biharmonic equation with boundary conditions on half upper disk and reflection method

Question

Suppose that for the solution $u$ to the biharmonic problem on a disk with radius $R$ around zero, i.e. $$\Delta^2u=f\, \text{on}\, B_R.$$ an estimation holds true. For instance, $$\int_{B_R}|\Delta^2 u|^2dx\le\dots$$ Now suppose the same problem on the half upper disk with the boundary conditions below $$\Delta^2u=f\, \text{on}\, G_R,$$ $$u=\frac{\partial u}{\partial \eta}=0, \text{on}\, \partial G_R.$$ My question: Is there any reflection to extend the solution u to the whole disk and use the result above?