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 \(\pdv{u}{n}\) on the boundary:

\[ \alpha u + \beta \pdv{u}{n} = w \text{ on } \partial\Omega, \]

where

• \(\alpha\) and \(\beta\) are given coefficients, which can be functions defined on \(\partial\Omega\),
• \(w\) is a given function defined on \(\partial\Omega\).

27.1 Objective

The goal is to find a function \(u\) such that

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

27.2 Formal definition

Find a function \(u: \overline{\Omega} \mapsto \mathbb R\) such that \[ \begin{align} \Delta u & = f \text{ in } \Omega \\ \alpha u + \beta \pdv{u}{n} & = w \text{ on } \partial\Omega \end{align} \]

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 \(\pdv{u}{n} = w/\beta\),

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.