Let Ω be a regular domain, for example be a rectangle.Is it true that solve system of PDE's like this: u + Δ w = 0 , w = b 1 , ∂ w ∂ n = b 2 , on ∂ Ω Δ u = 0 , with out boundary condition.

Question

Let   Ω be a regular domain, for example be a rectangle.Is it true that solve system of PDE&#039;s like this:  u  +  Δ  w  =  0  ,                                                                    w  =  b  1  ,               ∂      w              ∂      n        =  b  2  ,           on     ∂  Ω  Δ  u  =  0  ,                                                                             with out boundary condition.

Abigail Palmer · Accepted Answer

Yes, your formulation is fine.The idea is the following: just take the Laplacian of the first equation, since we know that   Δ  u  =  0 you get  Δ  Δ  w  =  0which is the biharmonic equation. It is well-known that as a fourth order elliptic operator, this problem on a domain is well-posed when you prescribe two boundary conditions. And as you have provided   w        |          ∂      Ω       and       ∂    n    w        |          ∂      Ω       you can convert the problem to a variational formulation and solve. (You would be minimizing the square norm of either the Hessian or Laplacian of   w under the boundary constraints.)After you solve the biharmonic equation for   w, you can easily recover   u as minus the Laplacian of   w using your original equation.

Let <mi mathvariant="normal">Ω be a regular domain, for example be a rectangle.

Answered question

Answer & Explanation

New Questions in Algebra I