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Solving a 4th order PDE: how to specify boundary conditions

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A fourth order PDE like the biharmonic equation: ∇^4 w = 0 can be placed in Comsol's standard forms by writing u = ∇^2 w, so that we have system of equations:

∇^2 v = w
∇^2 w = 0

where v = v(x,y) and w = w(x,y).

Suppose that in the original problem, I had the two boundary conditions:

w = 1 on D1
dw/dn = 0 on D1

w = unknown constant on D2
dw/dn = 1 on D2

I am not sure how I translate to the system now in (v, w). In particular, the value of w on the boundary D2 is unknown, but it is known that D2 is constant.

From my reading, it seems that I have to use the notion of Lagrange multipliers. But there is very little documentation on this in Comsol's manuals. I would appreciate if someone could help.




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