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Global integrability of supertemperatures

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Abstract

Let \(D\) be a Lipschitz domain or a John domain in \({{{\mathbb {R}}}^n}\) with \(n\ge 2\). We study the global integrability of nonnegative supertemperatures on the cylinder \(D\times (0,T)\). We show that the integrability depends on the lower estimate of the Green function for the Dirichlet Laplacian on \(D\). In particular, if \(D\) is a \(C^1\)-domain, then every nonnegative supertemperature on \(D\times (0,T)\) is \(L^p\)-integrable over \(D\times (0,{T^{\prime }})\) for any \(0<{T^{\prime }}<T\), provided \(0<p<(n+2)/(n+1)\). The bound \((n+2)/(n+1)\) is sharp.

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Author notes

  1. Hiroaki Aikawa

    Present address: College of Engineering, Chubu University, Kasugai, 487-8501, Japan

Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Hiroaki Aikawa & Takanobu Hara

  2. Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

    Kentaro Hirata

Authors
  1. Hiroaki Aikawa
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  2. Takanobu Hara
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  3. Kentaro Hirata
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Correspondence to Hiroaki Aikawa.

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H. A. and K. H. were supported in part by JSPS KAKENHI Grant numbers JP17H01092 and JP18K03333, respectively.

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Aikawa, H., Hara, T. & Hirata, K. Global integrability of supertemperatures. Math. Z. 296, 1049–1063 (2020). https://doi.org/10.1007/s00209-020-02467-y

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  • DOI: https://doi.org/10.1007/s00209-020-02467-y

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