27  Robin problems

The Robin problem is a boundary value problem for partial differential equations, where the boundary condition is a combination of the function value and its normal derivative. It is a generalization of both the Dirichlet and Neumann problems.

The Robin boundary condition combines the function value $u$ and its normal derivative $\frac{\partial u}{\partial n}$ on the boundary:

$$\alpha u+\beta \frac{\partial u}{\partial n}=w\text{ on }\partial \mathrm{\Omega },$$

where

• $\alpha$ and $\beta$ are given coefficients, which can be functions defined on $\partial \mathrm{\Omega }$,
• $w$ is a given function defined on $\partial \mathrm{\Omega }$.

27.1 Objective

The goal is to find a function $u$ such that

1. $u$ satisfies the Poisson equation $\mathrm{\Delta }u=f$ in the domain $\mathrm{\Omega }$,
2. $u$ satisfies the Robin boundary condition $\alpha u+\beta \frac{\partial u}{\partial n}=w$ on $\partial \mathrm{\Omega }$.

27.2 Formal definition

Find a function $u:\bar{\mathrm{\Omega }}\mapsto \mathbb{R}$ such that $$\begin{array}{rl}\mathrm{\Delta }u& =f\text{ in }\mathrm{\Omega }\\ \alpha u+\beta \frac{\partial u}{\partial n}& =w\text{ on }\partial \mathrm{\Omega }\end{array}$$

27.3 Special cases

• If $\beta =0$, the Robin condition reduces to the Dirichlet boundary condition $u=w/\alpha$,
• If $\alpha =0$, the Robin condition reduces to the Neumann boundary condition $\frac{\partial u}{\partial n}=w/\beta$,

27.4 Applications

The Robin problem is used in various physical contexts, such as:

• Heat transfer, where the boundary condition represents a combination of conduction and convection,
• DC resistivity problems, where Robin conditions model the behaviour of the problem at the boundary.

The Robin problem provides a more flexible model that can interpolate between purely Dirichlet or Neumann conditions and model more complex boundary interactions.