Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \Omega$ its boundary and consider for $L$ elliptic operator the following homogeneous Dirichlet boundary PDE for sufficiently regular functions $f$ $$ Lu = f \quad on \,\, \Omega $$ $$ u = 0 \quad on \, \, \partial \Omega$$
This problem is well-studied especially under the assumption of uniformly ellipticity.
Now, my question concerns the following: suppose we are on $R^2$ for simplicity and have explicitly a rectangular domain
$$ \Omega = (a,b) \times (c,d)$$
then the boundary is all four sides of the rectangle i.e.
$$ \partial \Omega = \partial \Omega_1 \cup \partial \Omega_2 \cup \partial \Omega_3 \cup \partial \Omega_4 $$
Where we defined for ease $\partial \Omega_1 = \big\{a\big\}\times [c,d]$, $\partial \Omega_2 = \big\{b \big\}\times [c,d]$, $\partial \Omega_3 = [a,b]\times \big\{c \big\}$ and $\partial \Omega_4 = [a,b]\times \big\{d\big\} $. Suppose, we have the same elliptic operator $L$ and consider the PDE:
$$ Lu = f \quad on \,\, \Omega $$
$$ u = 0 \quad on \,\, \partial \Omega_1 \cup \partial \Omega_2$$
And $u$ does not have prescribed data on $\partial \Omega_3$ nor $\partial \Omega_4$ and treated as unknown to be solved for.
Question: Is there a name for this type of "half" boundary problem? More to point, is the PDE even solvable with these boundary condition (assuming uniformly ellipticity)? If not, does imposing "symmetric boundary condition" in that $u|_{\partial \Omega_3} = u|_{\partial \Omega_4}$ i.e. for any $x \in [a,b]$ we have $u(x,c) = u(x,d)$, is it solvable now? A name and any references to this type of boundary problem would be highly appreciated.
