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cianfa72
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- 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.
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.
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