Dirichlet problem boundary conditions

[cianfa72]

TL;DR Summary

About boundary conditions for Dirichlet problem.

The Dirichlet problem asks for the solution of Poisson or Laplace equation in an open region $S$ of $\mathbb R^n$ with a condition on the boundary $\partial_S$.

In particular the solution function $f()$ is required to be two-times differentiable in the interior region $S$ and continuous on the boundary $\partial_S$. The boundary condition specifies its value $u$ at each point on it, hence $f=u$ at the boundary $\partial_S$.

Now my question is: which is the topology w.r.t one asks the function $f$ to be continuous in the closure $\bar S$ ? I believe it is the subspace topology on $\bar S$ from $\mathbb R^n$ as subset.

Is the above correct? Thanks.

 

[WWGD]

Id say that more than just topology, it's also an issue of Differential Structure, given you want your function to be (twice)differentiable. Given differentiability implies continuity, I suspect Differential Structure would be enough. I suspect you're dealing with the standard topology, but I'm not 100%.

 

[cianfa72]

Id say that more than just topology, it's also an issue of Differential Structure, given you want your function to be (twice)differentiable. Given differentiability implies continuity, I suspect Differential Structure would be enough. I suspect you're dealing with the standard topology, but I'm not 100%.

Yes, as far as can I understand, the region where the PDE is solved is an open set $S$ of $\mathbb R^n$ endowed with standard differential structure/topology. Then we have boundary condition on the boundary $\partial_S$.

The solution $f$ is required to be continuous on the closure $\bar S$. So the question is: which is the topology on $\bar S$ w.r.t. it is closed anf $f$ is required to be continuous ? My answer: it is the subspace topology from the superset $\mathbb R^n$.

 

[pasmith]

Unless otherwise stated, when we speak of $U \subseteq \mathbb{R}^n$ we mean the set together with its local affine structure (so that we can calculate derivatives) and the topology induced by the Euclidean inner product. If $U \neq \mathbb{R}^n$ then the restriction to the subspace topology is implied.

The smooth structure is the maximal atlas which contains the charts $(V, \mathrm{id})$ where $V$ is any open subset of $U$. (There are always at least two such structures, since $$(V, p \ni V \to \mathbb{R}^n : q \mapsto ((q_1 - p_1)^3,q_2 - p_2, \dots, q_n - p_n))$$ is not smoothly compatible with the identity chart.)

 

[cianfa72]

The smooth structure is the maximal atlas which contains the charts $(V, \mathrm{id})$ where $V$ is any open subset of $U$. (There are always at least two such structures, since $$(V, p \ni V \to \mathbb{R}^n : q \mapsto ((q_1 - p_1)^3,q_2 - p_2, \dots, q_n - p_n))$$ is not smoothly compatible with the identity chart.)

You mean fixed a point $p =(p_1, p_2 \dots p_n) \in V$, your map $$V \ni q \mapsto ((q_1 - p_1)^3,q_2 - p_2, \dots, q_n - p_n))$$ is not smoothly compatible with the identity map. However both structures are diffeomorphic (i.e. there exists a diffeomorphism between them).