spatial infinity. We show that quasinormal modes (QNMs) satisfying Dirichlet, Neumann, and Robin boundary conditions in the BTZ black hole can be interpreted in terms of ordinary QNMs defined with respect to an appropriately extended nozzle. We also discuss th…
Consider the Laplacian in a straight planar strip of width $\,d\,$, with the Neumann boundary condition at a segment of length $\,2a\,$ of one of the boundaries, and Dirichlet otherwise. For small enough $\,a\,$ this operator has a single eigenvalue $\,ε(a)\,$; we show that there are positive $\,c_1,c_2\,$ such that $\,-c_1 a^4 \le ε(a)- \left(π/ d\right)^2 \le -c_2 a^4\,$. An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length $\,2a\,$ in the common boundary.
The resonances for the Dirichlet and Neumann Laplacian are studied on compactly perturbed waveguides. An upper bound on the number of resonances near the physical plane is proven. In the absence of resonances, an upper bound is proven for the localised resolvent. This is then used to prove that the existence of a quasimode whose asymptotics is bounded away from the thresholds implies the existence of resonances converging to the real axis.
We rigorously derive non-equilibrium space-time fluctuation for the particle density of a system of reflected diffusions in bounded Lipschitz domains in $\mathbb R^d$. The particles are independent and are killed by a time-dependent potential which is asymptotically proportional to the boundary local time. We generalize the functional analytic framework introduced by Kotelenez [19, 20] to deal with time-dependent perturbations. Our proof relies on Dirichlet form method rather than the machineries derived from Kotelenez's sub-martingale inequality. Our result holds for any symmetric reflected d…
A numerical study of the nonlinear Schrödinger (NLS) equation subject to homogeneous Dirichlet, Neumann and Robin boundary conditions in the finite line is presented. The results are compared with both the exact analytical ones for the initial-value problem (IVP) of the NLS equation and the numerical ones for periodic boundary conditions. It is shown that initial solutions obtained by truncating the exact N-soliton solution of the IVP of the NLS equation into a finite interval develop solitary waves that behave as solitons, even after collisions with the boundaries. For periodic and homogeneou…
