论文标题
通过共形转换解决无界域的Dirichlet问题
Solving a Dirichlet problem for unbounded domains via a conformal transformation
论文作者
论文摘要
在本文中，我们解决了$ p $ -dirichlet问题的besov边界数据在无限制的统一域上具有边界边界的边界数据，当该域配备了满足庞加莱不平等的加倍措施时。这是通过研究一类变换来实现的，这些变换最近被证明在保持统一性的同时使域界定。这些转换以一种取决于到达域边界的距离以及量度的参数$ p $的方式将度量标准和度量变形。我们表明，转变的度量是加倍，而转变的领域支持庞加莱的不平等。这使我们能够将有界统一域的已知结果传输到无界域的结果，包括跟踪结果和ADAMS类型的不等式，最终导致解决BESOV类中边界数据的Dirichlet问题的解决方案。
In this paper, we solve the $p$-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter $p$. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.
