I Does the Poisson Equation Have a Unique Solution with Mixed Boundary Conditions?

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The Poisson equation can indeed have a unique solution under mixed boundary conditions, such as a combination of Dirichlet and Neumann conditions. Each specific set of boundary conditions leads to a unique solution, even if the conditions differ across the domain. Concerns arise about the consistency of solutions when applying different boundary types, but the uniqueness holds for the overall problem as defined. Examples illustrating this concept can clarify how mixed conditions interact to yield a singular solution. Understanding the interplay of boundary conditions is crucial for grasping the uniqueness of solutions in the context of the Poisson equation.
ajeet mishra
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My professor told that poission equation has a unique solution even for mixed boundary conditions( i.e. Dirichlet bc for some part and Neumann for the remaining part). But how is this possible? As different boundary conditions for the same problem will give different solutions.
 
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ajeet mishra said:
My professor told that poission equation has a unique solution even for mixed boundary conditions( i.e. Dirichlet bc for some part and Neumann for the remaining part). But how is this possible? As different boundary conditions for the same problem will give different solutions.
It has a unique solution for each specified set of boundary conditions.
 
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Chestermiller said:
It has a unique solution for each specified set of boundary conditions.
But how? As there are unique solutions for drichlet and neumann conditions separately and these, in general, may not be consistent.
 
ajeet mishra said:
But how? As there are unique solutions for drichlet and neumann conditions separately and these, in general, may not be consistent.
Can you provide a specific example (or examples) to illustrate what you are saying?
 

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