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Noncompactness of Integral Operators Modeling Diffuse-Gray Radiation in Polyhedral and Transient Settings

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Abstract

While it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for \({I-(1-\epsilon)K}\) being invertible, provided the emissivity \({\epsilon}\) does not vanish identically.

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Correspondence to Peter Philip.

Additional information

This research is supported by the DFG Research Center “Mathematics for Key Technologies” Matheon (FZT 86) in Berlin.

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Druet, PÉ., Philip, P. Noncompactness of Integral Operators Modeling Diffuse-Gray Radiation in Polyhedral and Transient Settings. Integr. Equ. Oper. Theory 69, 101–111 (2011). https://doi.org/10.1007/s00020-010-1821-8

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Mathematics Subject Classification (2010)

  • Primary 45P05
  • Secondary 45C05

Keywords

  • Radiosity equation
  • noncompact
  • integral operator
  • diffuse-gray radiation
  • polyhedral domain
  • transient setting