Given a continuous function on the boundary of a bounded open set in \(\mathbb {R}^d\) there exists a unique bounded harmonic function, called the Perron solution, taking the prescribed boundary values at least at all regular points (in the sense of Wiener) of the boundary. We extend this result to vector-valued functions and consider several methods of constructing the Perron solution which are classical in the real-valued case. Special emphasis is on the case where the codomain is a Banach lattice. In this case we investigate Perron’s classical construction via sub- and supersolutions. We also apply our results to solve elliptic and parabolic boundary value problems of vector-valued functions.

中文翻译：

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向量值方程的Perron解

给定\（\ mathbb {R} ^ d \）中有界开集的边界上的连续函数，则存在一个唯一的有界谐波函数，称为Perron解，至少在所有正则点（在维纳感）的边界。我们将此结果扩展到矢量值函数，并考虑了几种构造Perron解的方法，这些方法在实值情况下是经典的。特别强调的是共域是Banach晶格的情况。在这种情况下，我们通过子解和超级解来研究Perron的经典构造。我们还将我们的结果用于解决矢量值函数的椭圆和抛物线型边值问题。