论文标题
欧几里得空间中的边界价值问题
Boundary value problems in Euclidean space for Bosonic Laplacians
论文作者
论文摘要
玻色粒拉普拉斯是一个在欧几里得空间中定义的平滑函数上作用于平滑函数的同型不变的二阶差异操作员，并以更高阶段的不可减至的不可约合表示值。在本文中，我们研究了涉及上半部空间和单位球的玻色粒laplacian的边界价值问题。构建了上半个空间的泊松内核，并构建了单位球，这为我们提供了$ l^p $边界数据的Dirichlet问题的解决方案，$ 1 \ leq p \ leq \ leq \ infty $。我们还证明了解决方案问题的唯一性与玻色粒laplacians的连续数据有关，并为谐波laplacians的无谐波溶液提供了某些谐波功能的类似物，例如，凯奇（Cauchy）的估计值，均值，均值属性，Liouville的定理等。
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.
