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What are the basic things that i have to keep in mind to change the PDE having non unique solution to the PDE have unique solution. For example say i have $\Delta u=0 $ for $x\in \Omega$ and $u=1$ in $|x|=1$ Define $\Omega=x: |x| \ge1$

Here we can clearly see that it has infinite solutions . say $u=a(|x|-1)+1$.

How do i argue to put a condition on the given PDE so that the solution turns out unique. Turning it into dirichlet boundary condition gives uniqueness right ? Thanks for help.

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The solution of Dirichlet problem for the Laplace operator is unique in a bounded domain. You have an unbounded domain (or a set with two components, if in 1 dimension). To achieve uniqueness here, one has to impose a growth bound on the solution, such as o(|x|) in one dimension or $o(\log|x|)$ in two dimensions. If more is known about the domain, the restriction on solution may be less restrictive - see the Phragmen-Lindelof principle.

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To guarantee uniqueness, additional conditions must generally be specified on the boundary of the region where the solution is defined. Look here.

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