Global integrability of supertemperatures
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Mathematische Zeitschrift
Global integrability of supertemperatures Hiroaki Aikawa1,2 · Takanobu Hara1 · Kentaro Hirata3 Received: 9 May 2018 / Accepted: 3 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let D be a Lipschitz domain or a John domain in Rn with n ≥ 2. We study the global integrability of nonnegative supertemperatures on the cylinder D × (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 × (0, T ) is L p -integrable over D × (0, T ) for any 0 < T < T , provided 0 < p < (n + 2)/(n + 1). The bound (n + 2)/(n + 1) is sharp. Keywords Integrability · Supertemperature · Green function · Lipschitz domain · John domain Mathematics Subject Classification 35K05 · 35B09 · 31B05
1 Introduction In 1972, Armitage [6] showed that every nonnegative superharmonic function on a bounded domain of bounded curvature in Rn is L p -integrable up to the boundary for 0 < p < n/(n − 1). Here a domain D is said to have bounded curvature if there exists r0 > 0 such that each boundary point ξ ∈ ∂ D has an interior ball of radius r0 in D and an exterior ball of radius r0 in the complement of D touching at ξ . It is known that a domain has bounded curvature
H. A. and K. H. were supported in part by JSPS KAKENHI Grant numbers JP17H01092 and JP18K03333, respectively.
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Hiroaki Aikawa [email protected] Takanobu Hara [email protected] Kentaro Hirata [email protected]
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Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
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Present Address: College of Engineering, Chubu University, Kasugai 487-8501, Japan
3
Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
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if and only if it is a C 1,1 -domain ([5, Lemma 2.2]). Since then, the global integrability of nonnegative supersolutions has attracted many mathematicians. Hereafter let D be a bounded domain in Rn with n ≥ 2. It has turned out that the global integrability depends on the estimate of the Green function G D for the Dirichlet Laplacian on D. For simplicity we suppress the subscript D if it can be understood from the context. Fix x0 ∈ D and let g(x) = min{G(x, x0 ), 1}. Let k > 0. A bounded domain whose boundary is locally given by the graph of a kLipschitz function is called a k-Lipschitz domain. We simply say that D is a Lipschitz domain if D is a k-Lipschitz domain for some k > 0. It is known that for a k-Lipschitz domain D, there exist constants α ≥ 1 and C > 0 depending on n and k such that g(x) ≥ Cδ D (x)α for x ∈ D, (1.1) √ where δ D (x) = dist(x, ∂ D). Moreover, if 0 < k < 1/ n − 1, then α can be taken as 1 ≤ α < 2. It is also known the upper estimate g(x) ≤ Cδ D (x)β for x ∈ D with 0 < β ≤ 1 and different C > 0. See Remark 3.1 in Sect. 3. Maeda and Suzuki extended Armitage’s result to a Lipschitz domain by giving an estimate of p of L p
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