A structurally damped plate equation with Dirichlet-Neumann boundary conditions

Abstract

We investigate sectoriality and maximal regularity in L^p–L^q-Sobolev spaces for the structurally damped plate equation with Dirichlet–Neumann (clamped) boundary conditions. We obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space, in the half-space, and in bounded domains of class C⁴. It turns out that the first-order system related to the scalar equation on Rⁿ is sectorial only after a shift in the operator. On the half-space one has to include zero boundary conditions in the underlying function space in order to obtain sectoriality of the shifted operator and maximal regularity for the case of homogeneous boundary conditions. We further show that the semigroup solving the problem on bounded domains is exponentially stable.