Solution of the Robin problem for the Laplace equation

For open sets with a piecewise smooth boundary it is shown that we can express a solution of the Robin problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series.     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

‐solution of the Neumann,Robin and transmission problem for the scalar Oseen equation

We find necessary and sufficient conditions for the existence of an ‐solution of the Neumann problem, the Robin problem and the transmission problem for the scalar Oseen equation in three‐dimensional open sets. As a consequence we study solutions of the generalized jump problem.     

The Robin problem for bending of elastic plates

The solution of the Robin problem in a finite domain for the system of equations modeling the bending of elastic plates with transverse shear deformation is approximated by means of a generalized Fourier series method closely connected to the structure of the boundary integral equation treatment of the problem. The theory is exemplified by numerical computation that shows a high degree of accuracy and efficiency.     

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two‐sided estimates for this term in a variety of situations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On solution strategies to Saint-Venant problem

Different solution strategies to the relaxed Saint-Venant problem are presented and comparatively discussed from a mechanical and computational point of view. Three approaches are considered; namely, the displacement approach, the mixed approach, and the modified potential stress approach. The different solution strategies lead to the formulation of two-dimensional Neumann and Dirichlet boundary-value problems. Several solution strategies are discussed in general, namely, the series approach, the reformulation of the boundary-value problems for the Laplace's equations as integral boundary equations, and the finite-element approach. In particular, the signatures of the finite-element weak solutions—the computational costs, the convergence, the accuracy—are discussed considering elastic cylinders whose cross sections are represented by piece-wise smooth domains.     

The Robin problem for the Laplace equation in a three-dimensional starlike domain

The internal and external Robin problems for the Laplace equation in bounded starlike domains are addressed. We show how to derive the relevant solutions by using a suitable Fourier series-like method. Numerical results are specifically obtained considering three-dimensional domains whose boundary is defined by a generalization of the so-called “superformula” introduced by Gielis. By using the computer algebra code Mathematica©, truncated series approximations of the solutions are determined. Our findings are in good agreement with the theoretical results on the Fourier series due to Carleson.     

Heat Content Asymptotics for Riemannian Manifolds with Zaremba Boundary Conditions

The existence of a full asymptotic expansion for the heat content asymptotics of an operator of Laplace type with classical Zaremba boundary conditions on a smooth manifold is established. The first three coefficients in this asymptotic expansion are determined in terms of geometric invariants; partial information is obtained about the fourth coefficient.      

Robin Boundary Value Problem for One-Dimensional Landau–Lifshitz Equations

In this paper, we are concerned with the existence and uniqueness of global smooth solution for the Robin boundary value problem of Landau-Lifshitz equations in one dimension when the boundary value depends on time t. Furthermore, by viscosity vanishing approach, we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.     

Determination of the Robin coefficient in a nonlinear boundary condition for a steady‐state problem

In this paper, a constant heat transfer coefficient present in a nonlinear Robin‐type boundary condition associated with an elliptic equation is reconstructed uniquely from a single boundary energy measurement. Two types of such boundary energy measurement are considered, and solvability theorems for the solution of the resulting nonlinear inverse problems are provided. Further, one‐dimensional numerical results are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

一类催化反应Robin问题的渐近解

研究了一类非线性催化反应微分方程Robin问题.在一定的条件下,先利用摄动方法求出了原Robin问题的外部解,然后用伸长变量和幂级数理论分别构造了解的第一和第二边界层校正项,从而得到了Robin问题解的形式渐近展开式.最后利用微分不等式理论,证明了问题解的渐近表示式的一致有效性.     

Best estimates of RBF-based meshless Galerkin methods for Dirichlet problem

In this paper, Galerkin methods based on the radial basis functions to deal with the partial differential equations are discussed. The best error estimates for this method are obtained.     

Reduced multiscale computation on adapted grid for the convection-diffusion Robin problem

We propose a reduced multiscale finite element method for a convection-diffusion problem with a Robin boundary condition. The small perturbed parameter would cause boundary layer oscillations, so we apply several adapted grids to recover this defect. For a Robin boundary relating to derivatives, special interpolating strategies are presented for effective approximation in the FEM and MsFEM schemes, respectively. In the multiscale computation, the multiscale basis functions can capture the local boundary layer oscillation, and with the help of the reduced mapping matrix we may acquire better accuracy and stability with a less computational cost. Numerical experiments are provided to show the convergence and efficiency.     

A nonlinear problem for the Laplace equation with a degenerating Robin condition

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution.     

Robin Approximation of Dirichlet Boundary Value Problems

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L²(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H¹ and L² norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.     

Asymptotics for mixed Dirichlet–Robin problems in irregular domains

We study the asymptotic behaviour of the solutions of two-dimensional elliptic problems with Robin boundary conditions on the “prefractal” curves approximating the Koch curve type fractals.     

A priori estimates of the solution for the Dirichlet problem

We study the Dirichlet problem defined by u = f in and u =g in when is the half-space or the unitary rectangle, obtainingan a priori estimate of the solution. Furthermore, in both casesa concrete numerical estimation is arrived at. First we getthe a priori estimate in the case of the half-space. The problemis resolved for the rectangle by initially translating it intothe half-space and using the results we had obtained for it,so that the problem can then be reduced back to the rectangle.Because of this we first establish an extension of the Sobolevspace of second order in the rectangle to the one defined inthe half-space.     

Existence and Nonuniqueness of Solutions to a Robin Boundary Problem for Semilinear Elliptic Equations

Sufficient conditions for existence and nonuniqueness of radially symmetric solutions to the Robin boundary problem of the form Δu + a(||x||)|u|^{-p} = 0 \qquad in B ⊂ R^N \frac{∂u}{∂n} + λu = -α \qquad on ∂B are obtained.     

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.