2
$\begingroup$

I have a few questions about Newton boundary conditions for a second-order partial differential equation:

$$-\text{div}(a(x,u,\nabla u)) + c(x,u,\nabla u)$$

considered on a bounded connected Lipschitz domain $\Omega \subset \mathbb{R}^{n}$. Here $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ and $c: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$.

I understand that this could admit very many solutions so we prescribe boundary conditions to the solution.

One option is to prescribe simply the trace $u|_{\Gamma}$ of $u$ on the boundary, i.e. $$u|_{\Gamma} = u_{D}\text{ } \text{ on } \Gamma$$ with $u_{D}$ a fixed function on $\Gamma$. This condition is reffered to as a Dirichlet boundary condition.

Then the alternative natural possibility is to prescribe a local equation for the "boundary flux" $\nu \cdot a$, i.e. $$\nu \cdot a(x,u,\nabla u) + b(x,u) = h \text{ on } \Gamma$$ where $\nu = (\nu_{1},...,\nu_{n})$ denotes the unit outward normal to $\Gamma$ and $h: \Gamma \rightarrow \mathbb{R}$ and $b: \Gamma \times \mathbb{R} \rightarrow \mathbb{R}$. This condition is referred to as a Robin condition. If $b = 0$, it is called a Neumann boundary condition.

My question is regarding the Newton/Robin boundary condition. It seems that the Newton boundary condition is only concerned with the principle part $a$ of the pde with the addition of functions $b$ and $h$, is this the usual way of defining the Newton boundary condition?

Then if we consider the pde above but with the omission of function $c$, this gives: $$-\text{div}(a(x,u,\nabla u)) = g$$

Could we again take the newton boundary condition as $$\nu \cdot a(x,u,\nabla u) + b(x,u) = h \text{ on } \Gamma$$

Would the newton boundary condition remain the same for this omission of function $c$?

$\endgroup$
  • 1
    $\begingroup$ The answers to your questions are both: yes :) $\endgroup$ – daw Apr 9 '14 at 12:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy