Sensitivity analysis of a parabolic-elliptic problem

Gebauer, Bastian

doi:10.1090/S0033-569X-07-01072-4

Author: Bastian Gebauer
Journal: Quart. Appl. Math. 65 (2007), 591-604
MSC (2000): Primary 35K65, 35B40, 35M10
DOI: https://doi.org/10.1090/S0033-569X-07-01072-4
Published electronically: August 2, 2007
MathSciNet review: 2354889
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Abstract: We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative $L^\infty$-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous justification for modeling a small thermal capacity by setting it to zero. We also characterize weak directional derivatives of the temperature with respect to capacity as solutions of related parabolic-elliptic problems.

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