论文标题
海森伯格集团的非本地Harnack不平等现象
Nonlocal Harnack inequalities in the Heisenberg group
论文作者
论文摘要
我们处理Heisenberg-Weyl组$ \ Mathbb {h}^n $中的一系列非线性界面差异问题，其原型是$ p $ - 分离的sublaplace方程的dirichlet问题。这些问题在量子力学，铁磁分析，相变问题，图像分割模型中等于量子力学的许多不同情况下出现，等于非欧几里得几何框架和非局部远程相互作用确实会自然发生。我们证明了对相关弱解决方案的一般不平等现象。同样，在增长指数为$ p = 2 $的情况下，我们调查了分数sublaplacian操作员的渐近行为，以及上述HARNACK估计的鲁棒性，因为可怜性指数$ s $ to $ 1 $。
We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $p=2$, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent $s$ goes to $1$.
