Dirichlet‐to‐Neumann Map for a Nonlinear Diffusion Equation

A nonlinear diffusive equation with moving boundaries is analyzed by constructing the corresponding Dirichlet‐to‐Neumann map. In particular, the Dirichlet boundary value and the initial condition are used to derive the unknown Neumann boundary value. Then, a contraction‐mapping technique is used to prove existence and uniqueness of the solution for small times.     

On the Global Relation and the Dirichlet‐to‐Neumann Correspondence

The recently developed Fokas method for solving two‐dimensional Boundary Value Problems (BVP) via the use of global relations is utilized to solve axisymmetric problems in three dimensions. In particular, novel integral representations for the interior and exterior Dirichlet and Neumann problems for the sphere are derived, which recover and improve the already known solutions of these problems. The BVPs considered in this paper can be classically solved using either the finite Legendre transform or the Mellin‐sine transform (which can be derived from the classical Mellin transform in a way similar to the way that the sine transform can be derived from the Fourier transform). The Legendre transform representation is uniformly convergent at the boundary, but it involves a series that is not useful for many applications. The Mellin‐sine transform involves of course an integral but it is not uniformly convergent at the boundary. In this paper: (a) The Legendre transform representation is rederived in a simpler approach using algebraic manipulations instead of solving ODEs. (b) An integral representation, different that the Mellin‐sine transform representation is derived which is uniformly convergent at the boundary. Furthermore, the derivation of the Fokas approach involves only algebraic manipulations, instead of solving an ordinary differential equation.     

Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator

In this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary conditions on a Banach space X and its associated (abstract) Dirichlet‐to‐Neumann operator N acting on a “boundary” space . Our approach is based on similarity transformations and perturbation arguments and allows to split A into an operator A₀₀ with Dirichlet‐type boundary conditions on a space X₀ of states having “zero trace” and the operator N. If A₀₀ generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that A generates an analytic semigroup on X if and only if N does so on . Here we assume that the (abstract) “trace” operator is bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.     

Determination of a time‐ and space‐dependent heat flux relaxation function by means of a restricted Dirichlet‐to‐Neumann operator

We consider an inverse problem to recover a space‐ and time‐dependent relaxation function of heat flux in a three‐dimensional body on the basis of the restriction of the Dirichlet‐to‐Neumann operator of the related equation of heat flow onto a set of Dirichlet data of the form of a product of a fixed time‐dependent coefficient and a free space‐dependent function. Uniqueness of the solution of this inverse problem is proved. Copyright © 2004 John Wiley & Sons, Ltd.     

Positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions

In this paper, we investigate the existence of positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions involving Sobolev‐Hardy critical exponents and Hardy terms by using the concentration compactness principle, the strong maximum principle and the Mountain Pass lemma. We also prove, under complementary conditions, that there is no nontrivial solution if the domain is star‐shaped with respect to the origin.     

A Hardy inequality in a twisted Dirichlet–Neumann waveguide

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 Kříž [5]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Initial‐to‐Interface Maps for the Heat Equation on Composite Domains

A map from the initial conditions to the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with n interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed‐form solution to the interface problem.     

The Klein–Gordon Equation in a Domain with Time‐Dependent Boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

The Large‐Time Solution of Burgers' Equation with Time‐ Dependent Coefficients. II. Algebraic Coefficients

In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients where x and t represent dimensionless distance and time, respectively, while , are given continuous functions of t ( > 0). In particular, we consider the case when the initial data has algebraic decay as , with as and as . The constant states and are problem parameters. We focus attention on the case when (with ) and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to the initial‐value problem over all parameter values.     

The Inhomogeneous Neumann Problem in Lipschitz Domains

Abstract

Existence and uniqueness is established for the solution to the inhomogeneous Neumann problem for Laplace's equation in Lipschitz domains with data in L ^P Sobolev spaces.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

Remarks on Nonlinear Neumann Problems in Periodic Domains

We study a semilinear elliptic equation Au = f(x, u) with nonlinear Neumann boundary condition Bu = φ(ξ, u) in an unbounded domain Ω ? ?ⁿ, the boundary of which is defined by periodic functions. We assume that f and φ and the coefficients of the operators are asymptotically periodic in the space variables. Our main result states the existence of an asymptotically decaying, nontrivial solution of this problem with minimal energy. The proof is based on the concentration-compactness principle.     

The Fokas Method: The Dirichlet to Neumann Map for the Sine‐Gordon Equation

We study initial boundary value problems for the sine‐Gordon equation on the half‐line via the Fokas method, known as an extension of the inverse scattering transform. The method is based on the simultaneous analysis of both parts of the Lax pair and the global algebraic relation that couples known and unknown boundary values. One of most difficult steps of the method is to characterize the unknown boundary values that appear in the spectral functions. We derive the Dirichlet to Neumann map by using the global relation and the asymptotics of the eigenfunctions. Furthermore, employing perturbation expansion, we present an effective characterizations of the unknown boundary value in terms of the given initial and boundary values, and we then derive the first few terms of the expansions of the Neumann boundary value up to the third order.     

Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star‐shaped domain in ?^d, d ≥ 2. Conventional boundary‐based methods require a root search in eigenfrequency k, hence take O(N³) effort per eigenpair found, where N = O(k^d?1) is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to k the spectrum of a weighted interior Neumann‐to‐Dirichlet (NtD) operator for the Helmholtz equation. Approximations to the square roots k_j of all O(N) eigenvalues lying in [k ? ?, k], where ? = O(1), are found with O(N³) effort. We prove an error estimate with C independent of k. We present a higher‐order variant with eigenvalue error scaling empirically as O(?⁵) and eigenfunction error as O(?³), the former improving upon the “scaling method” of Vergini and Saraceno. For planar domains (d = 2), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For d = 2 we compute robustly the spectrum of the NtD operator via potential theory, Nyström discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error 10^?10, we show that the method is 10³ times faster than standard ones based upon a root search. © 2014 Wiley Periodicals, Inc.     

Dynamics of Vortex Rossby Waves in Tropical Cyclones,Part 1: Linear Time‐Dependent Evolution on an f‐Plane

Analyses of observational data on hurricanes in the tropical atmosphere indicate the existence of spiral rainbands which propagate outward from the eye and affect the structure and intensity of the hurricane. These disturbances may be described as vortex Rossby waves. This paper describes the evolution of barotropic vortex Rossby waves in a cyclonic vortex in a two‐dimensional configuration where the variation of the Coriolis force with latitude is ignored. The waves are forced by a constant‐amplitude boundary condition at a fixed radius from the center of the vortex and propagate outward. The mean flow angular velocity profile is taken to be a quadratic function of the radial distance from the center of the vortex and there is a critical radius at which it is equal to the phase speed of the waves. For the case of waves with steady amplitude, an exact solution is derived for the steady linearized equations in terms of hypergeometric functions; this solution is valid in the outer region away from the critical radius. For the case of waves with time‐dependent amplitude, asymptotic solutions of the linearized equations, valid for late time, are obtained in the outer and inner regions. It is found that there are strong qualitative similarities between the conclusions on the evolution of the vortex waves in this configuration and those obtained in the case of Rossby waves in a rectangular configuration where the latitudinal gradient of the Coriolis parameter is taken into account. In particular, the amplitude of the steady‐state outer solution is greatly attenuated and there is a phase change of across the critical radius, and in the linear time‐dependent configuration, the outer solution approaches a steady state in the limit of infinite time, while the amplitude of the inner solution grows on a logarithmic time scale and the width of the critical layer approaches zero.     

Complex Powers of the Neumann Laplacian in Lipschitz Domains

Let L^q_r(Ω) be the usual scale of Sobolev spaces and let Δ_N be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does map isomorphically onto L^q_r(Ω)/ℝ?     

The Large‐Time Solution of Burgers' Equation with Time‐Dependent Coefficients. I. The Coefficients Are Exponential Functions

In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients where x and t represent dimensionless distance and time, respectively, and , are given functions of t. In particular, we consider the case when the initial data have algebraic decay as , with as and as . The constant states and are problem parameters. Two specific initial‐value problems are considered. In initial‐value problem 1 we consider the case when and , while in initial‐value problem 2 we consider the case when and . The method of matched asymptotic coordinate expansions is used to obtain the large‐t asymptotic structure of the solution to both initial‐value problems over all parameter values.     

Neumann Problem in Domains with Outlets of Bounded Diameter

We study the Neumann problem for the Laplace equation in domains having quasicylindrical outlets. The aim is to collect and sort known results spread over the literature and to fill in the gaps of the theory which still exist. Namley, solutions with finite Dirichlet integral are found applying on the one hand the Helmholtz decomposition and on the other hand the Riesz representation theorem. We investigate generalizations for the non-Hilbert space setting and collect several types of Saint Venant estimates.