论文标题
在harnack不平等的反对称$ s $ harmonic功能上
On the Harnack inequality for antisymmetric $s$-harmonic functions
论文作者
论文摘要
我们证明了反对称$ s $ harmonic功能的Harnack不平等，更普遍地用于一般域中具有零订单项的分数方程的解决方案。这可以与移动平面的方法结合使用，以获得由分数laplacian驱动的半连线方程的对称性和过度确定问题的定量稳定性结果。 证明分为两个部分：远离对称平面的内部harnack不平等，以及靠近对称平面的边界harnack不等式。我们通过首先确定内部和边界案例中亚溶液的局部溶解和局部界限的弱竖琴不平等，证明了这些结果。 En passant，我们还获得了反对称$ s $ harmonic功能的新的平均值公式。
We prove the Harnack inequality for antisymmetric $s$-harmonic functions, and more generally for solutions of fractional equations with zero-th order terms, in a general domain. This may be used in conjunction with the method of moving planes to obtain quantitative stability results for symmetry and overdetermined problems for semilinear equations driven by the fractional Laplacian. The proof is split into two parts: an interior Harnack inequality away from the plane of symmetry, and a boundary Harnack inequality close to the plane of symmetry. We prove these results by first establishing the weak Harnack inequality for super-solutions and local boundedness for sub-solutions in both the interior and boundary case. En passant, we also obtain a new mean value formula for antisymmetric $s$-harmonic functions.
