An extension of the spectral fractional Laplacian to non-homogeneous boundary condition on rectangular domains, with application to well-posedness for plate equation with structural damping

Let Δ be the Dirichlet Laplacian on a rectangular domain R ⊂ R N . We study the mapping properties of an extension of the spectral fractional Laplacian, ( − Δ ) α , for α ∈ [ 0 , 1 ) , when applied to functions satisfying non-homogeneous boundary conditions. A symmetry formula is proven. As an application, we prove well-posedness results for the structurally damped plate equation u t t + Δ 2 u + ρ ( − Δ ) α u t = 0 , x ∈ R , t > 0 , with non-homogeneous boundary conditions u | ∂ R = f , Δ u | ∂ R = 0 , f ∈ L 2 ( ∂ R × [ 0 , ∞ ) ) . Other non-homogeneous boundary conditions are also considered. • Spectral fractional Laplacian extended to functions with non-zero boundary conditions. • Mapping properties are proven. • An integration by parts formula is proven. • Well-posedness of structurally damped vibrating plate with Dirichlet control.