# Introduce second derivative region boundary condition for 4th order PDE

I am trying to play around with the Foppl von Karman plate equations

and solve for $$w[x,y]$$. I want to consider a plate with a free edge, i.e. no forces or moments at the edge. As such I want to introduce $$\partial_{x,x}w[x,y]==0, \partial_{y,y}w[x,y]==0, \partial_{x,y} w[x,y]==0$$, as moments are zero, curvature (which is the second derivative of the vertical displacement) must also be zero here. I am aware of Dirichlet boundary conditions for boundary deflections (e.g.Dirichlet[w[x,y]==0, {x,y} \Element[] \Rectangle[]]) and Neumann for first derivative, however these don't seem to apply in my case from what i've understood. What can I use for second derivative boundary conditions for 2D problems, while being able enforce this for the entire boundary and not just specific points like D[w[x,y],{x,2}]==0/.{x->-a} for instance, which you seem to be able to do neatly for a dirichlet condition as you just specify x,y as being in a region you define?

• Can you re-write as a system of first order equations, like here Commented Jul 7, 2021 at 18:47
• @user21: It looks like the BCs in your link are $\Delta w=0$, whereas this question wants all second derivatives of $w$ to vanish. It's not obvious to me how to adapt that. Commented Jul 7, 2021 at 19:36
• Re-wrting as a system of two second order PDEs will also give two dependent variables. On the first variable one can apply DirichletCondition and NeumannValue and the same is true for the second dependent variable. Now, the second dependent variable would represent the 2nd and 3rd derivatives. If the above PDE can not be transformed into this form then currently (V12.3) there is not way to solve this with FEM. Commented Jul 8, 2021 at 5:09