Higher-order regularity for a structurally damped plate equation on rough domains

TL;DR

This paper establishes well-posedness and higher-order regularity for a structurally damped plate equation on rough domains using weighted Sobolev spaces, enabling treatment of complex boundary conditions without compatibility constraints.

Contribution

It introduces a novel approach employing weighted Sobolev spaces to analyze regularity of damped plate equations on irregular domains, avoiding compatibility conditions.

Findings

Proves well-posedness of the damped plate equation on rough domains.

Establishes higher-order regularity results using weighted Sobolev spaces.

Handles complex boundary conditions without unnatural compatibility constraints.

Abstract

We prove well-posedness and higher-order regularity for a linear structurally damped plate equation with inhomogeneous Dirichlet--Neumann boundary conditions on the half-space and on bounded domains. To this end, we study maximal regularity properties of the related first-order system on weighted Sobolev spaces of arbitrarily high smoothness. In particular, we consider Sobolev spaces with power weights that measure the distance to the boundary. This allows us to avoid unnatural compatibility conditions for the data and treat the plate equation with rough inhomogeneous boundary conditions on bounded $C^{1,\kappa}$-domains, where $\kappa\in (0,1)$ depends on the exponent of the spatial power weight, but is independent of the smoothness of the data. Our methods can serve as an example to treat more complicated mixed-order systems as well.