加权Sobolev空间上Laplace算子的最优半群估计与泛函演算
Optimal semigroup estimates and functional calculus for the Laplacian on weighted Sobolev spaces
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中文总结 AI 辅助
该研究推广了Lindemulder等人的成果，证明半空间上带Dirichlet和Neumann边界条件的Laplace算子在幂权Sobolev空间上的最优半群估计，及边界满足相容性条件时具有有界$H^\infty$-泛函演算，且该条件不可省略。
中文摘要 AI 辅助
本文研究带Dirichlet和Neumann边界条件的半空间上的Laplace算子，在衡量到边界距离的幂权Sobolev空间上对这些算子进行分析。我们证明了预解算子及对应热半群的最优估计，还证明了在边界满足一定相容性条件时，Dirichlet和Neumann Laplacian在Sobolev空间上具有有界$H^\infty$-泛函演算，且该相容性条件一般不可省略。本文结果是Lindemulder、Lorist、作者及Veraar在[J. Funct. Anal., 289(8):110985, 2025]中所得结果的直接推广。
英文摘要
In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. These operators are studied on Sobolev spaces with power weights measuring the distance to the boundary. We prove optimal estimates for the resolvent operators and the corresponding heat semigroups. In addition, it is proved that the Dirichlet and Neumann Laplacians admit a bounded $H^\infty$-functional calculus on Sobolev spaces with certain compatibility conditions at the boundary. We show that these compatibility conditions cannot be omitted in general. The results in this paper are a direct extension of those obtained by Lindemulder, Lorist, the author, and Veraar in [J. Funct. Anal., 289(8):110985, 2025].
