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

Abstract

Let Δ be the Dirichlet Laplacian on a rectangular domain $R\subset R^N$. We study the mapping properties of an extension of the spectral fractional Laplacian, ${(-\Delta )}^{\alpha }$, for $\alpha \in [0,1)$, 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$u_{tt}+{\Delta }^{2}u+\rho {(-\Delta )}^{\alpha }{u}_t=0,x\in R,t>0,$ with non-homogeneous boundary conditions$u{|}_{\partial R}=f,\Delta u{|}_{\partial R}=0,f\in L^2(\partial R\times [0,\infty \left)\right).$ Other non-homogeneous boundary conditions are also considered.