A structurally damped plate equation with Dirichlet-Neumann boundary conditions

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2015
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Schnaubelt, Roland
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Zusammenfassung

We investigate sectoriality and maximal regularity in Lp–Lq-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.

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Fachgebiet (DDC)
510 Mathematik
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Structurally damped plate equation, Clamped boundary condition, R-sectoriality, Optimal regularity, Operator-valued Fourier multipliers, Exponential stability
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ISO 690DENK, Robert, Roland SCHNAUBELT, 2015. A structurally damped plate equation with Dirichlet-Neumann boundary conditions. In: Journal of Differential Equations. 2015, 259(4), pp. 1323-1353. ISSN 0022-0396. eISSN 1090-2732. Available under: doi: 10.1016/j.jde.2015.02.043
BibTex
@article{Denk2015struc-31760,
  year={2015},
  doi={10.1016/j.jde.2015.02.043},
  title={A structurally damped plate equation with Dirichlet-Neumann boundary conditions},
  number={4},
  volume={259},
  issn={0022-0396},
  journal={Journal of Differential Equations},
  pages={1323--1353},
  author={Denk, Robert and Schnaubelt, Roland}
}
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    <dcterms:abstract xml:lang="eng">We investigate sectoriality and maximal regularity in L&lt;sup&gt;p&lt;/sup&gt;–L&lt;sup&gt;q&lt;/sup&gt;-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 C&lt;sup&gt;4&lt;/sup&gt;. It turns out that the first-order system related to the scalar equation on R&lt;sup&gt;n&lt;/sup&gt; 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.</dcterms:abstract>
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