紧集上调和函数的狄利克雷问题
The Dirichlet Problem for Harmonic Functions on Compact Sets
摘要：
对于任意紧集$K\subset \mathbb{R}^n$，我们发展了构成$\mathcal{O}_K$的Jensen测度和次调和峰点的理论，研究了$K$上的狄利克雷问题.首先我们考虑$K$上的函数空间$h(K)$，它可以被$K$的邻域内的调和函数一致逼近为可能解.与经典理论一样，我们的定理8.1给出了对于$\mathcal{O}_K$闭的紧集的$C(\mathcal{O}_K)\cong h(K)$。然而，一般来说，即使对于$\rO_K$上的连续数据，也不能期望连续解，如定理8.1所示。因此，我们表明，解决方案可以找到一类精细的调和函数。此外，通过定理8.7，与经典情形完全类似，对于所有的紧集$K$，该类与$C_b(\mathcal{O}_K)$等距同构。
英文摘要：
For any compact set $K\subset \mathbb{R}^n$ we develop the theory of Jensen measures and subharmonic peak points, which form the set $\mathcal{O}_K$, to study the Dirichlet problem on $K$. Initially we consider the space $h(K)$ of functions on $K$ which can be uniformly approximated by functions harmonic in a neighborhood of $K$ as possible solutions. As in the classical theory, our Theorem 8.1 shows $C(\mathcal{O}_K)\cong h(K)$ for compact sets with $\mathcal{O}_K$ closed. However, in general a continuous solution cannot be expected even for continuous data on $\rO_K$ as illustrated by Theorem 8.1. Consequently, we show that the solution can be found in a class of finely harmonic functions. Moreover by Theorem 8.7, in complete analogy with the classical situation, this class is isometrically isomorphic to $C_b(\mathcal{O}_K)$ for all compact sets $K$.
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