论文标题
反对称函数在非局部方程中的作用
The role of antisymmetric functions in nonlocal equations
论文作者
论文摘要
我们证明了针对零订单项的分数laplacian的dirichlet问题的反对称超级分析的HOPF型引理。 作为应用程序，我们将这种HOPF型引理与移动平面的方法结合使用，以证明对半线性分数平行表面问题的对称性。也就是说，我们证明，如果其级别表面与$ω$相似，则必须在有限的开放式$ω\ subset \ mathbb r^n $中对分数laplacian的非负解决方案进行径向对称。反过来，$ω$必须是一个球。 此外，我们讨论了分数环境中反对称函数的最大原理和HARNACK不等式，并在仅在解决方案上施加“局部”假设时对这些定理提供反述。
We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set $Ω\subset \mathbb R^n$ must be radially symmetric if one of their level surfaces is parallel to the boundary of $Ω$; in turn, $Ω$ must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only `local' assumptions are imposed on the solutions.
