In bounded domains of $\mathbb{R}^2$, the Dirichlet problem has a unique solution: the equation $\triangle u=0$ with prescribed boundary value has a unique solution. This is not true if the domain is unbounded.
Is there any result for existence and uniqueness for unbounded domains under further assumptions on the domain and the function? If there is a reference where this question is treated it would be of help.
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In physics actually the Dirichlet problem works on Poisson equation in all the infinite void space free from conductors (Laplace equation), even though the infinite space is obviously unbonded provided that the electrostatic potential V approaches 0 as r approaches infinite at least as r^-1, so it is possibile to extend the Dirichlet problem to unbonded domain.
yes you can use the flux through an imaginary free surface as your boundary condition like neumann or just have the domain in the unbounded exterior. see for example the Raeder Tomultz Kritalovski solution for the evolution of the surface tension components of the stress tensor along the free boundary of a nuclear flow with time varying viscosities.