月曜解析セミナー： Global integrability of supertemperatures

開催日時

2018年5月21日 15時00分 ～ 2018年5月21日 16時30分

場所

理学部3号館202

講演者

相川 弘明（北海道大学）

Ever since Armitage showed that every nonnegative superharmonic function on a bounded domain of bounded curvature (\(=C^{1,1}\) domain) in \(R^n\) is \(L^p\)-integrable up to the boundary for \(0\lt p\lt n/(n-1)\), the global integrability of nonnegative supersolutions has attracted many mathematicians.

In this talk we consider a parabolic counterpart. We study the global integrability of nonnegative supertemperatures on the cylinder \(D\times(0,T)\), where \(D\) is a Lipschitz domain or a John domain. 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 bounded \(C^1\)-domain, then every nonnegative supertemperature on \(D\times(0,T)\) is \(L^p\)-integrable over \(D\times(0,T’)\) for any \(0\lt T’\lt T\), provided \(0\lt p\lt (n+2)/(n+1)\). The bound \((n+2)/(n+1)\) is sharp.

Joint work with Hara and Hirata.