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

Robert Denk, Floris Roodenburg

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 $C^{1,κ}$-domains, where $κ\in (0,1)$ 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.

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

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 -domains, where 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.
Paper Structure (24 sections, 36 theorems, 179 equations)

This paper contains 24 sections, 36 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. R-sectoriality and maximal regularity
  5. R-sectoriality
  6. Maximal regularity
  7. Interpolation-extrapolation spaces
  8. Domains and localisation
  9. Weighted Sobolev spaces
  10. Weighted Sobolev spaces with boundary conditions
  11. Trace characterisations on the half-space
  12. Trace characterisations on special domains
  13. Trace characterisations on bounded domains
  14. Properties of weighted Sobolev spaces
  15. R-sectoriality for the elliptic operator on the half-space
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $p,q\in (1,\infty)$, $\kappa\in (0,1)$, $\gamma\in ((1-\kappa)p-1, p-1)$ and let $k\geq 2$ be an integer. Suppose that $1-\frac{1}{q}\neq \frac{\gamma+1}{2p}$ and $\frac{1}{2}-\frac{1}{q}\neq \frac{\gamma+1}{2p}$. Furthermore, let $\mathcal{O}$ be a bounded $C^{1,\kappa}$-domain. Then for all there exists a unique solution to eq:plate_eq_intro. Moreover, this solution $u$ depends continuousl

Theorems & Definitions (72)

  • Theorem 1.1
  • Definition 2.1: ($R$-)sectoriality
  • Definition 2.2: Maximal $L^q(v)$-regularity
  • Proposition 2.3
  • Proposition 2.4: Ha06
  • Proposition 2.5: Am95
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8: LLRV25
  • Remark 3.1
  • ...and 62 more