Robin Boundary Value Problem Research Articles

Robin boundary value problems on arbitrary domains

We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establishLpL_p-LqL_q-estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.

On a class of Robin boundary value problems in physical geodesy

In physical geodesy one considers several (exterior) Robin boundary value problems for the Laplace equation in three dimensions. The ellipsoidal Stokes problem, which occurs in context of gravimetric determination of the geoid, belongs to this class. Up to now, this and related problems have been treated with high order series expansions of spherical and spheroidal harmonics. In this article we investigate the nullfield method for this class of boundary value problems. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. For the case that the underlying surface is a triaxial ellipsoid or an oblate spheroid, explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided.

The nullfield method for the ellipsoidal Stokes problem

The ellipsoidal Stokes problem is one of the basic boundary-value problems for the Laplace equation which arises in physical geodesy. Up to now, geodecists have treated this and related problems with high-order series expansions of spherical and spheroidal (ellipsoidal) harmonics. In view of increasing computational power and modern numerical techniques, boundary element methods have become more and more popular in the last decade. This article demonstrates and investigates the nullfield method for a class of Robin boundary-value problems. The ellipsoidal Stokes problem belongs to this class. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. Explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided.

A uniqueness theorem for a Robin boundary value problem of physical geodesy

We get a uniqueness theorem for a Robin type boundary value problem for the Laplace equation arising in Physical Geodesy in the context of the gravimetric determination of the geoid. The boundary is an oblate ellipsoid of revolution and we have uniqueness of solutions provided that its eccentricity is (approximately) less than 0.526428.

ON UNIFORMLY VALID ESTIMATE OF SOLUTIONS TO SINGULAR PERTURBATION ROBIN BOUNDARY VALUE PROBLEM

In this paper we study the Robin boundary value problem with a small parameter ɛy″=f(t, y, ω(ɛ)y′, ɛ),a0y(0)+b0y′(0)=ξ(ɛ), a1y(1)+b1y′(1)=η(ɛ),where the function ω(ε) is continuous on ε⩾0 with ω(0)=0. Assuming all known functions are suitably smooth, f satisfies Nagumo's condition, fy>0, a42+b42≠0, (-1)iaibi⩽0 (i=0,1) and the reduced equation 0 = f(t, y, 0, 0) has a solution y(t) (0⩽t⩽1), we prove the existence and the uniqueness of the solution for the boundary value problem and give an asymptotic expansion of the solution in the power ɛ12 which is uniformly valid on 0⩽t⩽1.

Modified Green's functions and the third boundary value problem for the Helmholtz equation

It is well known that the application of boundary integral equation methods to the exterior Dirichlet and Neumann boundary value problems for the Helmholtz equation results in Fredholm equations which fail to have unique solutions at eigenvalues of the interior adjoint problem. As has often been pointed out, the non-uniqueness is an artifact of the integral equation method since the exterior boundary value problems are themselves uniquely solvable. A number of devices have been proposed to produce solutions to these exterior boundary value problems using integral equations which do have unique solutions (see, for example, Brakhage and Werner [ 11. Burton and Miller 121, Jones [3], Kleinman and Roach [4], Leis 151, and Ursell 161). Recently Kleinman and Roach [7,8] have discussed modified Green’s functions for the Dirichlet and Neumann problems and have shown how, by appropriate choice of parameter, one may not only produce a modification of the kernel which is free from real eigenvalues but also is a best possible choice according to one of a number of specified criteria, e.g., closest in the L, operator norm to the true Green’s function. In this note we extend this analysis to the more complicated case of the Robin or third boundary value problem. In doing so we follow the analysis of our earlier paper [9] in which we treated this boundary value problem for generalized L, boundary data. We begin with a more precise description of the problem and a summary of the results of this latter paper which we will find useful.

An Asymptotic Theory for a Class of Nonlinear Robin Problems. II

Various asymptotic phenomena exhibited by solutions of singularly perturbed Robin boundary value problems are studied in the case when the right-hand side grows faster than the square of the derivative.

An asymptotic theory for a class of nonlinear Robin problems. II

Various asymptotic phenomena exhibited by solutions of singularly perturbed Robin boundary value problems are studied in the case when the right-hand side grows faster than the square of the derivative.