论文标题
与$ p $ harmonic函数相关的perron方法在精美的打开集上
The Perron method associated with finely $p$-harmonic functions on finely open sets
论文作者
论文摘要
鉴于$ v $的精细边界上有一个有界的打开套件$ v $和功能$ f $，我们将四种类型的上perron解决方案引入了$ p $ - 能量最小化器的非线性dirichlet问题，$ 1 <p <\ iffty $，$ f $ as bounce as bouncordage。这些解决方案作为$ v $的$ p $ -superminimizers的合适家族的infima给出。我们（在自然假设下）表明，四个上层perron解决方案是同等的，并且它们是$ p $ - 能量积分的$ p $ - 米数。此外，我们还表明，上下的Perron溶液在Sobolev和均匀连续的边界数据（即这种边界数据是均得到共线的情况下，都重合了sobolev和均匀连续的边界数据。对于统一的连续边界数据，Perron解决方案也被证明是连续的，因此是$ p $ harmonic的。我们证明了我们的结果，其中一个完整的度量空间$ x $，配备了双重措施，支持$ p $-poincaré不平等，但它们也是新的，也是未加权的$ \ mathbf {r}^n $。
Given a bounded finely open set $V$ and a function $f$ on the fine boundary of $V$, we introduce four types of upper Perron solutions to the nonlinear Dirichlet problem for $p$-energy minimizers, $1<p<\infty$, with $f$ as boundary data. These solutions are given as pointwise infima of suitable families of fine $p$-superminimizers in $V$. We show (under natural assumptions) that the four upper Perron solutions are equal quasieverywhere and that they are fine $p$-minimizers of the $p$-energy integral. We moreover show that the upper and lower Perron solutions coincide quasieverywhere for Sobolev and for uniformly continuous boundary data, i.e.\ that such boundary data are resolutive. For the uniformly continuous boundary data, the Perron solutions are also shown to be finely continuous and thus finely $p$-harmonic. We prove our results in a complete metric space $X$ equipped with a doubling measure supporting a $p$-Poincaré inequality, but they are new also in unweighted $\mathbf{R}^n$.
