4
$\begingroup$

When reading papers, I sometimes see initial boundary value problems where the differential equation is defined on the closure of the domain, rather than only on its interior. An example is in the figure below.

Is this correct or should the differential equation be defined only on the set interior? My guess is that for a solution $u\in C^1$ the differential equation is valid in the set closure because lateral limits of the derivatives should exist.

Sugestions of references clarifying this point are welcome.

Example

$\endgroup$

1 Answer 1

Reset to default
6
$\begingroup$

Technically, a differential equation itself should be defined on an open set, because the definition of derivative involves a two-sided limit, not a one-sided limit. Thus if $u(x)$ is only defined on $[0,1]$, $u'(0)$ does not exist. However, in a boundary value problem the differential equation is to be valid in the interior of the region, boundary values are set on the boundary of the region, and it is assumed that the function is continuous on the closure of the region.

$\endgroup$
2
  • $\begingroup$ But then what about Neumann boundary condition? $\endgroup$
    – zooond
    Commented 13 hours ago
  • $\begingroup$ Good point. Again, it's really about the limit of a derivative as you approach the boundary. $\endgroup$
    – Robert Israel
    Commented 12 hours ago

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .