A Robin boundary value problem for the Cauchy–Riemann operator in a ring domain

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
İlker Gençtürk ◽  
Yankis R. Linares
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Ring Domain ◽  
Riemann Equation ◽  
Robin Condition ◽  
Cauchy Riemann ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

Abstract In this paper, we study a Robin condition for the inhomogeneous Cauchy–Riemann equation w z ¯ = f {w_{\bar{z}}=f} in a ring domain R, by reformulating it as a Dirichlet boundary condition.

scholarly journals Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

2019 ◽  
Vol 8 (1) ◽  
pp. 1252-1285
Author(s):  
Yibin Zhang ◽  
Lei Shi
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Unit Vector ◽  
Large Exponent ◽  
Concentrating Solutions ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

Abstract Let Ω ⊂ ℝ2 be a bounded domain with smooth boundary and b(x) > 0 a smooth function defined on ∂Ω. We study the following Robin boundary value problem: $$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\ &u>0 &\quad& \text{in }{\it\Omega},\\ &\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &\quad& \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$ where ν denotes the exterior unit vector normal to ∂Ω, 0 < λ < +∞ and p > 1 is a large exponent. We construct solutions of this problem which exhibit concentration as p → +∞ and simultaneously as λ → +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.

Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauchy–Riemann equation

2009 ◽  
Vol 139 (1) ◽  
pp. 157-181 ◽  
Author(s):  
Alip Mohammed ◽  
M. W. Wong
Keyword(s):  
Boundary Condition ◽  
Linear Equations ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
System Of Linear Equations ◽  
Riemann Equation ◽  
Cauchy Riemann ◽  
Robin Boundary ◽  
Poincaré Problem ◽  
Well Posed

The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.

REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER

2011 ◽  
Vol 21 (05) ◽  
pp. 1153-1192 ◽  
Author(s):  
JINGYU LI ◽  
KAIJUN ZHANG
Keyword(s):  
Boundary Condition ◽  
Shear Modulus ◽  
Poisson Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Robin Boundary Condition ◽  
Oscillation Period ◽  
The Poisson Equation ◽  
Robin Boundary

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

scholarly journals On Discrete Solutions for Elliptic Pseudo-Differential Equations

2021 ◽  
Vol 103 (3) ◽  
pp. 117-123
Author(s):  
O.A. Tarasova ◽  
◽  
A.V. Vasilyev ◽  
V.B. Vasilyev ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Discrete Analogue ◽  
Continuous Solutions ◽  
Pseudo Differential Equation

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

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