Abstract
It is shown that, for solid caps D of heat balls in ℝd + 1 with center z 0 = (0, 0), there exist Borel measurable functions w on D such that inf w(D) > 0 and ∫ v(z)w(z) dz ≤ v(z 0), for every supertemperature v on a neighborhood of D̅. This disproves a conjecture by N. Suzuki and N.A. Watson. On the other hand, it turns out that there is no such volume mean density, if the bounded domain D in ℝd × (−∞, 0) is only slightly wider at z 0 than a heat ball.
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Research supported by CRC-701, Bielefeld.
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Hansen, W., Netuka, I. Volume mean densities for the heat equation. Potential Anal 41, 1111–1126 (2014). https://doi.org/10.1007/s11118-014-9411-z
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DOI: https://doi.org/10.1007/s11118-014-9411-z