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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    $$a&gt;0,b&gt;0,a\ne b)$$  a &gt; 0 , b &gt; 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.

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