Dirichlet Boundary Problem Research Articles

THE INITIAL BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR SCHRÖDINGER-POISSON EQUATIONS

In this article, the author studies the initial-(Dirichlet) boundary problem for a high-field version of the Schrödinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schrödinger equations on the unit cube. A global existence and uniqueness is established for a solution to this problem.

Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in the multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed.

The Benney hierarchy and the Dirichlet boundary problem in two dimensions

A theoretical connection between reductions of the Benney hierarchy and the Dirichlet problem for Laplace's equation in the plane is made. The connection is used to deduce general formulas for the uniformizations of two spectral functions associated with N-parameter reductions of the hierarchy. Two types of reduction are considered: one type has been considered by previous authors using alternative arguments, the second type is new. The formulas are general and are expressed in terms of the modified Green's function (for Laplace's equation) in arbitrary N-connected, reflectionally-symmetric, planar domains. The Benney moments are found to be purely geometrical quantities associated with these domains.

The Schrödinger operator in a periodic waveguide on a plane and quasiconformal mappings

We consider the magnetic Schrödinger operator with a variable metric in a two-dimensional simply connected periodic waveguide. All the coefficients are assumed to be periodic along the waveguide. We investigate the Dirichlet and Neumann boundary problems, as well as the boundary problem of the third type. Under wide conditions on the boundary of the waveguide providing a band structure of the spectrum, we prove the absolute continuity of the spectrum. Bibliography: 16 titles.

The Dirichlet boundary problem for a nonlocal Cahn–Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution for a nonlocal Cahn–Hilliard equation with Dirichlet boundary condition on a bounded domain. Under a nondegeneracy assumption the solutions are classical but when this is relaxed, the equation is satisfied in a weak sense. Also we prove that there exists a global attractor in some metric space.

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.

Square-function estimates for singular integrals and applications to partial differential equations

The purpose of the present paper is to continue the program of study of elliptic boundary-value problems on Lipschitz domains with boundary data in quasi-Banach Besov spaces $B_s^{p,p}(\partial \Omega)$, initiated in [13]. Introducing a modified square function which is well-adapted for handling data with a fractional amount of smoothness, we establish the well-posedness of the Dirichlet and Neumann boundary problems for the Laplacian in Lipschitz domains, for a range of indices which includes values of $p$ less than $1$. An important ingredient in this regard is establishing suitable square-function estimates for a singular integral of potential type.

The fluid flow due to two submerged sinks in a two layer stratified fluid

A boundary integral technique is developed to study the free surface flow of a steady, two‐dimensional, incompressible, irrotational and inviscid fluid flow which is produced by two submerged sinks (or sources) which are of equal strength, placed along a solid horizontal boundary with a stagnation point on the free surface in a two layer stratified fluid in the presence of gravity. A special form of the Riemann‐Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing non‐linear boundary integral‐differential equations which are solved for the fluid velocity on the free surface and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various values of the Froude number and sink locations on the solid horizontal boundary and we have also studied the largest value of the Froude number for which no convergent solutions are possible, namely the critical Froude number. We have found that the free surface profile is dependent on two parameters, namely the Froude number on the free surface and the non‐dimensional distance between the two sinks.

Comments on ?Solution of the Dirichlet and Stokes exterior boundary problems for the Earth's ellipsoid? by V.V. Brovar, Z.S. Kopeikina, M.V. Pavlova Journal of Geodesy (2001) 74: 767?772

Specific features of the Brovar-Kopeikina-Pavlova approach to solution of the Dirichlet and Stokes exterior boundary problems are discussed. The solution is based on the Molodensky method of determination of the exterior gravitational field and figure of the Earth without use of the Gauss-Listing geoid.

Gravitational fluid flow induced in a three layer stratified fluid by two submerged sinks or sources with stagnation points on the free surfaces

A boundary integral technique is developed to study the free surface flow of a steady, two-dimensional, incompressible, irrotational and inviscid fluid which is induced in both two and three layer stratified fluids in the presence of gravity by two submerged sinks or sources with stagnation points on the free surfaces. A special form of the Riemann–Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing nonlinear boundary integral–differential equations which have been solved for the fluid velocity on the free surfaces and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various Froude numbers, sink flow strengths and sink heights in both two and three layer fluids. Further, we have also studied the critical Froude numbers for which no convergent solutions are possible for any larger values of the Froude number. We have found that the free surfaces are dependent on six parameters, namely the Froude numbers, the flow strength of the sinks and the ratio of sink heights to the thickness of either the middle layer in a three layer system and the bottom layer in a two layer system.

Large scale correlations in normal non-Hermitian matrix ensembles

We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with a general non-Gaussian measure and in ensembles of general non-Hermitian matrices with a class of non-Gaussian measures. In both cases, the eigenvalues are complex and in the large N limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N, in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues.

Barotropic background currents in the Gulf of California

A field of barotropic background currents (BC) in the Gulf of California is constructed numerically. The definition of BC used here is based on suppositions about (i) the horizontal homogenization of potential vorticity in large-scale gyres, and (ii) the tendency of a current structure to evolve to a state with a minimum of system mechanical energy. In the barotropic case here, the numerical algorithm is reduced to the solution of the Dirichlet boundary problems for planetary, topographic and flowing integral streamfunction components separately. The results obtained are compared to observations and the results of other numerical models. The calculated barotropic BC realistically describe the main features of the circulation in the southern part of the Gulf of California, including its mouth. Cyclonic movement in the deep-water and anticyclonic movement near the coasts are observed. The numerical results for the central and northern parts of the gulf, where seasonal climatological differences are pronounced, coincide with observations in the upper layer only during the winter season; i.e., the circulation here is anticyclonic. Thus, there are reasons to suggest that mainly planetary-topographic forcing, especially in the southern part, forms the circulation in the lower layer. The seasonal variability of currents in the upper layer caused by non-stationary effects such as wind, heat flux and baroclinicity, is apparently in phase with these mechanisms during winter, and in antiphase during summer, particularly for the central and northern gulf.

Application of wavelets technique to the integral equations of the method of auxiliary currents

The wavelets technique is used to solve the 2D Dirichlet, Neumann or mixed boundary problems when scattering bodies are cylinders with elliptical or complicated cross-sections. The method utilizes the Haar's and linear B-spline wavelets functions and the method of moments to solve the Fredholm integral equation of the first kind with a smooth kernel. The practical application of the technique is illustrated by solving the scattering problems by cylinders with complicated cross-section in kD⪢1 region. We discuss the selection of the auxiliary contours as well as the stability and accuracy of the results.

EXISTENCE AND UNIQUENESS FOR A SEMILINEAR ELLIPTIC PROBLEM ON LIPSCHITZ DOMAINS IN RIEMANNIAN MANIFOLDS

ABSTRACT For a general class of semilinear equations in a Lipschitz domain , where M is a smooth compact Riemannian manifold, we establish the existence and uniqueness of solutions to Dirichlet and Neumann boundary problems. The main contribution of this paper is that we consider ‘rough’ boundary data that are typically just functions. Results of this type for the linear equation are typically obtained using singular integral techniques.

The flow induced in a three layer stratified fluid by a submerged sink or source with stagnation points on the free surfaces

A boundary integral technique is developed to study the free surface flow of a steady, two-dimensional, incompressible, irrotational and inviscid fluid which is induced in both two and three layer stratified fluids in the presence of gravity by a submerged sink or source with stagnation points on the free surfaces. A special form of the Riemann–Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing non-linear boundary integral–differential equations which have been solved for the fluid velocity on the free surfaces and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various Froude numbers and sink heights in both two and three layer fluids. Further, we have also studied the critical Froude numbers for which no convergent solutions are possible for any larger values of the Froude number. We have found that the free surfaces are dependent on two parameters, namely the Froude number and the ratio of sink height to the thickness of either the middle layer in a three layer system and the bottom layer in a two layer system.

Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model.

Removable singularities for quasilinear degenerate elliptic equations with absorption term

Let N≥1 and p>1. Let F be a compact set and Ω be a bounded open set of RN satisfying F⊂Ω⊂RN. We define a generalized p-harmonic operator Lp which is elliptic in Ω∖F and degenerated on F. We shall study the genuinely degenerate elliptic equations with absorption term. In connection with these equations we shall treat two topics in the present paper. Namely, the one is concerned with removable singularities of solutions and the other is the unique existence property of bounded solutions for the Dirichlet boundary problem.

Removable singularities for quasilinear degenerate elliptic equations with absorption term

Let N≥1 and p>1. Let F be a compact set and Ω be a bounded open set of RN satisfying F⊂Ω⊂RN. We define a generalized p-harmonic operator Lp which is elliptic in Ω∖F and degenerated on F. We shall study the genuinely degenerate elliptic equations with absorption term. In connection with these equations we shall treat two topics in the present paper. Namely, the one is concerned with removable singularities of solutions and the other is the unique existence property of bounded solutions for the Dirichlet boundary problem.

Spectral analysis of viscous static compressible fluid equilibria

It is generally assumed that the study of the spectrum of the linearized Navier-Stokes equations around a static state will provide information about the stability of the equilibrium. This is obvious for inviscid barotropic compressible fluids by the self-adjoint character of the relevant operator, and rather easy for viscous incompressible fluids by the compact character of the resolvent. The viscous compressible linearized system, both for periodic and homogeneous Dirichlet boundary problems, satisfies neither condition, but it does turn out to be the generator of an immediately continuous, almost stable semigroup, which justifies the analysis of the spectrum as predictive of the initial behaviour of the flow. As for the spectrum itself, except for a unique negative finite accumulation point, it is formed by eigenvalues with negative real part, and nonreal eigenvalues are confined to a certain bounded subset of complex numbers.

A note on finite difference discretizations for Poisson equation on a disk

AbstractA simple second‐order finite difference treatment of polar coordinate singularity for Poisson equation on a disk is presented. By manipulating the grid point locations, we can successfully avoid finding numerical boundary condition at the origin so that the resulting matrix is simpler than traditional schemes. The treatments for Dirichlet and Neumann boundary problems are slightly different by adjusting the radial mesh width. Furthermore, the present discretization can be applied with slight modifications to Helmholtz‐type equations. The numerical evidence shows that the second‐order accuracy can also be preserved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 199–203, 2001