Unformatted text preview: Thus, assume that D is an open and bounded subset of R n with boundary ∂D . Assume also that u ( x ) is a solution to Laplace’s equation Δ u = 0 in D and that u is continuous on ¯ D , twice di±erentiable in D . Show that sup ¯ D u = sup ∂D u 5. a. Prove that if there exists a solution of the Neumann problem Δ u = f for x ∈ D ⊂ R n , ∂u ∂n = h ( x ) for x ∈ ∂D, then it is unique up to adding an arbitrary constant. b. Consider the Robin problem Δ u = f for x ∈ D , ∂u ∂n + a ( x ) u = h ( x ) for x ∈ ∂D Show that its solutions are unique. 1...
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 Spring '08
 Tanveer,S
 Math, Laplace, Boundary value problem, Boundary conditions, Neumann boundary condition, damped wave equation

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