论文标题
在有界几何形状的度量空间中域上分数cheger-laplacian的解决方案的规律性
Regularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry
论文作者
论文摘要
我们研究了与完全加倍的度量度量空间$（x，d_x，μ_x）$满足$ 2 $-POINCARé不平等的完全加倍度量度量空间中的DIRICHLET问题的存在，唯一性和规律性。给定一个有界的域$ω\子集X $，带有$μ_x（x \setMinusΩ）> 0 $，在besov类中$ f $ $ f $ $ b^θ_{2,2,2}（x）\ cap l^2（x）$ $ x \setMinusΩ$和$ \ MATHCAL {E}_θ（U，U）\ Le \ Mathcal {e}_θ（H，H）$每当B^θ_{2,2}} in b in $ x \ x \ setminusph中$ h = f $ in b^θ_{2,2}（x）$时，每当$ h，h）$。我们表明这种解决方案始终存在，并且该解决方案是唯一的。我们还表明，该解决方案是$ω$的局部Hölder连续的，并且满足了非本地最大和强大的最大原理。本文的一部分结果扩展了Caffarelli和Silvestre在Carnot组中的欧几里得和弗兰奇和法拉利的工作。
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,μ_X)$ satisfying a $2$-Poincaré inequality. Given a bounded domain $Ω\subset X$ with $μ_X(X\setminusΩ)>0$, and a function $f$ in the Besov class $B^θ_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^θ_{2,2}(X)$ such that $u=f$ in $X\setminusΩ$ and $\mathcal{E}_θ(u,u)\le \mathcal{E}_θ(h,h)$ whenever $h\in B^θ_{2,2}(X)$ with $h=f$ in $X\setminusΩ$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on $Ω$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
