带有界系数的双复Vekua方程的Bergman空间
Bergman spaces for the bicomplex Vekua equation with bounded coefficients
AI总结：
该研究针对有界域上带界系数的双复Vekua方程，构建其广义$L_p$-解的Bergman空间理论，分析空间完备性、解正则性等性质，还探讨主Vekua方程与电导率方程的关联等内容。
摘要：
我们构建了有界域上双复Vekua方程$\overline{\boldsymbol{\partial}}W=aW+b\overline{W}$的广义$L_p$-解的Bergman空间理论，其中系数$a$和$b$为有界双复值函数。我们研究了Bergman空间的完备性、解的正则性以及赋值泛函的有界性。对于$p=2$的情形，我们证明了再生核的存在性，同时给出了利用所得再生核表示的到Bergman空间的正交投影，以及正交补的显式表达式。此外，我们分析了主Vekua方程（$a=0$、$b = \frac{\overline{\boldsymbol{\partial}}f}{f}$，其中$f$为非零复值函数），并给出了其与一对电导率方程的关联、亚调和共轭的构造以及Runge性质的相关结果。
英文摘要：
We develop the theory for the Bergman spaces of generalized $L_p$-solutions of the bicomplex-Vekua equation $\overline{\boldsymbol{\partial}}W=aW+b\overline{W}$ on bounded domains, where the coefficients $a$ and $b$ are bounded bicomplex-valued functions. We study the completeness of the Bergman space, the regularity of the solutions, and the boundedness of the evaluation functional. For the case $p=2$, the existence of a reproducing kernel is established, along with a representation of the orthogonal projection onto the Bergman space in terms of the obtained reproducing kernel, and an explicit expression for the orthogonal complement. Additionally, we analyze the main Vekua equation ($a=0$, $b = \frac{\overline{\boldsymbol{\partial}}f}{f}$ with $f$ being a non-vanishing complex-valued function). Results concerning its relationship with a pair of conductivity equations, the construction of metaharmonic conjugates, and the Runge property are presented.
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