Volume mean densities for the heat equation |
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Authors: | Wolfhard Hansen Ivan Netuka |
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Institution: | 1. Fakult?t für Mathematik, Universit?t Bielefeld, Bielefeld, 33501, Germany 2. Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 18675, Praha 8, Czech Republic
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Abstract: | It is shown that, for solid caps D of heat balls in ? d + 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 D?. 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 ? d × (?∞, 0) is only slightly wider at z 0 than a heat ball. |
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