On the Green function of an orthotropic clamped plate in a half-plane

First, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form $$(Q(\partial ')-a^2\partial _n^2)(Q(\partial ')-b^2\partial _n^2)$$ ( Q ( ∂ ′ ) - a 2 ∂ n 2 ) ( Q ( ∂ ′ ) - b 2 ∂ n 2 ) (where $$\partial '=(\partial _1,\dots ,\partial _{n-1})$$ ∂ ′ = ( ∂ 1 , ⋯ , ∂ n - 1 ) and $$a>0,b>0,a\ne b)$$ a > 0 , b > 0 , a ≠ b ) with respect to Dirichlet boundary conditions at $$x_n=0.$$ x n = 0 . The Green function $$G_\xi $$ G ξ is represented by a linear combination of fundamental solutions $$E^c$$ E c of $$Q(\partial ')(Q(\partial ')-c^2\partial _n^2),$$ Q ( ∂ ′ ) ( Q ( ∂ ′ ) - c 2 ∂ n 2 ) , $$c\in \{a,b\},$$ c ∈ { a , b } , that are shifted to the source point $$\xi ,$$ ξ , to the mirror point $$-\xi ,$$ - ξ , and to the two additional points $$-\frac{a}{b}\xi $$ - a b ξ and $$-\frac{b}{a}\xi ,$$ - b a ξ , respectively.